characterising the different forms of correlations shared by the constituents of a composite quantum system is essential for the theoretical understanding and for the operational exploitation of the quantum system itself .correlations which can not be amenable to a classical description , in particular , exhibit a rich variety in mixed states of bipartite and multipartite quantum systems .nowadays , the notion of quantum correlations refers not only to entanglement , but to more general forms of correlations which are conventionally identified with the quantum discord , and capture for instance the necessary disturbance induced on quantum states by any local measurement ( which is nonzero even in all separable states apart from so - called classical - quantum states ) .correspondingly , the portion of correlations which are left in the state after a minimally disturbing local measurement can be identified with the classical correlations originally shared by the subsystems .  unlike entanglement , for which a resource theory is well established , proposals to quantify quantum and classical correlations in this more general paradigm are still relatively scarce , and the mathematical requirements that any such proposal has to obey to be regarded as a valid measure are still to be completely formalised . yetan interesting phenomenology associated to these correlations is being uncovered in different physical contexts .for example , from a foundational perspective the sudden transition from a decay to a plateau regime for classical correlations between a quantum system and its measurement apparatus has been interpreted as characterising the finite - time emergence of the pointer basis in the apparatus  . moreover , quantum correlations between noninteracting qubits have been shown to dynamically revive despite of decoherence thanks to memory effects of the local environment , independently of the quantum or classical nature of the environment  .quantum correlations of ground states also appear to play an important role in the characterisation of exotic phases of quantum many - body systems  . more generally , from an operational viewpoint various forms of quantum correlations , including and beyond entanglement , can and typically do provide fundamental resources for quantum technologies  . consequently , rigorously addressing the quantification of correlations is of paramount importance .  the notion of distance defined on the convex set of states of a quantum systempaves the way for several geometric approaches to the quantification of correlations  .we refer in particular to refs . for a comparison among these approaches with respect to the quantification of quantum correlations . in this paper, we focus on bipartite systems and we follow the approach of ref . , according to which the minimum distance between a state and the set of states that do not possess a particular kind of correlations is a quantifier of that kind of correlations .hence , the minimum distances between the state of a bipartite system and the sets of product , classical - quantum and separable states represent , respectively , the amount of total correlations , quantum correlations and entanglement of .furthermore , the minimum distance between the set of closest classical - quantum states to and the set of product states represents the classical correlations of .this geometric approach to the quantification of correlations manifests several appealing features .first , it is unifying , thus allowing for a direct comparison among all the above mentioned notions of correlations .second , it readily suggests generalisations to the multipartite setting  .in this paper we use specifically the bures distance on the set of states to define geometric quantifiers of correlations .the bures distance is defined as where and are two arbitrary states while is the uhlmann fidelity }^2.\ ] ] the reason for choosing this distance instead of others stems from its quite peculiar , but desirable , properties .bures distance is at the same time contractive under completely positive trace - preserving maps , locally riemannian and its metric coincides with the quantum fisher information  , thus playing a crucial role in high precision interferometry and quantum metrology .moreover , the minimum bures distance between a state and the set of classical - quantum states is simply related to the maximal success probability in the ambiguous quantum state discrimination of a family of states and prior probabilities depending on  .the task of minimal error quantum state discrimination plays a fundamental role both in quantum communication and cryptography and has been realised experimentally using polarised light . on the contrary ,e.g. , the hilbert - schmidt distance is locally riemannian but not contractive  , the trace distance is contractive but not locally riemannian  and the relative entropy , although widely used in information theory , is technically not even a proper distance as it is not symmetric .  herewe derive closed formulae for classical and total correlations of bell - diagonal states of two qubits according to the bures distance . together with the known corresponding formulae for entanglement  and discord - type quantum correlations  , these allow us to gain a complete and unifying view of bures correlations for bell - diagonal states .we then provide two applications of these results .we first report the explicit expressions of the bures correlations for two special subclasses of bell - diagonal states , namely werner states and rank-2 bell - diagonal states . finally , we consider a dynamical system made of two independent qubits locally interacting with a bosonic non - dissipative channel andshow that both quantum and classical correlations measured by bures distance can alternatively freeze during the evolution , joining the ranks of other faithful correlation quantifiers .it is worthwhile to note that the freezing analysis was addressed in ref . with similar methods for the trace distance , but with a different definition of classical correlations .the paper is organised as follows . in section [ sec : quantumcorrelations ] we review some known results concerning bures quantum correlations and entanglement of bell - diagonal states . in sections[ sec : classicalcorrelations ] and [ sec : totalcorrelations ] we provide , respectively , the closed formulae for bures classical and total correlations of bell - diagonal states . in section [ sec : examples ] we compute the correlations of two particular classes of bell - diagonal states , i.e. , werner states and rank-2 bell - diagonal states . in section[ sec : dynamics ] we analyse the dynamics of correlations between two noninteracting qubits initially prepared in a bell - diagonal state and subject to identical local pure dephasing channels .we conclude in section [ sec : conclusions ] with a summary and outlook .quantum correlations stem from two peculiar ingredients of quantum mechanics , the superposition principle and the tensor product structure of the hilbert space associated to a composite quantum system .they are completely characterised by entanglement in the case of pure states , whereas in the case of mixed states entanglement constitutes only a part of the quantumness of correlations  . as a result ,for any pair of comparable quantifiers of general quantum correlations and entanglement , the quantum correlations of a state should intuitively be always greater or equal to the corresponding entanglement , being equal if the state is pure .this is nicely captured by the aforementioned geometric approach .specifically , in this paper , the quantum correlations of a state are quantified by the minimum bures distance of to the set of classical - quantum states , namely where is the set of classical - quantum states , i.e. states of the form with being a probability vector , an orthonormal basis of qubit and any state of qubit and is any of the closest classical - quantum states to .the entanglement of is measured by the minimum bures distance of to the set of separable states , namely where is the set of separable states , i.e. states of the form with being a probability vector , and any state of qubit and , respectively , while is any of the nearest separable states to .as the set of classical - quantum states is contained in the set of separable states , we immediately have that for every .we shall restrict ourselves to the relevant but structurally simple class of bell - diagonal ( bd ) states of two qubits , which are diagonal in the `` magic basis '' of the four maximally entangled bell states . as a result, bd states are represented in the standard computational basis by the following matrix where and , , are the identity and the pauli matrices , respectively . the coefficients  ] .the values of and that maximise this fidelity are shown to coincide with the values given by eq .( [ eq : optimalainthefirstcaseplus ] ) .similar plots may be created by perturbing the ansatz states in such a way that the index is no longer , i.e. , but rather or .these plots both show two maxima , but the fidelity in these two maxima never exceeds the maximal value corresponding to ., , , , obeying condition ( [ item : condition1 ] ) with , and product states obtained as perturbations of the ansatz from eq .( [ eq : ansatzproductstates ] ) , by considering any value of and belonging to  ] .werner states are therefore a subclass of bd states , with eigenvalues given simply by and . combining the results of with the above analysis ,the bures distance based correlations of , as shown in figure [ fig : wernercorrelations ] as functions of , are given overall by    all the correlations vanish for , when the state reduces to the maximally mixed ( uncorrelated ) state , and are monotonically increasing functions of .however , entanglement and total correlations are not smooth functions of , due to non - analyticities at and , respectively . for , the closest product state to not the maximally mixed product of the marginals , and becomes instead increasingly less mixed with increasing , eventually reaching a pure state ( e.g.  ) when .we finally note that for every we have , saturating the bound only trivially for where .  .the figure displays total correlations ( solid blue line ) , classical correlations ( dashed red line ) , quantum correlations ( dot - dashed purple line ) and entanglement ( dot - dot - dashed black line ) .the sum of classical and quantum correlations ( dotted orange line ) is also plotted.,scaledwidth=60.0% ]  we next consider a class of rank-2 bd states , whose eigenvalues take the form , and , with .the bures distance correlations , shown in figure [ fig : rank2correlations ] as functions of , are in this case  in this case the non - additivity of correlations is evident , due to the fact that classical correlations and total correlations are identically equal even though quantum correlations are non - vanishing .furthermore , general quantum correlations of the discord type are equal to entanglement even though is not pure .  .the figure displays total correlations ( solid blue line ) , equal to the classical correlations , and quantum correlations ( dot - dashed purple line ) , which are equal to the entanglement .the sum of classical and quantum correlations ( dotted orange line ) is also plotted.,scaledwidth=60.0% ]in this section we study the dynamics of bures distance correlations between two initially correlated but noninteracting qubits , each undergoing local pure dephasing due to the coupling with a bosonic bath at zero temperature and with super - ohmic spectrum , whose characteristics do not depend on the qubit  .specifically , the dynamics of each qubit and the bosonic reservoir is governed by the following hamiltonian ( ) where is the qubit frequency , the frequencies of the reservoir modes , the pauli operator along the -direction , the bosonic annihilation operators , the bosonic creation operators and the coupling constants between the qubit and each reservoir mode .moreover , in the continuum limit we have , where is the reservoir spectral density , which in the super - ohmic case is given by with denoting the cut - off reservoir frequency .if the two - qubit state is initially prepared in bd form with coefficients , then the ensuing dynamics preserves the bd form of the state , whose correlation coefficients evolve in time according to the following expressions , where the decay characteristic function at zero temperature is given by , with dephasing factor and dephasing rate } ^{-s/2}\gamma[s]\sin\left[s \arctan(\omega_c t ) \right],\ ] ] where $ ] is the euler gamma function .we are interested in generally non - markovian dynamics since it can prolong the existence of quantum correlations between the qubits .consequently , in the following we will consider , where represents the zero - temperature crossover between markovian and non - markovian dynamics  .+  +  figure [ fig : dynamics ] shows the behaviour of bures distance total , classical and quantum correlations as a function of the dimensionless time , when the two qubits are initially prepared in a rank- bd state with coefficients , , and each locally interacts with a zero - temperature super - ohmic bosonic reservoir with .this particular dynamics leads to paradigmatic _ freezing _ phenomena for both quantum and classical correlations , which can stay constant in certain time intervals despite the state physically changing ; this was originally reported by analysing entropic measures of correlations .bures measures of correlations are here shown to capture the same phenomenology . in our specific example , the freezing of quantum correlations can be seen up until a certain time and the freezing of classical correlations from that exact time .it is worth noting that classical and quantum correlations exactly coincide at time , which is known as the instant of sudden transition from classical to quantum decoherence .this property also happens for correlations quantified by relative entropy and hilbert - schmidt distance . for correlations quantified by trace distance , although quantum and classical correlations alternately freeze in the same time intervals as the other measures , they are generally not coincident at the crossing time .the entanglement , instead , decays monotonically and completely disappears at a finite time exhibiting the well - known sudden death phenomenon .total correlations , finally , decay smoothly and asymptotically reach a non - vanishing value .however , the aforementioned sudden switch between freezing of quantum and classical correlations may not happen at all if the two qubits are initially prepared in a suitable bd state .this is displayed in figure [ fig : dynamics2 ] for initial coefficients .in fact , in the latter case , quantum correlations are frozen indefinitely and never coincide with classical correlations .this peculiar time behaviour is in accordance with ref . , in which this phenomenon was originally investigated by considering the entropic quantum discord .we also point out that , instead , the entanglement presents again a sudden death despite the indefinite freezing of general quantum correlations .this is an interesting point showing once more the very different nature of quantum correlations , as captured by the concept of quantum discord , and conventional entanglement : quantum correlations can be in principle completely unaffected by decoherence in spite of the fast disappearance of entanglement . for a more general study of the sudden change phenomena for quantum correlationssee ref .  .in this paper we obtained exact expressions for the classical and total correlations of two - qubit bell - diagonal states according to the bures distance .these expressions , together with the corresponding known formulae for entanglement and discord - type quantum correlations , allow for the completion of the hierarchy of bures distance based correlations in the physically relevant family of bell - diagonal states .we proved that the closest bell - diagonal classical - quantum state and the tensor product of the marginals of any bell - diagonal state achieve the infima in the definition of classical correlations .we have shown that the nearest uncorrelated state to every bell - diagonal state is not always the product of its marginals and that , in general , the total correlations are strictly smaller than the sum of quantum and classical correlations .these results are in agreement with those obtained with the trace distance but not with the hilbert - schmidt and relative entropy distances .finally , we have shown that , for two independent qubits under local bosonic pure - dephasing non - markovian channels and suitable initial conditions , bures distance quantifiers of correlations can also manifest : ( i ) the freezing of quantum correlations before a certain time and the freezing of classical correlations after the same time ; ( ii ) the indefinite freezing of quantum correlations notwithstanding the contemporary occurrence of entanglement sudden death .another peculiar feature of quantum but not classical geometric correlations , namely the possibility of double sudden changes ( under different dynamical conditions ) as observed theoretically and experimentally for trace distance quantifiers , has also been theoretically shown to occur for the bures measure of discord .overall , with the contributions brought forward by this paper , we have reached a fairly comprehensive understanding of the different components of correlations in generally mixed states of archetypical bipartite systems and their dynamical characteristics .experimental demonstrations with quantum optics and nuclear magnetic resonance setups have also verified the predicted resilience of classical and quantum correlations to particular decoherent evolutions .however , a satisfactory understanding of these phenomena from first principles is still missing .this is perhaps to be traced back to a still incomplete formalisation for the requirements that measures of classical and general quantum correlations have to satisfy .future work will be devoted to the theoretical and experimental understanding of the physical origin of the freezing under nondissipative decoherence common to any known _ bona fide _ measure of quantum correlations before a certain time ( including entropic , trace , bures , and skew - information based measures as studied in ) , and to any currently known _bona fide _ measure of classical correlations after the same time ( counting entropic , trace and now bures classical correlations ) .the ultimate implications of such extreme robustness to decoherence , not observable in entanglement , and the ensuing possibilities opened for quantum technologies clearly deserve further investigation .10 url # 1#1urlprefix[2][]#2 horodecki r , horodecki p , horodecki m and horodecki k 2009 _ rev .phys . _ * 81 * 865

Abstract: the notion of distance defined on the set of states of a composite quantum system can be used to quantify total , quantum and classical correlations in a unifying way . we provide new closed formulae for classical and total correlations of two - qubit bell - diagonal states by considering the bures distance . complementing the known corresponding expressions for entanglement and more general quantum correlations , we thus complete the quantitative hierarchy of bures correlations for bell - diagonal states . we then explicitly calculate bures correlations for two relevant families of states : werner states and rank-2 bell - diagonal states , highlighting the subadditivity which holds for total correlations with respect to the sum of classical and quantum ones when using bures distance . finally , we analyse a dynamical model of two independent qubits locally exposed to non - dissipative decoherence channels , where both quantum and classical correlations measured by bures distance exhibit freezing phenomena , in analogy with other known quantifiers of correlations .


counterparty credit risk has recently moved to the forefront of attention of the financial industry and government regulatory agencies . prior to 2007 ,financial institutions used to make provisions for possible counterparty defaults based on _ ad hoc _ assumptions and proprietary metrics . as a result of the financial crisis of 2007 - 2008 andthe subsequent stricter regulatory requirements , careful analysis and more rigorous modeling methodologies of counterparty credit risk have become a centerpiece of prudent risk management practice . in particular , these developments led to the establishment of special business units within financial institutions whose sole purpose is to monitor and mitigate counterparty credit risk .summaries of various approaches to counterparty credit risk modeling can be found in , , .  in a counterparty credit risk model ,one considers two parties , namely a dealer ( `` a bank '' ) and a client ( `` a counterparty '' ) such as an asset manager , hedge fund or corporate , which have a portfolio of over the counter ( otc ) transactions between them . over the time horizon of the transactions , a default of either the bank orthe counterparty may occur .the objective of the model is to put a dollar value on this default risk and develop strategies to mitigate it .the portfolio of transactions is typically structured as a collection of netting sets . from the risk management perspective, the cash flows within the same netting set are allowed to partially offset each other .generally , counterparty credit risk is managed through maintaining appropriate reserves of cash and high quality financial assets . in the case of credit support annex ( csa ) transactions , the two parties exchange and manage collateral .additionally , a fully fledged counterparty credit risk model takes into account the impact of margining and funding costs .  in this paper , we are concerned with a general approach to counterparty credit risk modeling .a natural modeling framework for counterparty credit risk , which we adopt in our discussion , is provided by backward stochastic differential equations ( bsdes ) . in particular, this approach allows for an efficient and fully consistent description of counterparty default valuation along with its hedging strategy , and can be easily implemented numerically .bsdes , in the context of finance , have been studied in the literature since the 1990 s , see , and have recently been applied to counterparty credit risk modeling , see e.g , .our approach extends the partial differential equations ( pde ) framework developed earlier by burgard and kjaer , see also , , .their methodology is based on a dynamic replication strategy involving the underlying assets as well as both parties credit sensitive zero coupon bonds , and it culminates in the fundamental pde for the counterparty credit sensitive value of an instrument .they also provide closed form solutions to the fundamental pde in a number of relevant cases . here , we extend their approach to a more general multivariate diffusion setting including stochastic interest rates and default intensities .  following a similar strategy, we formulate the counterparty credit risk model in purely probabilistic terms .the dynamic replication approach leads to a jump diffusion bsde with a random time horizon .the reason why a jump may occur in the bsde is that the possible default of either the counterparty or the bank itself ( and associated changes in the value of the portfolio ) can occur at a random time .equations of this type are usually very challenging to solve numerically .motivated by the method developed in , we reduce our fundamental jump diffusion bsde to a continuous bsde with a fixed terminal time .  