long - run asymptotics are a powerful tool to obtain explicit formulas in portfolio choice and derivatives pricing but their use has been mostly ad hoc in the absence of general results .this paper develops a method to derive optimal portfolios and risk premia explicitly in a general diffusion model for an investor with power utility and in the limit of a long horizon .the market has several risky assets and is potentially incomplete .investment opportunities are driven by , and partially correlated with , state variables that follow an autonomous diffusion .investment and pricing problems share a reputation for mathematical complexity .this common trait is not an accident ; the central message of duality theory is that the two problems are indeed equivalent , as state - price densities are proportional to the marginal utilities of optimal payoffs . in spite of this conceptual equivalence , portfolio choice and derivatives pricinghave followed largely different strands of literature , each of them with its own terminology .the portfolio choice literature focuses on finding the _ intertemporal hedging _ component of optimal portfolios .long - run asymptotics have appeared in this literature under different names : the _ risk - sensitive control _ approach , turnpike results and large deviations criteria are all efforts to achieve tractability by means of the long - run limit .the derivatives pricing literature strives to identify martingale measures that are optimal in the sense of the _ minimax martingale measure _ of or the _ least favorable completion _ of .power utility leads to the -optimal measure ( , ) which embeds several other martingale measures ; it reduces to the minimal measure for , to the minimal entropy measure for and to the variance - optimal measure for .the advantages of long - run asymptotics are their tractability and accuracy .long - run portfolios and risk premia are much simpler than their finite - horizon counterparts and allow explicit expressions even in cases in which the latter do not . in general ,long - run policies are identified by the quasilinear partial differential equation ( [ emainvpde ] ) which admits explicit solutions in several models of interest . in the case of a single state variable, this equation reduces to an ordinary differential equation , which is furthermore linear if the state variable has a constant correlation with excess returns .the accuracy of the long - run approach stems from the bounds ( [ cebou ] ) which estimate the duality gap at any horizon and hence the potential departure from the unknown finite - horizon optimum ._ long - run optimality _ holds ( definition [ lrdef ] ) when long - run policies are approximately optimal over long horizons .the main result of this paper gives a sufficient condition for long - run optimality in a general multidimensional diffusion .furthermore , this condition is sharp for certain models and a calibration to the parameters estimated by shows that it is satisfied for reasonable levels of risk aversion .two duality insights are central to our results .first , the usual duality between payoffs and martingale densities extends to their stochastic logarithms , which are portfolios and risk premia .second , long - run asymptotics become easier in a duality context because candidate long - run risk - premia yield an upper bound on the maximal expected utility and vice versa .this observation allows to overcome some difficulties arising in the verification theorems of the _ risk - sensitive control _ literature .an important concept arising in long - run analysis is the _ myopic probability _ ; a long - run investor with power utility under the original probability behaves like a logarithmic ( or myopic ) investor under the myopic probability .this probability plays an important role both for long - run analysis and for finite - horizon bounds and its existence is crucial for the long - run optimality result.=-1  the rest of the paper is organized as follows .section [ secmodel ] describes the model in detail , introducing notation .section [ seclongrun ] contains the main result : a general method to obtain long - run policies in closed form .it also provides sufficient conditions , adapted from , for the existence of solutions to the associated ergodic bellman equation .section [ interpret ] discusses the various implications of these results for portfolio choice and derivatives pricing , and the connections with the stochastic control and large deviations approaches .section [ secapp ] derives long - run portfolios and risk premia in two models of interest .the last one combines stochastic interest rates , drifts and volatilities , and still admits simple closed form solutions .section [ secconc ] concludes .all proofs are in the appendices .consider a financial market with a risk - free asset and  risky assets .investment opportunities ( interest rates , expected returns and covariances ) depend on state variables model their change over time , cumulative excess returns and state variables follow the diffusion where and are multivariate brownian motions .this setting provides a flexible framework that nests most diffusion models in finance , including the models of correlation risk considered by in which is a function of a state variable .the law of determines the drifts and the covariation matrices , and , where the prime sign denotes matrix transposition . by contrast ,the matrices are identified only up to orthogonal transformations .the market defined by ( [ rateeq])([correq ] ) is in general incomplete and the covariance matrix gauges the degree of incompleteness of the market , highlighting two extremes : _ complete markets _ for and _ fully incomplete markets _ for .let be an open connected set .denote by [ resp ., the class of -valued functions on with continuous ( resp . , locally -hlder continuous ) partial derivatives of order .the superscripts are dropped for or , so that is denoted by .the following assumption prescribes that the coefficients and are regular and nondegenerate .[ regmod ] , , , , and .the symmetric matrices and are positive definite for all .the region is typically of the form for some , as in the case of multivariate ornstein uhlenbeck processes , feller diffusions or a combination thereof . consider models in which is the open simplex in .to construct the processes , let endowed with the topology of uniform convergence on bounded intervals .let be the borel sigma algebra and let be the filtration generated by the coordinate process defined by for . for a second - order differential operator of the form in ( [ epoperatordef ] ) below , a solution to the martingale problem for on is a family of borel probability measures on such that , for each , ( i ) , ( ii )  and ( iii )  is a martingale for all .the next assumption ensures that and identify the law of . , with , denotes the starting points of and  .[ assmod ] there exists a unique solution to the martingale problem for \\[-8pt ] \nonumber \tilde{a } & = & \pmatrix { \sigma & \upsilon\vspace*{2pt}\cr \upsilon ' & a } , \qquad \tilde{b } = \pmatrix { \mu\cr b}.\end{aligned}\ ] ]  since for all the models considered in this paper , the measure in assumption [ assmod ] is simply denoted as .henceforth , consider the filtration defined as , that is , the right continuous envelope of . under assumption [ regmod ] for , is a martingale also with respect to ; cf . , section 5.4.  for consistency of notation , assumption [ assmod ] involves the joint diffusion process .however , it is essentially an assumption on the process . indeed , if assumption [ regmod ] holds , and if there is a unique solution to the martingale problem for the operator then assumption [ assmod ] holds . to see this fact , first consider the martingale problem for the operator associated to , where is an -dimensional standard brownian motion independent of , write where is a square root of and define the integrals for in ( [ reteq ] ) accordingly .   an investor trades in the market according to a portfolio , representing the proportions of wealth in each risky asset . since the investor observes the state variables and the asset returns  , the portfolio is adapted to the filtration generated by , and is -integrable .- integrability means each of the integrals is well defined . ]the corresponding wealth process follows : note that a positive initial capital implies a positive wealth at all times , that is , a.s . for all , thereby ruling out doubling strategies [ see , e.g. , ] .the investor s preferences are modeled by the power utility function denoting by the expectation with respect to , the goal is to maximize expected utility from terminal wealth . with a finite horizon ,the problem is .