in case of a counterparty or bank default , their assets are liquidated and the claims of their creditors are settled .different close out conventions are used to determine the value of the portfolio of transactions between the two parties at default . here , we discuss in detail two explicit close out conventions .the solutions to the corresponding fundamental bsdes are interpreted in terms of the commonly used valuation adjustments ( the `` xvas '' ) . in one of the cases , a closed form representation of the solution is available , while in the other one, we derive an accurate approximation to the solution .an important practical aspect is the computational feasibility of the solution to the fundamental bsde . in practice ,the portfolio of transactions between the dealer and the counterparty may consist of hundreds ( or thousands ) of different positions .naive modeling of such a portfolio leads to a high dimensional stochastic system .we discuss an approximation method that allows us to reduce the full model to model driven by a moderate number of risk factors .we then prove that reduced factor bsde indeed approximates the fundamental bsde .moreover , we briefly discuss the question of choosing an appropriate pricing measure .  for continuous diffusion bsdes ,efficient numerical schemes have been developed , see e.g. .we apply a scheme to numerically solve the reduced fundamental bsde which requires an efficient methodology to calculate conditional expected values . to this end, we use a variant of the longstaff - schwartz regression technique that is adapted to our setting . specifically , we propose the use of hermite polynomials as basis functions .hermite polynomials have various advantages , for example they exhibit an addition formula and a martingale property , which provide practical means for calculating conditional expectations .additionally , using the hermite architecture allows us to represent a part of the solution of the bsde in terms of explicit expressions .we continue the study of the numerical aspects of the problem in .  the paper is organized as follows .we present our model setup and develop the fundamental equations describing the value of a portfolio with and without counterparty credit risk in section [ bsdesec ] . the special case of deterministic interest rates and dividends , leading to a burgard - kjaer type of pde ,is covered in section [ sec : bkpde ] . in section [ sec : valueadj ] , we discuss the total value adjustments for the two most common close out values .we then turn our attention to practical issues such as the choice of pricing measure and risk factor reduction in section [ sec : pricingmeasurefactorreduction ] .finally , we propose a numerical algorithm to solving the fundamental bsde using monte carlo methods in section [ numsec ] .the key technical results are proved in the appendix .in this section we present a mathematical formulation of the problem of modeling counterparty credit risk between two counterparties : ( a `` bank '' ) and ( a `` client '' ) .the equations written below assume the perspective of the bank .we study contracts on an asset between the bank and its counterparty , both of which may default . in our setting neither of the two possible defaultshave an effect on the asset .  with this situation in mind, we consider a probability space and let the market filtration be generated by an -dimensional brownian motion and augmented by all -null sets .the information on default events is represented by the filtration to be specified later .we define the enlarged filtration by , for all .then is a filtered probability space satisfying the usual conditions .see for example ( * ? ? ?* vi.3 ) for the properties of progressive enlargements of filtrations .let denote the price process , namely an -dimensional markov process with infinitesimal generator .the dynamics of the asset under the measure are here , and are deterministic functions and is a standard -dimensional brownian motion .we assume the usual conditions for and to guarantee the existence and uniqueness of the strong solution to this sde .additionally , is an -dimensional real vector .we consider a netting set consisting of instruments which may comprise of derivatives and cash type with underlying asset . from the counterparty risk management perspective, transactions with a fixed counterparty are allowed to be offset against each other within a netting set .the composition of netting sets is set forth by bilateral agreements .we denote the time -value of the netting set by .additionally , we consider the riskless ( default - risk - free ) bank account .its dynamics is given by where is a stochastic credit riskless interest rate . in the following ,we assume that as well as all other rates and dividends are adapted with respect to the market filtration . from the financial perspective, is the rate paid by a theoretical `` central margin account '' , and can be proxied by the ois rate .  in our model we allow for both , the bank and its counterparty to default .we let and be the random default times of the bank and the counterparty respectively and denote their indicator processes by counting processes and the natural filtration generated by and constitutes the default event filtration , i.e. . both counting processes , are assumed to be cox processes , i.e. they have stochastic , time - dependent intensities , , i.e. ,\ ] ] and .\ ] ]  we denote the bank s and counterparty s default risky , zero - recovery zcbs and with respective maturities and . they follow the dynamics here , the adapted stochastic processes are the yields on and , respectively .the legal framework for otc derivative trades is provided by the _ isda master agreement _ which contains a common core and adjustable terms to be agreed on by both parties .the agreement aims at mitigating ( counterparty ) risk by documenting aspects like netting , collateral cash - flows , default events and the close out process ( see e.g. ) . in a world , where usually several transactions with a counterparty occur , netting allows the two parties to offset what they owe to one another .the financial crisis in 2008 fueled the need for derivative pricing methodologies that include aspects of counterparty credit risk .the goal is to find the value of a netted portfolio of derivatives on allowing for both the bank as well as its counterparty to default .more precisely , we let denote the time value of the netting set with time horizon , which is assumed to satisfy .the value of the netting set between the bank and the counterparty without counterparty default risk is denoted by .the difference between these two values , denoted by , is called the total valuation adjustment , i.e. .  to mitigate counterparty credit risk , the two parties exchange collateral in form of cash or high quality financial instruments .the mechanics of collateral in otc transactions are specified by the csa which is usually amended to the isda master agreement .typically , one distinguishes between two kinds of collateral , namely the initial margin and the variation margin .the former is posted by both counterparties without any netting taking place .the amount of initial margin is calculated using risk based methods such as var , covar or stress tests .these calculations involve historical simulations , typically using an observation window of 1 to 5 years , see e.g. . to ensure that the collateral can be retrieved in a default scenario ,it is segregated and can not be used to , for example , fund other positions .in contrast , variation margins , which we denote as , are calculated frequently based on the market value of the transactions , and they can be netted and rehypothecated . we denote the initial margin posted by the bank to the counterparty by , whereas the amount from the counterparty is denoted by .the margin value adjustment ( mva ) is the bank s cost of posting the initial margin to the counterparty over the time  ] is called capital value adjustment ( kva ) .another component of the total value adjustment is the funding value adjustment ( fva ) .essentially , the fva accounts for the costs of funding of uncollateralized ( or partly collateralized ) positions .there is a debate around how the funding adjustment should be treated in a framework like ours , see , pp .349 - 356 , and , .  in the followingwe first find the default free value of the contract in terms of a conditional expectation .we present the cash - flow netting at default and the cash flows associated with the portfolio that is set up to replicate the value .then the fundamental bsde describing the counterparty credit risky value is derived .note that , from now on , all interest rates are considered adapted stochastic processes unless stated otherwise .the value of the netting set without credit counterparty risk depends solely on the price process and is derived similarly to the price in the classical black - scholes model .as usual , we can set up a self - financing portfolio replicating .however , contrary to the classical pricing theory , the position in is not funded at a single riskless rate , but instead financed by a repurchasing agreement ( repo ) at a repo rate .the securities are pledged as collateral against cash to purchase the securities . in practice ,a haircut is applied to the amount of cash received against the collateral , that is the loan size is smaller than the current face value of the pledged securities .the amount of haircut depends on the quality of the securities . here, for simplicity , we shall assume a zero haircut ; it is straightforward to adapt our calculations to accommodate for non - zero haircuts .  the replicating portfolio consists of a position in units of , units in a repo cash account , and units of , i.e. we must have for all times  ] is the recovery rate in case defaults .we are not concerned here with recovery rate modeling and so , for simplicity , we assume that is deterministic . in reality ,recovery rates are an unknown random variable , not necessarily measurable with respect to . in summary , we see that the value at default , in case defaults , has the form   similarly , if the bank itself defaults , it has the right to proper fulfillment of the contract and hence , in addition to the collateral amount , it receives the outstanding balance . if , the bank pays a fraction of its own obligation , namely , where  ] , as where the pair of processes is the solution to the following bsde : . \end{split}\ ] ] we refer to this equation as the _ reduced fundamental bsde_. we emphasize here that it is enough to find the solution to the reduced bsde and then use to find the credit - risky portfolio value .hence will play a central role in the remainder of the paper .  as a simple yet instructive example of the above reduction, we consider the following jump bsde : where , , are constants and is an -measurable random variable .we find a closed form solution to this bsde by following the steps outlined above .the key is again to reduce the bsde with random time horizon and a jump into a bsde with fixed time horizon and without jumps .the corresponding reduced equation given by is a linear inhomogeneous ode .its solution reads as a consequence , applying theorem [ thm : transformedbsde ] gives the explicit solution of as notice that it would be hard to solve directly without the reduction step .in this section we derive the fundamental pde in the spirit of burgard and kjaer , see , .the pde approach requires that all the rates and dividends introduced above are deterministic . in that sensethe approach based on the fundamental bsde , which requires only that the rates and dividends are adapted to , is more general .we derive the burgard - kjaer pde starting with the reduced fundamental bsde .namely , we make the following ansatz : where is a smooth function \times\br^n\to\br ] using rate .the first term on the right hand side of reflects the difference in discounting in the classic black - scholes model and counterparty credit risky discounting as discussed above .the remaining terms on the right hand side can be identified as follows : ,\ ] ] represents the credit valuation adjustment ( cva ) , ,\ ] ] represents the debt valuation adjustment ( dva ) , ,\ ] ] represents the capital valuation adjustment ( kva ) , ,\ ] ] represents the margin valuation adjustment ( mva ) , and finally \ ] ] is the funding valuation adjustment ( fva ) .we now connect the solution of the reduced fundamental bsde , to the solution to the fundamental bsde .explicitly , we have the following relation between the two solutions : the expressions derived above extend the corresponding explicit formulas in .choosing the adjusted portfolio value as the close out value , we note that the generator of the reduced fundamental bsde has the following form : in contrast to the case of , the resulting reduced fundamental bsde is nonlinear in , and an explicit representation to its solution is not available .  instead , we can construct an approximate solution , assuming that the counterparty credit risk adjustment is small relative to .specifically , we use the approximation : this approximation is first order accurate in . substituting into , where satisfies the riskless equation , and using the above approximations , we obtain the following linear bsde for the adjustment , here , , where is defined in , is the following effective discounting rate : and   using the results summarized in appendix [ app : linbsde ] , we can solve this linear bsde explicitly .namely , we define the following stochastic exponential : then formula in appendix [ app : linbsde ] yields \\ & = \ee_t\big[\int_t^t d_{\fr}(t , s ) \gamma_{t , s } \big(g_s + r_sv_s\big ) ds\big ] .\end{split}\ ] ] explicitly , the expression above can be written as \\ & \quad - ( 1-r^\cc ) \ee_t\big[\int_t^t d_\fr(t , s)\ , \gamma_{t , s } \, ( r_s^\cc - q_s^\cc)(v_s - x_s - i^{fc}_s)^+  ds\big]\\ & \quad - ( 1-r^\cb ) \ee_t\big[\int_t^t d_\fr(t , s)\ , \gamma_{t , s } \ , ( r_s^\cb - r_s)(v_s - x_s+i^{tc}_s)^- ds\big]\\ & \quad + \ee_t\big[\int_t^t d_\fr(t , s)\ , \gamma_{t , s } \ , r^k_s k_s ds\big]\\ & \quad + \ee_t\big[\int_t^t d_\fr(t , s)\ , \gamma_{t , s } \,\big ( r_s^{fc } i^{fc}_s - ( r_s^{tc } + r_s ) i^{tc}_s + ( r_s^x + r_s ) x_s  \big ) ds\big]\\ & \quad + \ee_t\big[\int_t^t d_\fr(t , s)\ , \gamma_{t , s } \,(r^f_s - r_s ) ( v_s - x_s+i^{tc}_s ) ^- ds\big ]. \end{split}\ ] ] the individual terms in this expression can be interpreted in a fashion similar to the analogous terms in .notice that , compare to , the discount rate is replaced with .the first term on the right hand side of is an artifact of the difference in discounting in the classic black - scholes model and the counterparty credit risky discounting .the remaining terms on the right hand side have the following interpretation : ,\ ] ] represents the credit valuation adjustment ( cva ) , ,\ ] ] represents the debt valuation adjustment ( dva ) , ,\ ] ] represents the capital valuation adjustment ( kva ) , ,\ ] ] represents the margin valuation adjustment ( mva ) , and finally \ ] ] is the funding valuation adjustment ( fva ) .finally , we notice that the solution to the fundamental bsde is related to the riskless portfolio value via the following approximation : where is the total adjustment calculated above .we emphasize that , unlike , the relations above link the solution of the fundamental bsde to the solution of reduced fundamental bsde .up until now we have not addressed the issue of choosing a pricing measure .the bank s portfolio may consist of a large number of assets . in practice ,each of the asset classes is valued under its own martingale measure , which in turn depends on the appropriate choice of numeraire .for example , swaptions are priced under the forward swap measure , while equity options are priced under the rolling bank account measure . from the pricing perspectivethis approach is fully consistent , the choice of numeraire does not affect model valuations . however , the choice of pricing measure is crucial from the enterprise risk management perspective .the risk of a portfolio composed of various assets is not the sum of the risks of its components , as the dependences between assets may reduce or increase the total risk .it is thus important that the monte carlo simulations are carried out under a common pricing measure .there is no natural way of aggregating the different martingale measures into one pricing measure for the entire portfolio .we will not discuss this issue in detail here . in our model specification we simply assume a pricing measure that is not a martingale measure but rather behaves like a historical ( aggregate ) measure .the choice of is determined by the bank s risk appetite , regulatory requirements , and other factors , see e.g. , .  once the aggregate pricing measure has been selected , the next issue is model calibration .there are two categories of variables entering the model : ( i ) directly observable such as asset prices , interest rates , recovery rates , etc . , and ( ii )not directly observable variables , which have to be estimated from the market data , such as volatilities , correlations , default intensities , etc .. generally , for the indirectly observable model inputs , parameters inferred from cross - sectional market prices are associated with various martingale measures , while parameters inferred from historical time series are associated with physical measures .notice that parameters such as volatilities can be deduced from both types of calculations .their numerical values will differ depending on whether they are calculated as market implieds or by means of maximum likelihood estimation .however , typically the only practical way of calculating correlations is from historic time series . for default intensities , the cds market , whenever available , yields risk neutral default probabilities . for less liquid names , without a liquid cds market , historical default data , such as moody s dbs bank or various credit ranking models , can be used ( see e.g. ) .another practical issue is the choice of risk factors .a financial institution is likely to contain thousands of positions in a netting set , each of which subject to market and counterparty risk . from a practical perspective , an analysis of a system with such a large number of risk factors is infeasible . in order to bring the dimensionality of the problem to a manageable size , a methodology of reducing the number of risk factors is required .  in mathematical terms, we are facing the issue of approximating the solution to the following high - dimensional fbsde : under the usual lipschitz conditions on the coefficients , standard results guarantee the existence and uniqueness of the solution to this system ( see e.g. ) .  a common approach used in practice is principal component analysis ( pca ) . specifically , the instantaneous covariance of the price process has the spectral decomposition : where are the eigenvalues ordered by size , and are the spectral projections .generically , each of the eigenvalues is non - degenerate and each of the spectral projections defines a one - dimensional subspace . in general , the eigenvalues and spectral projections are stochastic and depend on the realization of the process and time .the left hand side of is estimated from suitable market data , as discussed above .reduction of risk factors is practical if only a small number of eigenvalues explain the covariance matrix , i.e. with where is a given tolerance level .we thus consider the projection operator onto the subspace spanned by the eigenvectors corresponding to the first eigenvalues .the key assumption ensuring that practicality of the factor reduction methodology is that is stable , and so its range persists regardless of market conditions .we can formulate this requirement heuristically as , \end{split}\ ] ] i.e. is approximately equal to its average over time .we refer to the orthonormal basis in defined by this projection as the principal factors .the existence of is a strong assumption and it is not true in a general mathematical set up .rather , it is an empirical fact indicating that the financial markets are driven by a relatively small number of persistent economic factors .we define the following quantity : \,du\big)^{1/2}\\ & = \big(\int_0^t \ee \big [ \tr \big(\sigma(u , s_u)^\top \sigma(u , s_u)-p\sigma(u , s_u)^\top \sigma(u , s_u)p \big ) \big ] \,du\big)^{1/2}. \end{split}\ ] ] in words , measures the average discrepancy between the true covariance of the assets and the truncated covariance given by the projection onto the principal factors .  the key fact , established below ,is that the price process of can be , to a good degree of accuracy , explained in terms of the principal risk factors .specifically , we consider the projection of the brownian motion on the principal factors . in general, is not a brownian motion .however , we can choose a standard -dimensional wiener process such that where is a constant -matrix with the property that and .we now consider a system driven by the principal risk factors , more precisely where the drift and diffusion coefficients are the same as in . as this equationcan be understood as an sde with a new diffusion coefficient , existence and uniqueness of the solution are obvious .we expect that the solution to this sde is approximately equal to the true process .we turn these intuitions into a mathematical statement as follows . for an adapted , matrix - valued process we introduce the following semi - norm : ^{1/2},\ ] ] and the norm then we have the following theorem .[ facredthm ] assume that and are lipschitz continuous : with constant and , and satisfy the standard growth conditions : with constant .  then has a unique strong solution , and where is a constant .the proof of this theorem is presented in appendix [ app : facredproof ] .the theorem above says that the price process driven by the truncated risk factors indeed approximates the true price process .the tightness of the approximation is given by , and it may degrade exponentially fast in the time horizon .we now consider the backward part of the system of equations driven by the principal risk factors : note that , unlike , the driving process in is not a standard brownian motion anymore but the martingale .there are several theoretical and practical aspects to be considered when working with this equation , which we will address in .for instance , existence of a solution to the above equation can be shown , however the solution is not unique . to see this ,recall that the process represents the delta hedging strategy , also compare . reducing the risk factorsleads to an incomplete market as one can not fully hedge one s position anymore .we may , for example , choose to be a minimum variance strategy . in order to prove uniqueness of the solution, we have to introduce another process , compare .  in order to measure the discrepancy between the exact bsde and its approximation, we find it convenient to introduce the following ( semi-)norms for adapted , vector - valued processes : ^{1/2},\\ \|x\|_{\beta,2 } & = \big ( \int_0^t \|x_u\|^2_\beta \ , du \big)^{1/2},\\ \|x\|_{\beta,\infty } & = \sup_{0 \leq t \leq t } \|x_t\|_\beta .\end{split}\ ] ] the following theorem shows that the solution to equation approximates the solution of the backward part of system .  [ facredthmbackward ] assume that the terminal value and the driver are lipschitz continuous : then there exist constants , depending on the time horizon , such that the following inequalities hold : du  & \leq c_3 \|\widetilde s - s \|_{2,\infty } .\end{split}\ ] ]  the proof of this theorem is presented in appendix [ app : facredproof ] .consequently , the solution can be approximated by the processes driven by the principal risk factors .moreover , the third of the inequalities in shows that the residual portion of is small .in this section we discuss a general numerical framework for solving continuous fbsdes using monte carlo methods .such equations include the reduced fundamental bsde discussed above .we propose an algorithm for finding the counterparty credit - risky value as an application .the method is then illustrated in a simple example .we briefly review a method for discretization of the forward backward system this method is classic and has been originally proposed by bouchaud and touzi in .  