\ ] ] since power utility is homothetic [ , it suffices to consider the case of unit initial wealth .henceforth , denotes the conjugate exponent to , to preserve economic intuition , recall that and , where is the investor s relative risk aversion .risk aversion increases as increases and as  decreases , and logarithmic utility corresponds to the limit .the martingale approach to utility maximization relies on the duality between final payoffs and pricing rules , represented by the related concepts of stochastic discount factors and martingale measures .a stochastic discount factor is a strictly positive adapted process , such that is a martingale =m_s s^i_s \qquad\mbox{for all } 0\le s\le t , 0\le i\le n.\ ] ] a martingale measure is a probability , such that and are equivalent for all and the discounted prices  equivalently , the excess returns are -martingales for all .martingale measures and stochastic discount factors are in a one - to - one correspondence through the relation , although their distinction is important in the present context of stochastic interest rates . except for a complete marketin which the martingale measure is unique , the description of a pricing rule requires the choice of unhedgeable risk premia . for any adapted , integrable process , a candidate ( local ) martingale measure given by , where and where .clearly , must be a martingale for  to be an equivalent local martingale measure .for such a , let denote the corresponding stochastic discount factor note that for any strategy and risk premia , by ( [ estrategywealthdynamics ] ) , the process is a super - martingale , even if the right - hand side of ( [ epredictrepeqn ] ) is only a local martingale . for power utility ,the following lemma applied to and for any shows that the duality bound is an immediate consequence of hlder s inequality and of the super - martingale property of the process  .[ lholderdualitybound ] let be random variables on a probability space such that -almost surely and }\leq1 ] and , for some ,   equation ( [ dualbound ] ) bounds the utility of any terminal wealth by a moment of any stochastic discount factor and vice versa .the first - order condition ( [ firstor ] ) is the usual alignment of marginal utilities with state - price densities .consider a finite horizon .lemma [ lholderdualitybound ] implies that a pair of a portfolio  and risk premia such that and is optimal if it satisfies ( [ firstor ] ) and =1 ] is approximately equal to a time - homogeneous function for most of the interval , departing from it as the horizon approaches .the question is whether using a time - homogenous policy throughout the interval  ] . however , by the bound ( [ dualbound ] ) the finite - horizon portfolio  satisfies \le \frac1pe_p^y[(m^{{\eta}}_t)^q]^{1-p}e^{-p\varphi_t t}.\ ] ] hence , is preferred to when ^{1-p}-\frac1t\log e_p^y[(x^{{\pi}}_t)^p ] } \biggr).\ ] ] for a long - run optimal pair , the limit of the right - hand side is zero .thus , for any minimal difference in fees , a long - run optimal portfolio is preferable for investors with sufficiently long horizons .the construction of long - run optimal portfolios and risk premia takes place in two steps . in the first step , theorem [ tpdemodel ] computes the finite - horizon performance of the long - run optimal `` candidates ''  and . in the second step ,theorem [ tlongrunopt ] establishes a sufficient condition for long - run optimality , requiring that the bounds found in the first step converge at long horizons .the candidate long - run optimal and crucially depend upon the solution of the quasi - linear partial differential equation ( pde ) in ( [ emainvpde ] ) , which acts as a long - run version of the hamilton jacobi bellman equation .thus , theorems [ tpdemodel ] and [ tlongrunopt ] are akin to verification theorems of stochastic control theory , but for the asymptotic objective in definition [ lrdef ] .an advantage of these results is that they only rely on the local properties of the processes  , avoiding the knowledge of the transition density of , which may be very complicated if known at all .the second part of this section studies the existence of solutions to the ergodic bellman equation in ( [ emainvpde ] ) .theorems [ tlambdastructure ] , [ tbellmansolution ] and proposition [ ptestfortightness ] below adapt the results of to the present setting and under some extra conditions in addition to the assumptions of theorems [ tpdemodel ] and [ tlongrunopt ] .their main message is that the quasi - linear pde generally admits only one candidate for long - run optimality . as shown with examples in section [ secapp ] , this candidate may or may not be long - run optimal .recall that , although their dependence on is omitted to alleviate notation , , , , , , and are functions of the state variable  .[ tpdemodel ] in addition to assumptions [ regmod ] and [ assmod ] , assume that :  [ itdiffexpression ] and solve the ergodic hjb equation ( cf .section [ sec3.2 ] ) \\[-8pt ] \nonumber & & \qquad { } + \nabla{v}'(b - q{\upsilon}'\sigma^{-1}\mu ) + \frac{1}{2}{\operatorname{tr}}(ad^2v)= \lambda;\end{aligned}\ ] ]  [ itwellposed ] there is a unique solution to the martingale problem for \\[-8pt ] \nonumber \hat{b } & = & \pmatrix { \displaystyle\frac{1}{1-p}(\mu+\upsilon\nabla v ) \vspace*{2pt}\cr b - q\upsilon'\sigma^{-1}\mu+ ( a - q\upsilon'\sigma^{-1}\upsilon)\nabla v } , \end{aligned}\ ] ] where is as in ( [ epoperatordef ] ) .  then , the pair given by satisfies the equalities & = & e^{\lambda t+v(y)}e_{\hat p}^y\bigl[e^{-v(y_t)}\bigr],\\ \label{duaest } e_p^y[(m^{{\eta}}_t)^q]^{1-p } & = & e^{\lambda t+v(y)}e_{\hat p}^y\bigl[e^{-({1}/{(1-p)})v(y_t)}\bigr]^{1-p}.\end{aligned}\ ] ]  equations ( [ priest ] ) and ( [ duaest ] ) provide lower and upper bounds on finite - horizon expected utility .indeed , the duality inequality ( [ dualbound ] ) yields &= & \frac1p e_p^y[(x^{{\pi}}_t)^p ] \le u_t(y)\le \frac1p e_p^y[(m^{{\eta}}_t)^q]^{1-p}\\ & = & \frac1p e^{\lambda t+v(y)}e_{\hat p}^y\bigl[e^{-({1}/{(1-p)})v(y_t)}\bigr]^{1-p}.\end{aligned}\ ] ] combining ( [ priest ] ) and ( [ duaest ] ) with ( [ celbou ] ) yields the central quantitative implication : an upper bound on the certainty equivalent loss ^{1-p}}- \frac1t\log{e_{\hat p}^y\bigl[e^{-v(y_t)}\bigr ] } \biggr).\ ] ]  theorem [ tpdemodel ] now reduces the long - run optimality ( definition [ lrdef ] ) of to the condition that the right - hand side in ( [ cebou ] ) converges to zero . theorem [ tlongrunopt ] below provides a criterion that covers most applications and proposition [ pfadsisok ] below shows a model in which this criterion is sharp , in that it holds for all the parameter values for which long - run optimality holds .[ tlongrunopt ] if , in addition to the assumptions of theorem [ tpdemodel ] :  the random variables are -tight - valued random variables is _ p - tight in _ if the induced measures form a tight family in , the space of borel measures on .thus , is -tight in if for each there exists a compact such that . ] in for each ;   , where is defined as   then the pair in ( [ lrport ] ) is long - run optimal .section [ secapp ] shows how to check conditions ( i ) and ( ii ) in typical classes of models .a sufficient condition for ( i ) above to hold is that there exists a nonnegative such that , for each , the level set is compact and that < \infty\mbox{for } \displaystyle\frac12 < q < 1,\mbox{for } -\!1 \leq q\le \displaystyle\frac12,q\neq0,\mbox{for } q<-1 ] ;  if and , the certainty equivalent loss is bounded ;  if and , the certainty equivalent loss diverges to .a calibration to real data shows that long - run optimality holds for typical levels of risk aversion , in the model with one asset and one state considered by and .the state variable represents the dividend yield and the asset is an equity index . in the notation of this section , they use the set of parameter values ( in monthly units ) , , , , , . then , condition ( [ efadstypeoucond ] ) is satisfied for , that is , for risk - aversion less than .  )setting equal to the long - run optimal risk premium , and equal to the long - run optimal portfolio ( solid line ) and to the myopic portfolio ( dashed line ) .risk aversion is equal to two ( , top ) and to five ( , bottom ) . ]figure [ longmyopic ] compares the finite - horizon performance of the long - run optimal portfolio to the one of its myopic component .the plots show the estimates of the correponding upper bounds in ( [ celbou ] ) : the myopic component prevails in the short run but its performance progressively deteriorates as the horizon increases. the break - even horizon significantly increases with risk - aversion , passing from nine years for a risk - aversion of two , to twenty - three years for a risk - aversion of five .also , the magnitude of the certainty equivalent loss increases with risk aversion : the differences are within one percentage point for a risk - aversion of two but increase to three precentage points for a risk - aversion of five .the next model features a single state variable following the square - root diffusion of , which simultaneously affects the interest rate [ ] , the volatilities of risky assets and their drifts .note that the model is neither affine nor quadratic ( due to the presence of the term with  ) and yet the long - run solution admits a simple expression where ; ; , and .the parameters satisfy the following .