for the forward process , we apply a standard discretization scheme ( see e.g. ) such as euler s or milstein s scheme ( for the latter , assuming suitable integrability conditions ) .let denote a regular time grid , where .in particular , for the euler scheme , the approximation takes the form of the following discretized forward process where and .  in order to approximate the backward part of the fbsde, we set , and .this leads to the following system : starting with the terminal condition we proceed with finding and for all .note that the s in are not adapted and depend on .these two problems can be solved by taking conditional expectations which leads to \\ & = \ee_i \left [ y_{i+1 } \right ] +  f(s_i , y_{i } , z_i ) \delta_i , \end{split}\ ] ] where we have used the notation =\ee _{ t_i } [ \ , \cdot \ , ] ] .we discuss this issue in the following section .a practical and powerful method of computing the conditional expected values in is the longstaff - schwartz regression method originally developed for pricing american options ( see also ) .we use a variant of this method that involves the hermite polynomials .this choice is natural as expressions involving conditional expectations of hermite polynomials of gaussian random variables lead to convenient closed form expressions .let , , denote the -th normalized hermite polynomial corresponding to the standard gaussian measure .for a multi - index , where each is a nonnegative integer , we define these functions form an orthonormal basis for the hilbert space , where is the standard gaussian measure in dimensions , .  the key property of is the following addition formula for  ] .the solution of is now obtained from solution to the reduced bsde as   we will now construct a numerical solution to the reduced bsde . more precisely , we compare the numerical solution to its explicit solution in the case of .we assume the time horizon of , and choose the following values of the parameters : we divide the time interval into subintervals and generate monte carlo paths .for estimating the conditional expected values we choose the hermite architecture , with .figure [ traj ] shows representative monte carlo trajectories simulating and .here , the black lines are the paths of the exact solution , while the red lines are the numerical approximations calculated according to the algorithm above .notice that the approximate path of is very close to the exact trajectory .however , the paths representing differ more .apparently the numerical solution to the process of a bsde converges slower than the numerical solution to the process .figure 1    on the other hand , the expected values of both and are close approximations of the exact solution of the bsde .this is shown in figure [ expvalues ] .figure 2    finally , figure [ relerror ] shows the relative error of the expected values of and versus the expected values of the exact solution .in this section we present general results on sdes and bsdes we have used throughout the paper .first we are interested how the fundamental bsde with jumps can be transformed into a reduced fundamental bsde .the fundamental bsde can more generally be expressed as an equation , \\y_{\tau \wedge t } & = \e1_{\tau > t } \xi + \e1_{\tau\leq t } ( \theta^1_\tau \e1_{\tau= \tau^1 } + \ldots + \theta^m_\tau \e1_{\tau= \tau^m } ) , \end{split}\ ] ] driven by an -dimensional brownian motion and a counting process .the solution to this bsde is the set of adapted stochastic processes that satisfies .the driver is a given deterministic function and denotes the first default time .the bsde has a possible random time horizon , more precisely its terminal value depends on whether a default event happens before the fixed time horizon . in that casethe bsde stops at the random time at an entry of the adapted stochastic process .otherwise the bsde carries on to the final time with an - measurable random variable as the final value .although jump bsdes are rather complex to handle , we are in the particular situation that only ever one jump occurs and it happens at the very end of the bsde .this is what we can use to transform the above random horizon jump bsde into an equation without jumps and with a fixed terminal time , i.e. , \\\cy_t & = \xi . \end{split}\ ] ] the following result is a generalization of theorem 4.3 in , giving us the possibility to express the solution to the jump bsde in terms of the solution of a continuous bsde .in the first case no default happens before the terminal time , i.e. . on we have from that , and for all  ] . then again from we have on \} ] .the definition of from gives meaning we have .the last case considers the situation when the default happens before or at time , i.e. . again from we have and thus on we get which is equation in integral form .continuous linear bsdes are equations with a driver that is linear in and , meaning we consider equations of the type ,\\ y_t&=\xi ,  \end{split}\ ] ] with -dimensional brownian motion , measurable random terminal value and , , being adapted stochastic processes .the solution of this equation is any pair of adapted processes that satisfies . these are some of the few bsdes for which at least the first part of the solution can be found explicitly . from (* proposition 2.2 ) we have \\ \end{split}\ ] ] where here , denotes the stochastic exponential of a stochastic process .we begin by rewriting the sdes for and in the integral form : consequently , their difference is given by and thus , by means of ito s isometry , \,du\big)^{1/2}. \end{split}\ ] ] note that \,du \\ & = \int_0^t \ee \big [ \tr \big (  \sigma(u , s_u)^\top \sigma(u , s_u ) - p \sigma(u , s_u)^\top \sigma(u , s_u ) p \big ) \big ] \,du \\ & = \int_0^t \delta(u)^2 du , \end{split}\ ] ] where is defined by . using lipschitz continuity ,this yields where is a constant , explicitly given as .we shall now invoke classic grnwall s inequality : if is a nonnegative continuous function with where is a non - decreasing function and , then squaring , and applying the inequality above to , we obtain where we have set . taking the supremum over yields the claim .   the difference between and the backward part of the systemis given by in the following we adapt the arguments used to prove the existence of a solution to a bsde ( see , ) .applying ito s formula to the process , where the constant will be chosen later , yields taking expectations on both sides of this equation leads to the following identity : du \\ & =  \|\xi ( \widetilde s_t ) - \xi ( s_t)\|_\beta^2 + 2 \int_t^t \ee\big[e^{\beta u } ( \widetilde y_u - y_u)\big ( f(u,\widetilde s_u , \widetilde y_u , \widetilde z_u ) - f(u , s_u , y_u , z_u ) \big)\big ] du.\\ \end{split}\ ] ] using lipschitz continuity of the terminal condition and driver , we obtain that du \\ & \leq k_\xi \|\widetilde s_t - s_t\|_\beta^2 + 2 k_f\int_t^t \ee\big [ e^{\beta u } |\widetilde y_u - y_u|\big ( |\widetilde s_u - s_u| + |\widetilde y_u - y_u| + |\widetilde z_u - z_u| \big)\big ] du .\end{split}\ ] ] using the elementary inequality where is a constant , we find that we thus arrive at the following key inequality : du\\ & \leq k_\xi \|\widetilde s_t - s_t\|_\beta^2 + k_f\int_t^t \|\widetilde s_u - s_u\|^2_\beta  du. \end{split}\ ] ] now , we choose sufficiently large so that , and then subsequently we choose so that .99 andersen , l. , duffie , d. , and song , y. : funding value adjustments , working paper ( 2016 ) .barles , g. , buckdahn , r. , pardoux , e. : _ backward stochastic differential equations and integral partial differential equations _ , stochastic and stochastic reports 60 , 5783 ( 1997 ) .basel committee on banking supervision _margin requirements for non - centrally cleared derivatives _ ( 2015 ) .basel committee on banking supervision _basel iii : the stardardised approach for measuring counterparty credit risk exposures : frequently asked questions _bertsekas , d. p. : _ dynamic programming and optimal control _ , vol .2 , athena scientific ( 2012 ) .bichuch , m. , capponi , a. , sturm , s. : _ arbitrage - free xva _ , http://arxiv.org/pdf/1608.02690v1.pdf[arxiv:1608.02690 ] ( 2016 ) .bouchard b. , touzi n. : _ discrete time approximation and monte - carlo simulation of backward stochastic differential equations _ , stochastic processes and their applications , vol 111 , 2 , 175 - 206 ( 2004 ) .brigo d. , morini m. , and pallavicini a. : _ counterparty credit risk , collateral and funding : with pricing cases for all asset classes _ , wiley , isbn : 978 - 0 - 470 - 74846 - 6 ( 2013 ) .burgard c. , and kjaer m. : _ pde representation of derivatives with bilateral counterparty risk and funding costs _ , the journal of credit risk , 7(3 ) , 1 - 19 ( 2011 ) .canabarro , e. , and duffie , d. : _ measuring and marking counterparty risk _ , asset / liability management for financial institutions , institutional investor books ( 2003 ) . crpey s. , bielecki t. , and brigo d. : _ counterparty risk and funding : a tale of two puzzles _ , chapman and hall / crc financial mathematics series , isbn : 978 - 1466516458 ( 2014 ) evans , l. c. : _ partial differential equations _ ,ams ( 1998 ) .geiss , c. , labart , c. : _ simulation of bsdes with jumps by wiener chaos expansion _ , http://arxiv.org/pdf/1502.05649v1.pdf[arxiv:1502.05649 ] ( 2015 ) .ghamami , s. , and goldberg , l. r. : _ stochastic intensity models of wrong way risk : wrong way cva need not exceed independent cva _ , working paper ( 2012 ) .green , a. : _ xva : credit , funding and capital valuation adjustments _ , wiley ( 2015 ) .green , a. , kenyon , c. : _ mva : initial margin valuation adjustment by replication and regression _ , https://arxiv.org/pdf/1405.0508.pdf[arxiv:1405.0508 ] ( 2014 ) .green , a. , kenyon , c. , and dennis , c. : _ kva : capital valuation adjustment _ , https://arxiv.org/pdf/1405.0515.pdf[arxiv:1405.0515 ] ( 2014 ) .gregory , j. : _ the xva challenge : counterparty credit risk , funding , collateral , and capital _ , wiley ( 2015 ) .gregory , j. : _ the impact of initial margin in otc derivatives _ , working paper ( 2016 ) .hull , j. , and white , a. : _ valuing derivatives : funding value adjustments and fair value _ , fin .j. , * 70 * , 46 - 56 ( 2014 ) .isda publications http://www.isda.org ( 2016 ) .el karoui n. , mazliak , l. : _ backward stochastic differential equations _ , crc press , isbn 9780582307339 ( 1997 ) .el karoui n. , peng s. , and quenez m.c .: _ bsdes in finance _ , mathematical finance , 7 ( 1 ) , 1 - 71 ( 1997 ) .kenyon c. , green a. , berrahoui , m. : _ which measure for pfe ?the risk appetite measure _ , preprint arxiv ( 2015 ) .kharroubi i. , lim t. and ngoupeyou a. : _ mean variance hedging on uncertain time horizon in a market with a jump _ , applied mathematics and optimization , 68 ( 3 ) , 413 - 444 ( 2013 ) .kloeden , p. , and platen , e. : _ numerical solution of stochastic differential equations _ , ( 1992 ) .lesniewski , a. , lewis , h. , and richter , a. : _ wrong way risk _ , working paper ( 2016 ) .lesniewski , a. , richter , a. : _ risk factor reduction in a bsde setting _ , working paper ( 2016 ) .longstaff , f. a. , and schwartz , e. s. : _ valuing american options by simulation : a simple least - squares approach _ , rev . fin ., 14 ( 1 ) , 113 - 147 ( 2001 ) . moody s : https://www.moodys.com/credit-ratings/dbs-bank-ltd-credit-rating-2922[_dbs credit rating _ ] , ( 2016 ) pardoux , e. , and rascanu , a. : _ stochastic differential equations , backward sdes , partial differential equations _ , springer ( 2014 ) .protter , p. : _ stochastic integration and differential equations _ , springer ( 2004 ) .stein , h. : _ fixing underexposed snapshots - proper computation of credit exposures under the real world and risk neutral measures _ , technical report , bloomberg lp , december , 1 - 23 ( 2013 ) .