[ assmodcir ] and .assumption [ assmodcir ] ensures that the state variable remains strictly positive , thereby satisfying assumption [ assmod ] with .guessing a form of the solution , the main ode ( [ emainvpde ] ) becomes multiplying the above equation by and setting the constant , linear and quadratic terms to zero leads to four candidate solutions , corresponding to any combination of signs in the terms and below : with when , \\[-8pt ] \nonumber \lambda & > & ( c - qa\rho'\nu_0)^2>0.\end{aligned}\ ] ] under , has again four possible dynamics : the choice of and ensures that satisfies assumption [ assmod ] under and is -tight in for each .the latter statement follows by the positivity of [ see , corollary 5.1.11 , and the discussion immediately after assumption [ boundedpotential ] ] . thus the candidate optimizer is the candidate long - run optimal policies are and the candidate long - run martingale measure is long - run optimality obtains under the following conditions .[ lcirisok ] let assumption [ assmodcir ] hold . for , long - run optimality holds if \\[-8pt ] \nonumber ( 1 - 2q\rho'\rho)\sqrt{\theta } + ( b+qa\rho'\nu_1 ) & > & 0.\end{aligned}\ ] ]  the main economic message of this parametric restriction is the same as in the previous example .long - run optimality holds if either one of the following conditions is satisfied : the covariation of risk premia with state shocks is small ( ) , the market is sufficiently incomplete ( ) or risk aversion is low ( ) .long - run analysis is a tractable and yet nontrivial framework for dynamic portfolio choice and derivatives pricing in incomplete markets , and yields simple expressions for portfolios and risk premia .long - run policies admit closed - form solutions even in cases in which finite - horizon policies do not , and the finite - horizon performance of long - run policies has a simple expression .long - run optimality entails that the certainty equivalent loss vanishes for long horizons and requires some joint restrictions on preferences and asset dynamics .it does not hold at the intersection of three extreme cases : risk premia highly co - varying with state shocks , a nearly complete market and high risk aversion .otherwise , long - run optimality holds and time - homogeneous portfolios are approximately optimal for long horizons .proof of lemma [ lholderdualitybound ] denote by . in the case , hlder s inequality with and yields } & = & { e_{p}[(xm)^pm^{-p}]}\leq { e_{p}[(xm)^{p\tilde{p}}]}^{1/\tilde{p}}{e_{p}[m^{-p\tilde { q}}]}^{1/\tilde{q}}\\ & = & { e_{p}[xm]}^{1/\tilde{p}}{e_{p}[m^q]}^{1-p}\leq { e_p[m^q]}^{1-p},\end{aligned}\ ] ] because }\leq1 ] and is proportional to .proof of theorem [ tpdemodel ] since the brownian motions and are partially correlated , the following orthogonal decomposition holds ( see the discussion following assumption [ assmod ] ) : where is a -dimensional brownian motion independent of and the matrix is defined by the identity . for , define the process by \\[-8pt ] \nonumber & & \hspace*{72pt}\quad{}- \int_0^{\cdot}q(\sigma^{-1}\mu+ \sigma^{-1}{\upsilon}\nabla v ) ' \sigma\bar{\rho } \,db_t\biggr)_t.\end{aligned}\ ] ] by assumption , the operators associated to the models and satisfy assumption [ assmod ] .since assumption [ regmod ] also holds , [ ( ) , theorem 2.4 , remark 2.4.2 ] implies that for each and ,  and are equivalent on with thus , is a -martingale . with this notation , it suffices to prove that given the solution pair to ( [ emainvpde ] ) , with as in ( [ lrport ] ) , the following  almost sure identities hold : indeed , if ( [ priestas ] ) and ( [ duaestas ] ) hold , then ( [ priest ] ) and ( [ duaest ] ) follow by taking expectations with respect to . consider first ( [ priestas ] ) . passing to logarithms, it suffices to prove that the first term on the left - hand side of ( [ epritemp1 ] ) is substituting , the decomposition and collecting terms the second term in the left - hand side of ( [ epritemp1 ] ) follows from ( [ ephatprnderiv ] ) , & & \hspace*{8pt}\qquad{}- \frac{1}{2}\nabla v'\bigl(a+(q^2 - 2q)\upsilon'\sigma^{-1}\upsilon\bigr)\nabla v \biggr)\,dt \nonumber \\[-10pt ] \\[-10pt ] \nonumber & & { } + \int_0^t\bigl(\nabla v'a - q(\mu+ \upsilon\nabla v)'\sigma^{-1}\sigma\rho\bigr)\,dw_t\\[-2pt ] & & { } - q\int_0^t(\mu+ \upsilon\nabla v)\sigma^{-1}\sigma\bar{\rho}\,db_t.\nonumber\end{aligned}\ ] ] subtracting ( [ epritemp4 ] ) from ( [ epritemp2 ] ) yields & & \qquad= \int_0^t\biggl(pr - \frac{1}{2}q\mu'\sigma^{-1}\mu - q\mu'\sigma^{-1}\upsilon\nabla v + \frac{1}{2}\nabla v'(a - q\upsilon'\sigma^{-1}\upsilon)\nabla v\biggr)\,dt \\[-2pt ] & & \qquad\quad{}-\int_0^t \nabla v'a \,dw_t.\end{aligned}\ ] ] now , it s formula allows us to substitute and the claim ( [ epritemp1 ] ) follows by recalling that solves ( [ emainvpde ] ) .consider now the equality ( [ duaestas ] ) .again , by taking logarithms it suffices to show that the first term in the left - hand side is equal to ( plugging and ) & & \hspace*{174pt } { } + \int_0^{\cdot}{\eta}'a\,dw_t\biggr)_t\\[-2pt ] & = & -q\int_0^t\biggl(r + \frac{1}{2}\mu'\sigma^{-1}\mu+\frac { 1}{2}\nabla v'(a-\upsilon'\sigma^{-1}\upsilon)\nabla v\biggr)\,dt\\[-2pt ] & & { } + q\int_0^t\bigl(\nabla v'a -(\mu+ \upsilon\nabla v)'\sigma^{-1}\sigma\rho\bigr ) \,dw_t\\[-2pt ] & & { } - q\int_0^t(\mu+ \upsilon\nabla v)\sigma^{-1}\sigma\bar{\rho}\,db_t\end{aligned}\ ] ] and subtracting ( [ epritemp4 ] ) yields as in the previous case , ( [ eduatemp1 ] ) now follows by substituting ( [ eitov ] ) and recalling that solves ( [ emainvpde ] ) .the proof of theorem [ tlongrunopt ] requires two lemmas .[ ltightnesslemma ] let , and let be a tight family of probability measures on . then   by monotonicity , it suffices to prove the result for bounded .let be an increasing sequence satisfying and since is bounded above , the limit in ( [ etempeqn ] ) is nonpositive .since the measures are tight , they are relatively compact with respect to the topology of weak convergence .thus , up to a subsequence , there exists a probability measure  on  such that because is continuous , bounded and positive .thus , for any , there is a such that implies hence , taking it follows that the limit in ( [ etempeqn ] ) is indeed .  [ lgeneratorupperbound ]let be a solution to the martingale problem for the operator on , where if , , then \leq f(y ) + ( 0\vee\sup_e lf)t ] , let . by construction there exist , satisfying and in .set . by ( [ emlbounds ] ), it follows that on  similarly , setting and using ( [ emlbounds ] ) it follows that on  therefore , ( a ) holds by first applying lemma [ llupde ] and then lemma [ lksthm26lemmas](i ) for .furthermore , since can be taken arbitrarily large , ( b ) holds as well . regarding ( c ) , let and let be the associated function solving and let . by ( [ emlbounds ] ) where the last equality follows by , theorem 4.4.5 .thus .regarding theorem , note that by ( [ eahatarel ] ) , remark holds for , and the argument in lemma carries over exactly , up to obvious changes in notation .  before proving theorem [ tbellmansolution ] ,the following definitions and results are needed from the theory of large deviations for occupancy times of diffusions .let assumptions [ regmod ] , [ regmoddomain ] and [ boundedpotential ] hold . to make the dependence upon specific , for let be the measure in theorem [ tpdemodel ] and let be the operator associated to .let denote the space of borel probability measures on .define the function by where   it is clear that is nonnegative ( ) and lower semi - continuous with respect to the weak topology on ( is bounded ) .set and .denote by the occupation measure for on , which satisfies for compact , it follows that for all [ , section 7 ] , furthermore , the following facts hold .[ lifunctprop ] let assumptions [ regmod ] , [ regmoddomain ] and [ boundedpotential ] hold . if there exists a such that , then for all , and possesses a density , such that , where is the formal adjoint to .  that for all follows by repeating the argument in lemma up to , and through the point where for a.e . . by [ ( ) , theorem 1.15.1 ] , the same conclusion can be extended to all .  as for the second statement , the argument in lemma can be repeated up to and including the point where it is shown that implies for all .thus the conclusions follow from , page 181 .in particular , .proof of theorem [ tbellmansolution ] as in the proof of theorem [ tlambdastructure ] , the proof of theorem [ tbellmansolution ] adapts the results in to the present setting , now extending theorem and theorem .theorem yields that for , is -positive recurrent , while theorem states that the solution corresponding to is unique up to an additive constant .theorem carries over with only notational changes , in the light of ( [ eahatarel ] ) and lemma [ lksthm26lemmas ] .theorem is composed of three parts . the first part ( proposition ) states that if is transient under , then there exists such that for all , , and large enough \leq c(y)e^{-\alpha t}.\ ] ] the second part uses the first part to show that is recurrent under .the third part states that is actually positive recurrent .the argument used to prove the second part ( recurrence ) carries over unchanged , hence , details are provided on the first and third parts .  as for the first part ( proposition ) assume that is transient under . consider the set . to construct the constants , set for the function from assumption [ generalcase ] .note that by ( [ ewdropoff ] ) , let be as in ( [ eahatarel ] ) .set and .let and set since , it follows that and hence , the set is relatively compact ( weak topology ) for large enough , so that and .set where is such that . then , repeating the arguments in proposition , it follows that for any [ equation , \leq\sup_{e}|f|\ \hat { p}^{\lambda , y}[\mu_t\in\mathcal{c}_m , t < \tau ] + \sup_{e}|f/\bar{\phi}|\ \bar{\phi}(y)e^{-m t}.\ ] ] by ( [ edvub ] ) it follows , after taking that \leq -\inf_{\mu\in\bar{\mathcal{c}}_m } i^{\lambda}(\mu).\ ] ] now , since is lower semi - continuous and is compact , for some .it is now shown that .suppose , by contradiction , that and that is transient under .since , lemma [ lifunctprop ] implies that :   = 1 ] for all .this conclusion contradicts ( a ) above .thus , and , in view of ( [ ekstranscondlimit ] ) , the inequality in ( [ ekstranscond ] ) holds for any and large enough .  to show the third part [ positive recurrence for under ] , the same steps as in part one can be repeated to obtain ( [ ekstranscondlimit ] ) , as these steps do not require that is transient .now , if , then for all , ( [ ekstranscond ] ) implies that \,dt < \infty,\ ] ] in which case is transient under [ , chapter 4.2 ] .but part two implies that is recurrent .thus , there is a such that .since is a probability measure , ergodicity follows by lemma [ lifunctprop ] .proof of proposition [ ponestatenice ] the invariant density for under is where and is from ( [ emnudef ] ) .equations ( [ priest ] ) and ( [ duaest ] ) from theorem [ tpdemodel ] become & = & e^{\lambda t+v(y)}e_{\hat p}^y[\phi _c(y_t)^{-\delta}],\\ e_p^y[(m^{{\eta}}_t)^q]^{1-p } & = & e^{\lambda t+v(y)}e_{\hat p}^y\bigl[\phi_c(y_t)^{-{\delta}/{(1-p)}}\bigr]^{1-p}.\end{aligned}\ ] ] since , , it follows that provided that and .these conditions are equivalent to those in ( [ eqrhocondnice ] ) .thus , by the ergodic result ( [ eergodicassumptions ] ) it holds that & = & \int_e\phi_c^{-\delta}\phi_c^2m_{\nu}\,dy \equiv k_1,\\ \lim_{t\uparrow\infty}e_{\hat p}^y\bigl[\phi_c(y_t)^{-{\delta } /{(1-p)}}\bigr]^{1-p } & = & \int_e\phi_c^{-\delta/(1-p)}\phi_c^2m_{\nu}\,dy\equiv k_2\end{aligned}\ ] ] and long - run optimality follows .furthermore , in light of ( [ celbou ] ) , the conclusion in ( [ ecledecaynice ] ) follows with .proof of theorem [ teolin ] if and , symmetric solve ( [ vzeq ] ) and ( [ vueq ] ) , respectively , then solves ( [ emainvpde ] ) for from ( [ vleq ] ) . under assumption [ assmodlindif ] , the condition ( [ ewdropoffalt ] ) holds , hence ,theorems [ tlambdastructure ] and [ tbellmansolution ] imply that , if for the associated process is -tight in  for each , then is the desired solution .the riccati equation ( [ vueq ] ) admits the form with where is assumed to be the unique symmetric positive definite square root of . is a real valued matrix when . for a real valued square matrix , write if is strictly positive definite . if , then the real part of each of its eigenvalues is strictly positive . to see this ,let such that .then ( with denoting the complex conjugate of ) .\ ] ] from [ ( ) , lemma 2.4.1 ] , if there exist two matrices and such that and , then there exists a unique solution such that .since and , the choice of and suffices .the condition yields therefore , has eigenvalues with strictly positive real part and is invertible .thus from ( [ vzeq ] ) is well defined .it remains to prove that is -tight in . under , has the dynamics setting the dynamics takes the form where is such that and . for follows that since is -tight in , which follows because any compact set is contained in a closed ball around the origin , and by ( [ etempyzcalc ] ) if  is in a closed ball around , then so is . using the methods derived in it follows that under , is positive recurrent . to show this ,let and denote the maximum and minimum eigenvalues of and let denote the minimum eigenvalue of . since and by assumption [ assmodlindif ] , it follows that and .furthermore , thus , with the notation of [ ( ) , theorem 6.2 ] , and where the last inequality follows for large enough so that the numerator is negative . therefore , [ ( ) , theorem 6.2 ] applies and is positive recurrent .proof of proposition [ pgeneralou ] in light of theorems [ teolin ] and [ tlongrunopt ] , it suffices to show that for the following quantity is bounded as a function of : \\[-8pt ] \nonumber & & \hspace*{53pt}\qquad{}+ \tfrac{1}{2}q\rho'\rho(v_0-v_1y)^2\bigr)e^{-v_0 y + ( { 1}/{2})v_1 y^2}.\end{aligned}\ ] ] note that , for , in ( [ ethetaval ] ) satisfies therefore , and ( [ eboundquant ] ) is bounded over only if the quadratic term is negative but from ( [ egeneraloucond ] ) , the quantity is negative , while is positive by ( [ eouthetabound ] ) .therefore , the leading quadratic term is negative and the result follows .proof of corollary [ cfadstypeou ] when condition ( [ egeneraloucond ] ) reduces to set and consider the continuous function on . for a fixed consider the implicit equation for  obtained by setting . for can show there are no such that . for , only along the curve . for a fixed and large positive  and for large negative  therefore , plugging back in for , for the restriction is necessary .proof of proposition [ pfadsisok ] when , by corollary [ cfadstypeou ] , long - run optimality holds for .for long - run optimality holds if or .consider now , which is equivalent to since .when the solution and simplify considerably to ; ; and . under , has the dynamics   for , it follows that with , and . therefore , = e[\exp(\mathbf{a}y_t^2 + \mathbf{b}y_t)],\ ] ] where , . for , \nonumber\hspace*{-35pt } \\[-8pt ] \\[-8pt ] \nonumber & & \qquad= \cases { ( 1 - 2\mathbf{a}\sigma^2)^{-1/2}\exp\bigl((1 - 2\mathbf{a}\sigma ^2)^{-1}(\mu^2\mathbf{a } + \mu\mathbf{b } + \frac{1}{2}\sigma^2 \mathbf{b}^2)\bigr),\vspace*{2pt}\cr  \hspace*{39pt}\mathbf { a } < \displaystyle\frac{1}{2\sigma^2},\vspace*{5pt}\cr \infty,\qquad \mathbf{a } \geq\displaystyle\frac{1}{2\sigma^2}.}\hspace*{-35pt } \ ] ]  therefore , <\infty ] if which is true for all because only when .thus , ^{1-p } \\ & & \qquad= \bigl(1-(1-q)(1-e^{-b t})\bigr)^{-{(1-p)}/{2}}\\ & & \qquad\quad{}\times\exp\biggl((1-p)\biggl(\frac{1}{2}b(1-q)y^2e^{-b t } - 4q(1-q)\rho ' \nu_0 e^{-{b}/{2}t}y \\ & & \hspace*{150pt}{}+ \frac{8}{b}q^2(1-q)^2(\rho'\nu _0)^2(1-e^{-b t})\biggr)\\ & & \hspace*{2pt}\qquad\qquad\qquad{}/{\bigl(1-(1-q)(1-e^{-b t})\bigr)}\biggr).\end{aligned}\ ] ] if , \sim ke^{\frac{b}{2}t} ] for large  , so the certainty equivalent loss is bounded by , proving ( ii ) in proposition [ pfadsisok ] .if , then \sim e^{k_1e^{b t}} ] for large , where are positive constants . in this case , the certainty equivalent loss diverges to with speed of the order of .this proves ( iii ) and the proof is complete .proof of proposition [ lcirisok ] since satisfy ( [ eciroptans ] ) , solves ( [ emainvpde ] ) . under dynamics of are and , as mentioned in section [ scir ] , the positivity of give that is -tight in for each .  therefore , long - run optimality will follow if the quantity for from theorem [ tlongrunopt ] is bounded over .specifying to this example , it is necessary to show that \\[-9pt ] \nonumber & & \hspace*{26pt}\qquad{}+ \frac{1}{2}q\biggl(\frac{v_0}{y}+v_1\biggr)a^2\rho'\rho y\biggl(\frac{v_0}{y } + v_1\biggr)\biggr)e^{-(v_0\log y + v_1 y)}\end{aligned}\ ] ] is bounded on .this expression admits the form   for from ( [ eciroptans ] ) , by ( [ ecompare ] ) it follows that and so ( [ ecirbound ] ) will follow only if . as for , & = & \frac{1}{2}qa^2\rho'\rho\frac{\delta^2}{a^4}\biggl(\sqrt{\lambda}- \biggl(b\theta - qa\rho'\nu_0-\frac{1}{2}a^2\biggr)\biggr)^2 \\[-2pt ] & & { } -\frac{1}{2}\frac{\delta}{a^2}\biggl(\lambda-\biggl(b\theta - qa\rho'\nu _ 0-\frac{1}{2}a^2\biggr)^2\biggr)\\[-2pt ] & = & \frac{1}{2}\frac{\delta^2}{a^2}\biggl(\sqrt{\lambda}-\biggl(b\theta - qa\rho ' \nu_0-\frac{1}{2}a^2\biggr)\biggr)\\[-2pt ] & & { } \times\biggl((2q\rho'\rho -1)\sqrt{\lambda } -\biggl(b\theta - qa\rho'\nu_0-\frac{1}{2}a^2\biggr)\biggr).\end{aligned}\ ] ] from ( [ parcir ] ) thus , since by ( [ ecompare ] ) as for , from ( [ parcir ] ) thus , , since by ( [ ecompare ] ) , paper benefited from the helpful comments of seminar participants at cornell university , hitotsubashi university , university of michigan , princeton university , university of texas at austin , the ams meeting in san diego , the oberwolfach workshop on stochastic analysis in finance and the sixth seminar on stochastic analysis at ascona .we are indebted to an anonymous referee who helped simplify the proof of the main result and improve its presentation .