Abstract: we discuss a general dynamic replication approach to counterparty credit risk modeling . this leads to a fundamental jump - process backward stochastic differential equation ( bsde ) for the credit risk adjusted portfolio value . we then reduce the fundamental bsde to a continuous bsde . depending on the close out value convention , the reduced fundamental bsde s solution can be represented explicitly or through an accurate approximate expression . furthermore , we discuss practical aspects of the approach , important for industry applications : ( i ) efficient numerical methodology for solving a bsde driven by a moderate number of brownian motions , and ( ii ) factor reduction methodology that allows one to approximately replace a portfolio driven by a large number of risk factors with a portfolio driven by a moderate number of risk factors .


the recent spread of wireless devices has brought heavy data traffic which video - streaming requests , such as youtube , occupy a dominant portion of .unfortunately , however , current wireless systems have limitations of resources to accommodate this tremendous video traffic .discovering new but inexpensive resources is considered a solution to cope with the explosive traffic . in the same vein ,memory for data caching arises as a new resource to exploit in wireless communications .the characteristic of video traffic facilitates utilization of memory to handle huge video traffic ; a few popular videos account for the majority of video traffic and hence network traffic to carry the videos to the end users can be significantly reduced by storing the top - ranked video files near the users who are likely to request them .although caching has been intensively studied in wired networks , also known as content delivery network ( cdn ) , it is transparent to wireless segments . without consideration of wireless aspects ,cdn is not enough to provide sufficient data rate to wireless end users . in this context, there have been recent studies figuring out wireless caching . for a given caching placement ,several transmission schemes to exploit the given memory contents have been studied in .based on appropriate user grouping , when each user requests multiple files , a groupcasting scheme was proposed in . with practical zipf popularity distribution , characterized the regimes with different coded multicasting gains .prefix caching for wireless video streaming was proposed and optimized in .caching was exploited in cooperative multi - point mimo transmission ( comp ) to reduce backhaul cost in . because where and what to cache determines the performance of wireless caching , caching placement has also attracted research interests , namely femto - caching and device - to - device ( d2d ) caching . in , proactive caching for femto - caching and d2d caching was studied . a caching scheme based on prediction of future demandswas proposed in .optimal caching placement is able to minimize the access distance and inter - cell interference in wireless cache networks . in ,an optimization problem in terms of the average delay , a function of caching placement , was formulated and proved to be np - complete when each user can access to a different set of femto base stations . with random caching placement for d2d communication , optimal cluster size and the parameter characterizing the random caching distributionwere studied in , where a square cell was composed of square clusters and d2d communication was activated if the file requested by user in a cluster was cached in any device inside the cluster . for d2dcoded caching placement , optimal portion of files to store and density of nodes caching the requested file were investigated in , respectively . combining coded multicasting with d2d communication , proposed caching and delivery schemes which can exploit spatial reuse gain as well as coded multicasting gain .caching placement optimizing a tradeoff between throughput and outage was explored in .when maximum - distance separable coding is used , optimal storage allocation under a total memory constraint was studied in .it was shown in that symmetric allocation achieved asymptotically optimal performance . replacing outage with delay, a similar tradeoff was revealed in a multi - hop network .  although the aforementioned studies on caching placement aim at wireless caching , the wireless channel models in those studies partially address effects of wireless channel fading on caching placement . in particular ,constant and identical channel links are assumed in most of the previous works .although the averaging effect of long file transmission justifies the assumption from a viewpoint of each link , random fading channels might reveal different aspects .for example , caching different files according to their popularity ( i.e. , _ file diversity _ ) , which increases the chance to access nearby caching nodes , is reported to be optimal because caching the same file in multiple helpers is redundant under constant and identical channels . however, caching the same file in different helpers might be able to offer _ channel diversity _ and hence caching the same file can be rather beneficial in random fading channels .although channel fading is considered in , the fundamental tradeoff was not clearly shown since they focused on proposing suboptimal or asymptotic solutions .  in this context , under random channel fading , this paper revisits and studies the optimal caching placement to minimize average bit error rate ( ber ) when there exist multiple helpers storing files .we identify the tradeoff between channel diversity and file diversity and derive optimal caching placement optimizing the tradeoff .our main contributions are as follows :  * we propose a greedy algorithm to find the optimal caching placement in terms of ber . exploiting the property of the optimal caching placement which we derive ,the proposed algorithm has considerably reduced search space .it is shown that optimal caching placement is not focusing on the file diversity gain only , contrary to ; optimal caching placement is a balance between the file diversity gain and the channel diversity gain . *we identify the conditions in terms of the popularity factor , i.e. , zipf exponent , that the optimal placement can be found without running the proposed greedy algorithm , and derive the corresponding optimal caching placement in closed form .in particular , we derive two special thresholds of popularity seeking an extreme of either file diversity gain or channel diversity gain , respectively . furthermore , given enough proximity and transmit power ,we show that optimal caching placement is parametrized solely with popularity .the rest of this paper is organized as follows : section [ system_model ] presents our system model .an optimization problem is formulated in section [ problem_formulation ] .we propose the algorithm to find a solution in section [ optimal_caching_placement ] and prove the optimality of the proposed algorithm . in section [ characterization ] ,we identify and analyze the tradeoff between the file diversity gain and the channel diversity gain .numerical results are presented in section [ numberical resutls ] .finally , section [ conclusion ] concludes this paper .in this paper , we consider a cell composed of multiple clusters . in each cluster , the users can access helpers which are capable of storing files in their memories .there is a file library whose size is which is strictly larger than .i.e. , .each helper can store one file from the library in advance , and the users request a file in the library independently with probability according to the zipf distribution with exponent , of which probability mass function is given by in each cluster , only one user is served for each file delivery period , which corresponds to orthogonal multiple access .assuming orthogonal frequency allocation across clusters , inter - cluster interference is not considered in this paper , and thus we consider one cluster in isolation for tractable analysis . in addition , for simplicity , we assume that based on open loop power control , all of the users in a cluster have the same average received signal - to - noise ratio ( snr ) from helpers regardless of their positions , but they still suffer from independent small - scale fading effects . as a result , the multi - user scenario is simplified into single - user one where a user who requests a file is served at every moment .if the requested file is already stored in any helper s memory in a cluster , the requested file is transmitted from the helper which cached the file . throughout this paper, we call this type of communication _ cluster communication_. if the requested file is cached by more than one helper , a helper which has the largest channel gain is selected by the requesting user , using perfect channel state information . fig .[ fig_system_model ] shows an example of caching placement for a system with seven helpers , where the instantaneous received snr from helper which caches the file is denoted by ; the average of is represented as . if the first file is requested by the user , since there are three helpers which has the file , the requesting user selects the helper whose received snr is the maximum among the three helpers and informs the corresponding helper to deliver the the first file . on the other hands ,if the file requested by the user is not stored in the cluster , the bs serves the requesting user with average received snr , ; namely , the file is transmitted through _cellular communication_. we denote the ratio between the average received snrs of cluster and cellular communication as such that , and owing to closer distances to caching helpers than to the macro base station , is reasonably assumed to be greater than one. however , instantaneous channels gains from helpers and bs follow rayleigh distribution .  packet error rate ( per ) and frame error rate ( fer ) are practical performance metrics .however , it is hard to mathematically analyze the impacts of caching placement and channel fading with per or fer because an expression of per or fer incorporating all affecting factors , such as channel coding and upper layer parameters , is usually not available . on the other hand , fer or peris in general represented as a function of ber .therefore , minimizing ber results in reducing fer or per , albeit not linearly proportional . in this context, we study optimal caching placement to minimize ber , which will also proportionally reduce per or fer . assuming arbitrary coherent modulation / demodulation , the instantaneous ber in general form is given as  , where is the instantaneous received snr , , and are appropriate constants for each type of modulation , and is q - function defined as for simplity, we assume and , which corresponds to qpsk , but generalization for arbitrary and is straightforward and the assumption of and does not lose any insights .then , a file is transmitted over many coherent periods and the corresponding average ber of the file is obtained as .because caching placement affects the average ber , in this section , we derive the average ber as a function of caching placement . let us first derive the average ber of the file , denoted by , for given caching placement  ]  * return *   in general , a greedy algorithm is not global optimal ; however , we show that the proposed greedy algorithm can find an optimal solution . through the following lemma and theorem , we prove the optimality of the proposed greedy algorithm .