Abstract: this paper develops a method to derive optimal portfolios and risk premia explicitly in a general diffusion model for an investor with power utility and a long horizon . the market has several risky assets and is incomplete . investment opportunities are driven by , and partially correlated with , state variables which follow an autonomous diffusion . the framework nests models of stochastic interest rates , return predictability , stochastic volatility and correlation risk .  in models with several assets and a single state variable , long - run portfolios and risk premia admit explicit formulas up the solution of an ordinary differential equation which characterizes the principal eigenvalue of an elliptic operator . multiple state variables lead to a quasilinear partial differential equation which is solvable for many models of interest .   the paper derives the long - run optimal portfolio and the long - run optimal pricing measures depending on relative risk aversion , as well as their finite - horizon performance .  .


the term _ oscillator _ is typically used to denote any physical system which , operating on its own ( independent of neighboring oscillators ) , exhibits limit cycle behavior .when such oscillators are coupled to each other , they can spontaneously synchronize with each other . a simple , yet truly powerful model describing synchronization in oscillator assemblies is the kuramoto model , which has been successfully used in many biological , chemical , physical and social contexts . this model for coupled phase oscillators and its variationshave been widely studied in the literature . under specific conditions, it has been observed to exhibit complex behavior . while extensive work has been performed for all - to - all coupled oscillators , real - world assemblies of oscillatorsare seldom globally connected to one other . spiking neurons , for instance , are connected by a complex network structure ; synchronization of such neuronal systems has been modeled using the kuramoto model modified to account for the network topology .kuramoto oscillators with structured underlying network topologies are increasingly being investigated in the literature ( e.g. ) .we consider a generic system of _ non - identical _ phase oscillators connected by a network structure , and explore the computer - assisted reduction of the system dynamics .coarse - graining is feasible when there is an inherent _ separation of time scales _ in the system , i.e. , when constituent processes of the system dynamics occur at very different rates .networks with spectral gaps ( big jumps in the eigenspectrum of their graph laplacian ) can endow the coupled oscillator dynamics with this kind of time scale separation .our illustrative example is a simple network structure containing a spectral gap after a relatively small number of eigenvalues ( sorted in ascending order ) of the graph laplacian .the small number of leading eigenvalues before the spectral gap endows the system with _ low - dimensional _ long - term dynamics .the eigenvectors associated with these eigenvalues ( corresponding to slow modes " ) are used to define the coarse variables useful in model reduction .such coarse variables take into account the network structure , but do not account for the fact that the oscillators in the network are _ non - identical _ in terms of their angular frequencies .we will discuss how this additional heterogeneity ( intrinsic to the oscillators , as opposed to the heterogeneity associated with their coupling connections in the network ) can also be accounted for in the selection of a set of coarse variables .once appropriate coarse observables are identified , one typically obtains a reduced set of equations ( approximately ) describing the evolution of these observables . in this paperwe will circumvent this step using the so - called _ equation - free _framework ; in this approach , short bursts of detailed system simulation are used to estimate the coarse time - derivatives ( actions of coarse jacobians etc . ) required to compute solutions with the coarse variables .the use of this approach is illustrated in more detail in the appendix .the remainder of this paper is structured as follows : section [ sec : dyn ] describes our illustrative example and outlines its relevant dynamic behavior .section [ sec : cg ] discusses possible approaches to coarse - graining the system dynamics , focusing on the selection of appropriate coarse variables ( observables ) . a first round of results of our coarse - grained computations is presented in section [ sec : res ] ; a quick review of the the equation - free framework employed for these computations is relegated to the appendix .sections [ sec : het ] and [ sec : hetc ] focus on the heterogeneity of the intrinsic oscillator frequencies , its effect on their states , and present an approach to account for these effects in coarse - graining .section [ sec : conc ] concludes with a summary and discussion of possible generalizations of the methods .our illustrative example is a network of oscillators with a single state variable ( _ phase _ ) associated with each oscillator .these phases evolve based on the kuramoto equations , taking into account the particular connectivity structure :   a sample graph with a spectral gap , , created using the procedure described in the text .( the image was created using the graphlayout package for matlab written by matthew dunham , university of british columbia . ) , scaledwidth=40.0% ]  here , is the total number of oscillators in the system , and are the phases and the intrinsic angular frequencies of the individual oscillators , and is the coupling strength , measuring the influence of every oscillator on the oscillators connected to it .the matrix , the adjacency matrix defining the network structure connecting the oscillators , is defined as follows : if oscillators and communicate with each other and otherwise .the first 100 eigenvalues of the graph laplacian corresponding to .,scaledwidth=50.0% ]     the evolution of the kuramoto order parameter over time at a coupling strength of .the results from direct simulation are the solid lines ( blue ) while the results from coarse projective integration ( time steps for simulation and for projection ) are the dots ( red ) .( the thickness of the plotted dots make the projection step appear shorter ) .the phase portrait in terms of the real and imaginary components of the first coarse variable ( see eq . [ eq : coarse2 ] ) is shown in the inset.,scaledwidth=50.0% ]   the evolution of the kuramoto order parameter over time at a coupling strength of .the results from direct simulation are the solid lines ( blue ) while the results from coarse projective integration ( time steps for simulation and for projection ) are the dots ( red ) .( the thickness of the plotted dots make the projection step appear shorter ) . the phase portrait in terms of the real and imaginary components of the first coarse variable ( see eq .[ eq : coarse2 ] ) is shown in the inset.,scaledwidth=50.0% ]  for our simulations , we use oscillators with their intrinsic frequencies sampled from a gaussian distribution with mean and standard deviation ; we will discuss the possibility of different distributions in section [ sec : conc ] . for the underlying connectivity we built a network with a spectral gap , which was observed to lead to low - dimensionality in the long - term system dynamics ; this provides the motivation for coarse - graining .the target graph for our illustration was created from a collection of subgraphs ( communities ) each containing nodes ; the total number of nodes ( oscillators ) in the final network was .each subgraph was created using the watts - strogatz model , which contains 2 parameters , - the average degree of the nodes and - the probability of rewiring .the values of for the subgraphs were assigned by uniformly sampling an even number in the interval [ 14,38 ] ( corresponding to an average degree of approximately 25 - 75% of the total number of nodes in the sub - graph ) .the values of for the individual subgraphs were sampled uniformly in a log scale between 0.001 and 1 ( i.e. , the values of are sampled uniformly between -3 and 0 ) .the watts - strogatz model was chosen to create the constituent subgraphs because it creates graphs ranging from poisson degree distribution ( random ) to power law distribution ( scale - free ) depending on the parameter .once all the sub - graphs ( or communities ) are created , a node is randomly chosen from each of the communities to be its _leader_. now , all the leaders are connected to each other resulting in a complete network of leaders ( with edges ) .we thus arrive at a connected graph , with communities and nodes in total ; a sample resulting graph is shown in fig .[ fig : graph ] for the case of and . in our simulations, we use a graph created using the same procedure with and .the normalized laplacian of the graph , denoted by , is defined as : where is the degree of node .the normalized laplacian ( in this paper , the term laplacian should always be taken to mean the normalized graph laplacian ) corresponding to our graph was computed and its first few eigenvalues ( arranged in the ascending order ) are plotted in fig .[ fig : leig ] ; there is a clear gap in the spectrum after the 10th eigenvalue .we perform direct simulations of the phase oscillator model ( eq . [ eq : kura ] ) at different values of coupling strength , with the network topology and oscillator frequencies chosen as described above ; the initial phases of the oscillators are sampled from a uniform distribution between and .all our results are reported after the instantaneous average system phase has been subtracted ( i.e. , in a frame that rotates along with the average system phase ) . at sufficiently large values of coupling strength observe ( as expected ) that the oscillators spontaneously synchronize their frequencies , and their phases lock " at steady state .representative phase evolution at such a high coupling strength is shown at successive time steps in fig .[ fig : phaseev ] ; note how the community structure of the oscillators quickly becomes visually apparent in the figure . a quantitative measure of _ phase synchronization _ ( or coherence in an oscillator population ), the so - called _ order parameter _ has been defined as : its values can range between and . the higher the value of the order parameter , the higher the degree of synchronization - a value of indicates the state where all oscillators have the exact same phase . the evolution of this order parameter is shown at a coupling strength of as solid lines in fig .[ fig : order1 ] .as expected ( and confirmed computationally ) the steady state value of the order parameter decreases with coupling strength until a critical value of ; below this critical value the oscillators do not synchronize any more , and limit cycle oscillations are observed initially ( in parameter space ) , resulting from a supercritical hopf bifurcation at .the evolution of the order parameter at depicted by the solid lines in fig .[ fig : order2 ] exhibits such steady limit cycle oscillations .our objective is to develop and implement a computer - assisted coarse - grained model of our illustrative coupled oscillator system .the most important step in coarse - graining probably consists of the selection of appropriate coarse observables : a reduced set of variables in terms of which a useful _ closed _ description can be obtained .given suitable coarse variables , it is sometimes possible to derive the reduced equations analytically ( in closed form ) . if , however , the closures required to write down " the reduced equations in closed form are not known , or can not be easily approximated , the so - called equation - free framework can be used to _ computationally _ implement the reduced model , circumventing its explicit derivation .a brief description of the main points of this modeling / computation framework for complex systems can be found in the appendix .  in order to motivate our selection of coarse variables , we first identify collective features in the detailed simulation results .consider the temporal evolution of the oscillator phases as shown in fig .[ fig : phaseev ] ( for ) . in our network , the first oscillators belong to the first community , the next to the second community and so on ; we define as the set containing the indices of all the oscillators in the community : } = \{(k-1)s+1,(k-1)s+2, ...ks\}. \label{eq : comm}\ ] ] two oscillators within the same community are connected more tightly " ( they share more common neighbors ) than oscillators in different communities .we _ observe _ in the simulations that the phases of all the oscillators within a community synchronize with each other at much shorter time scales than those over which the entire network synchronizes .this suggests that , because of the _ construction _ of our network topology , its structure can help rationalize the selection of coarse variables appropriate for the _ long - term _ dynamics , after initial transients quickly die out .that this separation of time scales leads to low - dimensionality in the system state can be seen in fig .[ fig : phaseev ] : the randomly initialized individual oscillator phases ( our microscopic " or fine scale " variables , in equation - free notation ) quickly evolve to respect " the coarse community structure of the network . as a result ,the following possibilities for coarse variables suggest themselves :   an obvious choice for a set of coarse variables , for our example is to use a single common ( time - varying ) phase angle for each community .the _ restriction operation _( fine states to coarse states in equation - free language ) is then defined by assigning the average phase , , of all the oscillators in the community as the single common phase of that community . the _ lifting operation _ ( coarse states to consistent fine ones ) is implemented by assigning this common phase as the phase angle of all the oscillators in that community . } \}. \label{eq : coarse1}\end{aligned}\ ] ]  this apparently intuitive coarse variable selection suffers from two drawbacks .firstly , partitioning the oscillators into different communities ( clustering " ) is nontrivial for a general network structure ( even though -due to the particular construction- it was easy for our example ) .even when community structures can be identified , however , this set of coarse variables does not take into account the differences _ between _ the different communities and the structure _ within _ the communities .this suggests an alternative set of coarse variables that uses the graph laplacian of the network .the components of a few eigenvectors of the laplacian of the graph are plotted in arbitrary units .the plot ( blue line ) at the bottom corresponds to a vector that is a linear combination of the first 10 eigenvectors ; it is clearly localized to oscillators in the fifth community only .the ten linear combination coefficients are [ 2.066 , 0.259 , 0.004 , 0.273 , -0.323 , 0.222 , -1.151 , 0.017 , 4.385 , -4.982].,scaledwidth=50.0% ]   consider the normalized laplacian matrix , defined in eq .[ eq : lap ] , for the graph .let be the eigenvalue and the corresponding normalized eigenvector of the graph laplacian . from fig .[ fig : leig ] , it can be seen that the first 10 eigenvalues are well separated from the rest ( in other words , a spectral gap exists ) .the components of the eigenvectors of the graph laplacian corresponding to the ten smallest eigenvalues are plotted as connected dotted lines in fig .[ fig : evect ] .these eigenvectors embody , in an alternative way , the coarse community structure of the entire network .linear combinations of these eigenvectors can be used to approximately represent any one of the individual communities ( as an example , a linear combination of the first 10 eigenvectors whose support lies ( approximately ) only on the oscillators in the fifth community is shown as a thick solid line in fig .[ fig : evect ] ) .when only a few eigenvectors capture the presence and structure of the different communities , they form a suitable basis to represent the long term dynamics of the system .a comparison of figs .[ fig : phaseev ] and [ fig : evect ] also suggests that a linear combination of these eigenvectors is likely to represent well the long term evolving states of the phase oscillators . in other words , after a short initial transient , the system becomes attracted to a _ low - dimensional manifold _ on which the individual oscillator phases can be represented using a lower dimensional basis formed by the first few ( here 10 ) laplacian eigenvectors of the network ; these eigevectors _ parametrize _ the low - dimensional manifold .the use of the graph laplacian in constructing ( coarse ) observables is , of course , not new . below we will _ use _ these variables not only as observables , but as the means to implement accelerated coarse - grained computations with the model .there is a clear analogy between such observables and the use of the finite fourier transform in solving initial - boundary value problems : instead of eigenfunctions of diffusion in _ physical space _ we have eigenvectors of the laplacian _ on a graph _ ; instead of evolution equations for time - dependent fourier mode amplitudes we envision evolution equations for time - dependent components of the system phases on the first few graph laplacian eigenvectors . finally , it is also worth pointing out that these eigenvectors are also the eigenvectors of the linearization of the model around the uniform solution of a nearby " problem : that of coupled _ identical _ oscillators .we start by defining an vector of _ complex _ phase angles of the oscillators , : .\label{eq : complex2}\ ] ] the phase angles should be defined modulo ; this complex phase vector correctly represents the phase variable on a unit circle ( described by and ) .we now choose as coarse variables ( ) the components , , of this phase vector , , along the direction of the first ten eigenvectors of the graph laplacian . , \label{eq : evect } \\ & u^{(2 ) } = \{z_{j \in [ 1,m ] } = \textbf{v}_j^{t } \mbox{\boldmath } \}. \label{eq : coarse2}\end{aligned}\ ] ] this projection is our _ restriction _ , while translation between the fine description ( phases of all the oscillators ) and the coarse description is governed by eq .[ eq : lift2 ] , our _ lifting _ operator . a third option for coarse graining , which includes additional heterogeneity considerations is discussed in sec .[ sec : het ] .using the set of coarse variables , we accelerated the network simulation using _ coarse projective integration _ as detailed in the appendix .representative results are shown in figs .