[ lemma_1 ] for , ber gain of file decreases as increases  please refer to appendix [ pf_lemma_1 ] .[ thm_1 ] for , the caching placement obtained by the proposed greedy algorithm , , is optimal for the -helper system.  please refer to appendix [ pf_theorem_1 ] .the proposed greedy algorithm has low complexity , however , we can reduce the complexity further with the following proposition .[ proposition_1 ] for non - uniform popularity ( i.e. , ) , the elements of optimal caching placement must satisfy   please refer to appendix [ pf_prop_1 ] .proposition [ proposition_1 ] implies that since a lower index file is requested more frequently , the placement which stores more files whose index is low can decrease the average ber more .it is shown in the proof of theorem 1 that caching placement of the iteration is optimal for the system with helpers .when there exist helpers , caching the file is strictly suboptimal since this implies one of the files more popular than the file is not cached .hence , for optimal caching placement . moreover , by proposition 1 , for .therefore , in the iteration which determines optimal caching placement of a system with helpers , it is unnecessary to compare the ber gain of the files less popular than the popular file .consequently , the number of comparisons in the iteration is reduced from the size of file library to . because iterations are required to find optimal caching placement of a system with helpers , the total number of comparisons decreases from to , which is independent of .theorem [ thm_1 ] ensures that the proposed greedy algorithm finds the optimal caching placement , given system parameters ( i.e. , ,  ,  ,  , and ) . in this section, we identify the conditions that the optimal placement can be found without running the proposed greedy algorithm , and derive the corresponding optimal placement . before analyzing further ,we first define the two types of gains which can be reaped by caching : _ file diversity gain _ and _ channel diversity gain_.  the amount of ber reduced by adding a new file which is not cached in any helper , is called file diversity gain .changing communication from cellular to cluster reduces average ber by means of the proximity between the helpers and the user .obviously , file diversity gain is proportional to the popularity of the file .hence , the file diversity gain of the file is defined as   the amount of ber reduced by adding a file cached already by some helpers , is called channel diversity gain .  as we consider the effect of wireless fading channel such as small - scale fading , if we increase the number of helpers which store a specific file , the selection pool of channel links enlarges and ber of the file correspondingly improves due to the channel diversity .the channel diversity gain of the file is given as a function of the number of helpers that caches the file and the popularity of the file .the channel diversity gain of the popular file is written as    in this subsection , we derive the condition when even placement , =\left[1, ...,1,0, ... ,0\right] ] .similarly , we can obtain an optimality condition of single - file caching .[ proposition_3 ] for , single - file placement is optimal if and only if , where is defined as   please refer to appendix [ pf_prop_3 ] .when the zipf exponent is between and , owing to the following lemmas , we can analyze the structure of optimal placement in high snr regime .[ lemma_2 ] as , and for   please refer to appendix [ pf_lemma_2 ] .lemma [ lemma_2 ] reveals that the channel diversity gain of multiple helpers decreases much faster than both of channel diversity gain of one helper and file diversity gain , in high snr regime .hence , if one more helpers cache the file , any file which was cached by a single helper or none of helpers has larger gain than any other files cached by multiple helpers , regardless of the popularity of files in high snr regime .now , we propose doubly caching placement with , in which up to the -th popular files are stored in two helpers and the other files starting from the + 1-th popular file are stored in one helper until all helpers are occupied . in the following proposition ,the proposed doubly caching placement is shown to be optimal for a certain range of the zipf exponent .[ proposition_4 ] as , , and , doubly caching placement with is optimal if and only if and doubly caching placement with is optimal if and only if , where   please refer appendix [ pf_prop_4 ] .  with propositions [ proposition_2 ] , [ proposition_3 ] and [ proposition_4 ] ,optimal caching placement can be obtained without running the greedy algorithm by comparing the zipf exponent of popularity , , with the thresholds , , .that is , if we have information about the file preference of users , then we can determine the optimal caching placement of helpers .until this point , we focused on the case when is larger than or equal to two .the condition is necessary for file diversity gain of each file to be larger than channel diversity gain of the file from one to two helpers .the regime where implies that the average received snr from helpers is larger than that from the macro base station by 3 db .given the proximity of femto base stations , it is known that the average received snr of a femto cell user is approximately 10 db higher than that of a macro cell user , so is typically achieved .we identified the optimal caching placement in the region of .for , fortunately , we can find optimal caching placement without running the proposed algorithm in high snr for .[ proposition_5 ] as , , and , doubly caching placement with is optimal.  please refer appendix [ pf_prop_5 ] .proposition [ proposition_5 ] verifies the analytic insights ; smaller brings less file diversity gain since the gain of first caching is higher as the snr difference between bs and helpers is larger . consequently , when is less than , channel diversity gain is more preferred . combining this fact with lemma [ lemma_2 ], we can naturally conclude that caching all files in two helpers becomes optimal for in high snr .table [ tbl_summary ] summarizes the results of section [ characterization ] ; optimal caching placement depending on the range of zipf exponent and .the optimal caching placement in the region of is identified . for , the solution when snr is not largeis unknown due to analytic intractability involved in lemma 1 but optimal caching placement is presented when snr is high .[ tbl_summary ]  .optimal caching placement depending on the range of zipf exponent [ cols="<,^,^,^",options="header " , ]   we compare ber performance of optimal caching placement with other caching placement for various system environments .  . ]  in fig .[ fig_sim1 ] , the average bers of different caching placement are shown for various snr of cluster communication under the default system parameters .this figure reveals that doubly caching placement achieves almost the same ber performance as optimal caching placement in all snr region .although even caching placement is known to be optimal without consideration of wireless aspects , it is rather outperformed by doubly caching placement in all snr region .   for snr [ db ] .]  the effects of the zipf exponent is considered in fig .[ fig_sim2 ] when [ db ] . as the zipf exponent increases , the frequency of requesting the most popular file increases and thus the ber of high - ranked file dominates the average ber .consequently , single - file caching placement shows better performance than even caching placement in high , but the opposite result is observed in low . on the other hand , doubly caching placement still show comparable ber performance with optimal caching placement for all values of the zipf exponent .we proposed an optimal greedy algorithm for caching placement in wireless femto - caching network .the proposed algorithm minimizes the average bit error rate with low computational complexity , exploiting the property of optimal caching placement which we derived .we also identified and explored the tradeoff between file diversity gain and channel diversity gain to minimize the average bit error rate .we derived two special thresholds of popularity seeking an extreme of either file diversity gain or channel diversity gain , respectively , which provide a useful caching placement guideline without running the proposed algorithm .furthermore , given enough proximity and transmit power , we showed that optimal caching placement was parametrized solely with popularity .we prove this lemma according to the range of : and . for , since zero is only integer which satisfies inequality , becomes using , and become obviously , we have , for , second , for the case when , we prove the equivalent statement that for .the equivalence is established by the two facts ; ( 1 ) and , obviously implies .( 2 ) conversely , can be extended to , and using mathematical induction , we can conclude for such that .now , we prove the equivalent statement . expressing the with , can be rewritten as where is defined as .since has a positive value for all , it is enough to show that using the given pdf formula of in , becomes  \\     & = \frac{n+2}{\bar{\rho } } \exp\left(-\frac{3x}{\bar{\rho}}\right)\left(1-\exp\left(-\frac{x}{\bar{\rho}}\right)\right)^{n-1 } ,      \end{aligned}\ ] ] which is greater than zero for arbitrary and .based on , after simple manipulation , channel diversity gain can be expressed as integral of beta function . using the two properties of the beta function , and , we have , for any , where is defined as hence , the ratio between and is represented by since numerator and denominator in have order of and order of for high snr regime , respectively , the limit becomes therefore , for any such that and .similarly , using and , the ratio between and is represented by where + since numerator and denominator in have order of and order of for high snr regime , respectively , the limit becomes therefore , for any such that and .this theorem is proved by induction . first , for , we can readily show that . supposing , we will show that by contradiction .suppose is not optimal and there exists optimal caching placement for the -helper system such that following the algorithm [ greedy_algorithm ] , the average ber of is given as hence , substituting ( [ eq_ber_opt ] ) into ( [ ineq_ber_n ] ) , we obtain the following inequality now , let us consider another caching placement for the helpers system , of which elements are the same as except that one element which least increases the average ber is reduced by one . in other words , comes from in a reverse - greedy way . where is defined as .we note that the average ber of must not be less than that of ; hence , we have in addition , from the definition of , by substituting ( [ eq_ber_tilde ] ) into ( [ ineq_ber_n-1 ] ) , ( [ ineq_ber_n-1 ] ) becomes if we combine ( [ ineq_max ] ) and ( [ ineq_min ] ) , we have the following inequality : now , to complete our proof by contradiction , we prove that can not be satisfied .since and , there exists at least one file index , such that . according to lemma [ lemma_1 ] and , the following inequality holds since , is smaller than the left - hand side of , and + is greater than the right - hand side of , we have which contradicts ( [ ineq_max_min ] ) . consequently , there can not exist another optimal caching placement which is not .this completes the proof .by contradiction , we prove proposition [ proposition_1 ] .suppose the optimal caching placement for the -helper system , , does not satisfy the proposition [ proposition_1 ] .then , there exists at least one which satisfies consider another caching placement that has the same element with except that and are switched with each other as follows : then , the difference of average ber between two caching placement schemes becomes since the -th popular file is more probable to be requested than the -th popular file , the first term on the right - hand side on is positive . indicates that we have more helpers which cache the -th popular file than helpers which cache the -th popular file .consequently , the average ber of the -th popular file is lower than that of the -th popular file because more helpers provide higher channel diversity gain which reduces ber .therefore , is positive ; we have this contradicts to optimal caching placement .first , we prove the necessary condition for optimality of even caching placement .the condition can be rewritten as note that means that the file diversity gain of the -th file ( i.e. , the least popular file among possibly cached files ) is larger than the channel diversity gain of the most popular file cached in a single helper .in addition to that , by the order of popularity , both of file and channel diversity gain decreases as popularity of file becomes lower .hence , for any , .similarly , for any , .