[ fig : order1 ] and [ fig : order2 ] , reported in the form of time - series of the order parameter for two values of the coupling strength , above and below respectively . forboth these cases , the magnitude of the projective step is equal to the duration of the full simulation used to estimate the coarse time derivatives ; that is , the full system , eq . [ eq : kura ] , is simulated for only of the overall evolution time .the coarse evolution in both cases clearly follows ( in the eye norm " ) the resolved full direct simulation results ; this demonstrates the accuracy of the equation - free approach and indirectly validates our selection of coarse variables .we performed coarse - fixed point calculations ( as outlined in the appendix ) using both choices of coarse variables and compared it to steady states calculated with the full model .given an initial guess of the coarse variable steady values , the coarse time - stepper was constructed by lifting , followed by full model simulation ( a representative time - stepper horizon was ) and restriction . fixed points of the coarse time - stepper were arrived at through a newton - krylov gmres iteration . to quantify accuracy , we calculated the pairwise linear correlation coefficient between the detailed ( fine scale " ) phases of the actual fixed point , and those lifted from the converged coarse fixed point values for each choice of coarse variables .the results are reported in table [ tab : fixed ] for three different coupling strengths .the first(respectively , second ) row corresponds to results obtained using (respectively , ) as coarse variables .clearly , gives a more accurate coarse description of the system fixed points compared to just using the average phases in the communities ( ) ..[tab : fixed]correlation between the detailed phases of the actual fixed point , and those lifted from the converged coarse fixed point values for each choice of coarse variables . [ cols="^,^,^,^",options="header " , ]   a coarse limit cycle computed from direct simulations at a coupling strength of is plotted using a solid ( blue ) line .the star ( black ) corresponds to the solution ( the point on a poincar map ) obtained using a coarse limit cycle solver .this point is then used as the initial condition for coarse projective integration ( red dots ) ; the coarse trajectory returns to that point after one period.,scaledwidth=50.0% ]   we have already discussed the existence of stable limit cycle oscillations below ( and close to ) the critical coupling strength , .the limit cycle found out from direct simulations for is plotted as a solid line in the phase space projection on the real and imaginary parts of in fig .[ fig : limit ] .we also find a ( coarse ) point on this limit cycle by locating the ( coarse ) fixed point of an appropriate ( coarse ) poincar map ; the point is represented as a star in fig .[ fig : limit ] .this point ( as well as the period of the coarse limit cycle ) is found by solving ( again through newton - krylov gmres ) eq .[ eq : limit ] for the appropriate set of values of the coarse variables ; the poincar map was defined in terms of the coarse variable . in these computations ,the full system was simulated for the entire poincar return time ; but the map , and the newton fixed point computation were performed in the _ reduced _ space of the coarse variables . in an extra validation step , the trajectory around the limit cycle was followed through coarse projective integration ( see fig . [fig : limit ] ) , and seen to coincide visually with the ( phase space projection of the ) full simulation limit cycle .these representative computations confirm that computational coarse - graining ( with the appropriate selection of coarse variables ) can be used to effectively perform computations with the ( explicitly unavailable ) coarse - grained model .coarse projective integration , coarse fixed point and limit cycle computations ( and also , easily , coarse stability and continuation computations ) can be implemented in the form of computational wrappers " around the full simulation .the choice of coarse variables ( and the associated lifting and restriction steps ) form a critical part of the approach ; an improvement on this process is presented below .in our coarse graining efforts , so far , we have focused on the structure of the network and ignored the effect of the heterogeneity in the angular frequencies of the oscillators . in this sectionwe present a systematic approach to taking this heterogeneity into account in the coarse - graining procedure .the motivation comes from the observation , in the literature , that for all - to - all coupling ( that is , in the absence of fine network structure ) the oscillator phases will , under certain conditions , become _ quickly _ correlated with their intrinsic frequencies .we have observed the same phenomenon in our , more structured , networks : the sequence of insets in fig .[ fig : slopet ] shows the transient evolution of the ( excess " ) phase of the oscillators in our network plotted against their individual frequencies .the term excess phase " , , here refers to the portion of the phase vector that is not captured by the projection on the first 10 laplacian eigenvectors ; using eq .[ eq : excess ] we plot the complex argument of each oscillator , _ viz ._ , against . what we see is that , even when the oscillators are initialized with random phases ( in the form of the cloud " seen in the first inset ) , they very quickly ( visibly by time and almost quantitatively by ) develop a _strong , stationary _ correlation with the intrinsic frequencies .these plots confirm the existence of a strong linear correlation between the excess phase " and the angular frequencies of the oscillators .the evolution of the slope of the best linear fit is plotted in fig . [fig : slopet ] ( a ) and ( b ) for different values of the coupling strength .a number of additional observations can be made from these plots .the time taken for the correlation slope to approach steady state is _ much less _ than the time scale in which the order parameters reach steady state ( compare with figs .[ fig : order1 ] and [ fig : order2 ] ) . even for the case of ,corresponding to stable limit cycle oscillations , the values of do not vary much once the stable limit cycle is approached .[ fig : slopek ] shows how this ( quickly achieved ) steady state correlation slope varies with the oscillator coupling strength . for a range of coupling strengths , this steady state value of obtained for three different frequency distributions ( the oscillator frequencies were sampled from the normal distribution with mean and standard deviation of , and respectively . ) the steady state correlation between excess phase and frequency appears to be _ independent _ of the range of ( or the variance in ) the oscillator frequencies .[ fig : slopek ] and its insets quantify the dependence of the correlation on ; the steady state values of are seen to decrease with coupling strength as one might expect ( since , at higher coupling strengths , the oscillator phases should exhibit less variance ) .in particular , an approximate inverse proportionality is computationally observed between the coupling strength , and the steady slope , quantified by the following fitted curve : we find it remarkable that our decoupled " procedure , which first considers heterogeneity in the network _ structure _ , and only then considers heterogeneity in the _ oscillator intrinsic properties _ gives us so robust features for the network dynamics .note that the constant applies only for the particular network used in the simulation shown . for different network structure realizations ,the constant will be different . based on these results, we can now integrate the effects of both network structure and oscillator frequency distribution in the coarse - graining of the oscillator phases ( in particular , in our _ lifting operator _ ) .the discussions in section . [ sec : het ] suggest that capturing the correlation between excess phase and intrinsic oscillator frequency can lead to a better set of coarse variables and a more accurate lifting operator for the coarse - graining process . for our illustrative example ,a single additional variable , the slope is , as we will show , sufficient in capturing this correlation .before we demonstrate this , however , we note the more general question : for arbitrary heterogeneity distributions ( not just gaussian as the one studied here ) , what is the nature and number of additional coarse variables necessary to quantitatively account for the frequency heterogeneity ? we will return to this question in the last section . using the single scalar slope as an additional coarse variable, we define the corrected " vector of complex phase angles , , similar to , but now taking the into account : , \label{eq : complex3 } \\ &\sum_{j=1}^{m } z_j \textbf{v}_j \rightarrow \mbox{\boldmath}. \label{eq : lift3}\end{aligned}\ ] ] our corrected lifting operation , going beyond the first ( here , 10 ) eigenvectors of the graph laplacian , is given by eqs .[ eq : complex3 ] and [ eq : lift3 ] . for the corrected restriction operation ,the vector of phase angles is initially projected on the first graph laplacian eigenvectors to obtain the ; the excess phases " are then used to estimate the slope through linear regression .our augmented set of coarse variables now reads : } = \textbf{v}_j^{t } \mbox{\boldmath } , c\}. \label{eq : coarse3}\ ] ] the results of computational coarse graining with the coarse variable set are , as one might expect , qualitatively similar to , _ but more accurate than _ , the results with the coarse set presented in sec .[ sec : res ] .note that since ( as we computationally observed ) quickly approaches an approximately constant value for each specific system realization , we can -in studying long term dynamics- _ fix _ its value at and not even consider it as an extra dynamic coarse variable .the constant value for different coupling strengths can also be inferred from formulas like eq .[ eq : slopefit ] .each successive coarse variable choice , and clearly includes more information about the system than the previous one : the coarse variable set just accounts for the presence of different communities , accommodates the structure of the different communities while set considers the influence of the heterogeneous frequencies of the oscillators as well . in order to quantify whether this additional information is also meaningful , we compared the results of coarse fixed point analysis using the three choices of coarse variables . the results in table [ tab : fixed ] demonstrate that the additional information included in successive choices of coarse variable sets actually leads to more accurate computation of the system features ( in particular , its coarse steady states )we have demonstrated an approach to coarse - graining the computations of the ( long - term ) dynamics of networks of coupled heterogeneous oscillators ; our approach was based on the equation - free framework , and was able to account -in separate steps- for the network structure and the oscillator intrinsic heterogeneity .the effect of the network structure on the evolution of the individual oscillator phases was first accounted for using the spectral properties of the network ( under the assumption that the network graph laplacian possesses a spectral gap ) . in a second step ,the effect of the heterogeneity in oscillator frequencies was accounted for by observing ( and then capturing ) a strong correlation between ( excess " ) phase angles and intrinsic oscillator frequency distribution .both steps were incorporated in the construction of a _ lifting _ and a _ restriction _ operator ( from coarse variables to detailed , fine scale state consistent realizations and vice versa ) .these operators can then be linked , in the equation - free framework , with algorithms such as coarse fixed point and coarse limit cycle computations , as well as with coarse projective integration , all of which were demonstrated .  even though the individual oscillator dynamics and the network topology used for illustration here were relatively simple , we are confident that the procedures demonstrated here can be extended to different , more complex individual oscillator dynamics and for other types of networks , as long as they possess spectral gaps as well. extensions to networks of spiking neurons , which can also be considered as coupled oscillators -but with much more complex , and especially _ directional _ coupling topologies- is probably too ambitious with only these tools . as pointed out in ref . , _ the information required to construct a detailed and specific configuration of neocortex containing some connections exceeds by far the roughly bits of information available in the genome for specification of the entire organism . on these groundsalone it appears that nature s strategy for construction of the neocortex must depend on the dynamic assembly of rather specific but simple modules _ " .this reasoning supporting module simplicity and specificity " provides hope and motivation for the deployment of reductionist approaches in such systems .one particular direction of extension for which we are more confident , and which we are currently pursuing , is the study of diverse heterogeneity distributions . in our illustrative example , the ( intrinsic frequency ) heterogeneity distribution was simply a normal one ( with different variances ) . a single scalar quantity ( the slope , ) of the correlation between heterogeneity and system state was sufficient to improve our coarse description here .this slope is but the first nontrivial coefficient of an expansion of the system state ( here , the excess phases ) as a function of a random variable ( here , the intrinsic frequency ) . in effect, this is a one - term " polynomial chaos expansion of a function of a random variable ( the oscillator frequency ) with a particular probability distribution .it is straightforward to use different expansions ( depending on the distribution of the random variable , different hierarchies of polynomials are applicable , see for example the askey scheme ) ; it is also straightforward to use more than one term in the expansion in a particular polynomial set if the correlation exhibits more structure than the straight line we observed here .this research avenue provides a direct link between existing and developing tools in the study of uncertainty quantification ( polynomial chaos approaches , and the associated collocation schemes ) with the study of coupled heterogeneous oscillator problems , even when heterogeneity arises in more than one properties of the coupled system .one particularly interesting direction for network dynamics arises when the oscillator behavior depends crucially on the _ degree _ of this oscillator in the overall network .if correlations between node degrees and oscillator states quickly develop in system startup transients , the tools we outlined above may well serve in successful coarse - graining of the overall network dynamics .the equation - free ( ef ) approach to modeling and computation for complex / multiscale systems has been developed for problems that can , in principle , be described at multiple levels . in particular , it is applicable to systems for which the evolution equations are available at a fine " ( atomistic , microscopic , individual - based ) scale , while the equations for the coarse " ( macroscopic , system level ) behavior , which is of interest , are not available in closed form .we will illustrate the ef approach through a brief description of _ coarse projective integration _ and coarse fixed point computation .the system of interest is completely specified at any moment in time by a set of fine or microscopic variables .we start with the assumption that an appropriate set of coarse variables ( observables in terms of which closed equations can in principle be written at the macroscopic level ) have been selected .we also assume that good _ lifting _ ( ) and _ restriction _ (  ] .2 .  evolve the oscillators using as the initial condition for their phases in the microscopic simulator for a time , long enough for the fast components of the dynamics to equilibrate , but short compared to the slow ( coarse ) system time scales ( see ) .the final state of this step is .3 .  evolve the microscopic variables , , for additional time steps , generating the values , to , i.e. ,  ] , to .projective step : estimate time derivatives from these restrictions , , to , and use any numerical scheme ( the simplest one would be forward euler ) to project " the macroscopic variables into the future " over a time interval to obtain .   using the lifting and restriction operations , and the fine - scale system simulator, one can define a _ coarse time - stepper _ ( eq . [ eq : coarsets ] ) that takes as input the coarse variables at a given time , , and outputs the coarse variables at a later time , . = \mathcal{r } [ \mathcal{e}_{\tau } [ \mathcal{l }  [ u(t ) ] ] ] .\label{eq : coarsets}\ ] ] one can also use such a coarse time - stepper to find the coarse fixed points , , by solving eq .[ eq : fixed ] for ( in principle ) any time , using matrix - free implementations of algorithms like newton - krylov - gmres to iteratively solve sets of nonlinear equations .\label{eq : fixed}\ ] ] with the help of coarse poincar maps , one can solve a similar equation to find a ( coarse ) point on the ( coarse ) limit cycle , , as well as its period , .\label{eq : limit}\ ] ] matrix - free implementations of eigensolvers ( e.g. matrix - free arnoldi procedures ) can ( and have been ) used to characterize the coarse linearized stability of coarse fixed points and limit cycles .33ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1'''' [ 0]secondoftwosanitize [ 0 ] + 12$12 & 12#1212_12%12[1][0] * * ,  ( ) _ _ ( , ) \doibase doi : 10.1016/j.physa.2004.12.046 [ * * ,  ( ) ] * * ,  ( ) link:\doibase 10.1126/science.254.5039.1750 [ * * ,  ( ) ] link:\doibase 10.1103/physrevlett.76.404 [ * * ,  ( ) ] link:\doibase  10.1038/35002660 [ * * ,  ( ) ] link:\doibase 10.1103/revmodphys.77.137 [ * * ,  ( ) ] \doibase doi : 10.1016/j.physd.2009.04.012 [ * * ,  ( ) ] \doibase doi : 10.1016/j.physd.2006.12.004 [ * * ,  ( ) ] * * ,  ( ) \doibase doi : 10.1016/j.physa.2007.11.008 [ * * ,  ( ) ] \doibase doi : 10.1016/j.physd.2009.04.002 [ * * ,  ( ) ] link:\doibase 10.1038/30918 [ * * ,  ( ) ] link:\doibase 10.1002/aic.10106 [ * * , ( ) ] * * ,  ( ) \doibase doi : 10.1016/j.acha.2005.07.004 [ * * ,  ( ) ] * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) link:\doibase 10.1137/0907058 [ * * ,  ( ) ] _ _ ( , ) * * ,  ( ) link:\doibase 10.1016/j.neunet.2009.07.011 [ * * ,  ( ) ] _ _ ( , ) _ _ ( , ) link:\doibase 10.1137/s1064827501387826 [ * * ,  ( ) ] _ _ ( , ) \doibase doi : 10.1016/j.cma.2007.11.033 [ * * ,  ( ) ] * * ,  ( ) \doibase http://dx.doi.org/10.1137/s1064827594277673 [ * * ,  ( ) ]

Abstract: we present a computer - assisted approach to coarse - graining the evolutionary dynamics of a system of _ nonidentical _ oscillators coupled through a ( fixed ) network structure . the existence of a spectral gap for the coupling network graph laplacian suggests that the graph dynamics may quickly become low - dimensional . our first choice of coarse variables consists of the components of the oscillator states their ( complex ) phase angles along the leading eigenvectors of this laplacian . we then use the equation - free framework , circumventing the derivation of explicit coarse - grained equations , to perform computational tasks such as coarse projective integration , coarse fixed point and coarse limit cycle computations . in a second step , we explore an approach to incorporating oscillator heterogeneity in the coarse - graining process . the approach is based on the observation of fast - developing correlations between oscillator state and oscillator intrinsic properties , and establishes a connection with tools developed in the context of uncertainty quantification .