+ furthermore , by lemma 1 which states that for , we can conclude for .since is a product of popularity and , we have for and . combining this result with, we have the following inequality : for which implies that lower bound of file diversity gain is larger than upper bound of channel diversity gain . as a consequence ,file diversity gain is always larger than the maximum of channel diversity gain , which concludes optimality of even caching placement .+ for the proof of converse , we will show the contraposition of the converse is true .the contraposition of the converse is that if , even caching placement is not optimal .similarly , implies then , obviously , before the -th popular file is cached , the most popular file will be cached in two helpers . as a consequence ,even caching placement can not be optimal .we first prove the sufficient condition for optimality : if , then single - file placement is optimal .the condition can be rewritten as by lemma [ lemma_1 ] , for which implies that if , channel diversity gain of the most popular file is always larger than file diversity gain of the second - popular file until helpers cache the most popular file .thus , the maximum channel diversity gain is always larger than maximum of file diversity gain for all iterations and consequently single - file placement becomes optimal . for proving the necessary condition, we prove the contraposition : if , then single - file placement is not optimal . is equivalent to which means the second - popular file will be cached before caching the first - popular one times .hence , single - file placement can not be optimal .   to prove the proposition [ proposition_4 ], we first show the special character of optimal caching placement : as , the optimal caching placement satisfies for , where is the number of helpers that cache the -th popular file . by lemma [ lemma_2 ] ,optimal caching placement consists of . because for , from proposition [ proposition_1 ] , for then , for optimal caching placement , feasible combinations of become .first , we consider the case when . in this case , by lemma [ lemma_2 ] , for .also , when , by proposition [ proposition_1 ] , for , and for since .therefore , the total number of helpers is bounded as this bound implies existence of at least one empty helper , which contradicts the fact that the optimal placement uses up all of helpers memories . consequently , the case when can not lead optimal caching placement .second , let us consider the case when . then , by proposition [ proposition_1 ] , for , and for .using these bound , the total number of helpers is bounded below by this lower bound implies that the caching placement requires at least helpers , which contradicts the fact that we have caching helpers .consequently , the case when is not feasible . as a result ,the optimal caching placement satisfies that for .+ using the above property , optimality of the doubly caching placement can be proven as follows . for each , we will prove there exists an unique range of zipf exponent which makes doubly caching placement with optimal .from the fact that we have just shown , as , optimal caching placement has to satisfy for therefore , optimal caching placement will select a pair between and which gives a higher gain .the total gains from each pair are given , respectively , as for , for . since is a common term for both of and , whether is larger than or not determines if or for optimal caching placement .for each , if , furthermore , using for and for , inequality leads to for and which implies that is preferred to for .consequently , for each , optimal caching placement satisfies now , let us consider for each .we can rewrite as in contrast to , inequality implies that optimal caching placement satisfies for .then , from proposition [ proposition_1 ] , from and , we can conclude that doubly caching placement with is optimal if .+ however , when , inequality is equivalent to , where . obviously , when there are helpers , caching -th file in two helpers violates proposition [ proposition_1 ] . to satisfy proposition [ proposition_1 ] ,is required when . as a result , doubly caching placement with is optimal if .+ for the converse , we prove that doubly caching placement with is optimal only if and doubly caching placement with is optimal only if . in order to prove the converse, we prove their contrapositions that if or , then doubly caching placement with is not optimal and if , doubly caching placement with is not optimal , respectively .+ when , is equivalently given by which means optimal caching placement satisfies rather than .consequently , optimal caching placement is to cache the -th popular file in a single helper . on the other hand , for , can be rewritten as which indicates that optimal caching placement is to cache the -th popular file in two helpers . hence , if or , optimal caching placement is to cache the -th popular file in a single helper or to cache the -th popular file in two helpers . therefore , doubly caching placement with is not optimal if or and doubly caching placement with is not optimal if .this proposition is proved by contradiction .suppose doubly caching placement with is not optimal. then , there exists another optimal caching placement .let us denote optimal caching placement as and doubly caching placement with as .since proposition [ proposition_4 ] characterizes optimal caching placement in whole range of in high snr regime , has to be doubly caching placement with for some .if we subtract the average ber of from the that of , the difference must be less than zero due to optimality , i.e. , now , we show the difference becomes larger than zero .the difference of the average ber is expressed as  is a set of files cached in but not cached in , and is a set of files cached doubly in but not doubly cached in , respectively .except for the files in and , other elements are the same in and .therefore , we can express the difference as a function of channel diversity gain and file diversity gain as ; to be cached only in , the popularity of the file has to be less than .also , the files which has lower popularity than can not be cached doubly due to proposition [ proposition_1 ] for optimal caching placement .thus , all the files in are less popular than all the files in .if we increase the popularity of the files in up to the popularity of , we can make a lower bound ; applying and to the definitions of both diversity gain and , we can get . is greater than zero due to , which contradicts the supposition that doubly caching placement with is not optimal .n. golrezaei , a. f. molich , a. g. dimakis , and g. caire , femtocaching and device - to - device collaboration : a new architecture for wireless video distribution , " _ ieee commun . mag .142 - 149 , apr . 2013 .maddah - ali and u. niesen , decentralized coded caching attains order - optimal memory - rate tradeoff , " in _ proc .ieee 51st annual allterton conference on communication , control , and computing ( allterton ) _ , oct .2013 , pp .421 - 427 .n. golrezaei , k. shanmugam , a. g. dimakis , a. f. molich , and g. caire , femtocaching : wireless video content delivery through distributed caching helpers , " in _ proc .ieee conference on computer communications ( infocom ) _ , mar .2012 , pp .1107 - 1115 .k. shanmugam , n. golrezaei , a. g. dimakis , a. f. molisch , and giuseppe caire , femtocaching : wireless content delivery through distributed caching helpers , " _ ieee trans .inf . theory _ ,59 , no . 12 , pp . 8402 - 8413 , dec .n. golrezaei , p. mansourifard , a. f. molich , and a. g. dimakis , base - station assisted device - to - device communications for high - throughput wireless video networks , " _ ieee trans .wireless commun ._ , vol . 13 , no . 7 , pp3665 - 3676 , jul . 2014 .n. golrezaei , a. g. dimakis , and a. f. molisch , wireless device - to - device communications with distributed caching , " in _ proc .ieee international symposium on information theory ( isit ) _ , july 2012 , pp . 2781 - 2785 .h. kang , k. park , k. cho , and c. kang , mobile caching policies for device - to - device ( d2d ) content delivery networking , " in _ proc .ieee conference on computer communications workshops ( infocom wkshps ) _ , apr .2014 , pp .299 - 304 .b. hong , and w. choi , optimal storage allocation for wireless cloud caching systems with a limited sum storage capacity , " _ submitted to ieee trans .wireless commun ._ , under minor revision .d. leong , a. g. dimakis and t. ho , distributed storage allocations , " _ ieee trans .inf . theory _4733 - 4752 , july 2012 .a. goldsmith , _ wireless communications _ , cambridge university press , 2005 .c . lee and y .- h .lee,cognitive radio based femtocell resource allocation , " _ international conference on information and communication technology convergence ( ictc ) _, 2010 , pp .274 - 279 .

Abstract: this paper investigates optimal caching placement for wireless femto - caching network . the average bit error rate ( ber ) is formulated as a function of caching placement under wireless fading . to minimize the average ber , we propose a greedy algorithm finding optimal caching placement with low computational complexity . exploiting the property of the optimal caching placement which we derive , the proposed algorithm can be performed over considerably reduced search space . contrary to the optimal caching placement without consideration of wireless fading aspects , we reveal that optimal caching placement can be reached by balancing a tradeoff between two different gains : file diversity gain and channel diversity gain . moreover , we also identify the conditions that the optimal placement can be found without running the proposed greedy algorithm and derive the corresponding optimal caching placement in closed form .