experiments have demonstrated the ability to use continuous weak measurements to control the behavior of single quantum systems in a variety of technologies : including optical systems , atom - optical systems , superconducting microwave systems , and nv diamond .more recently , experiments have shown that it is possible to reconstruct quantum trajectories from the experimental records generated by continuous weak measurements in superconducting microwave devices .  the use of continuous weak measurements to probe the behavior of individual quantum mechanical devices has been considered for over 30 years .this is often referred to as _quantum filtering_. that is , generating an estimate of the underlying state of the quantum system that is conditioned on the measurements that have been obtained .belavkin formulated the basic theory for quantum filtering in the early 1980 s , and related approaches evolved separately in the late 1980 s in a number of forms , see for relevant reviews .using classical signals generated by weak measurements to effect controls is often referred to as measurement - based quantum control or incoherent quantum feedback control , as opposed to coherent quantum feedback control see for recent comprehensive review of the subject .continuous weak measurements are normally derived by considering a direct ( projective ) measurement on a coupled quantum system , consisting of an environment and the quantum system of interest .if the environment is much larger than the system of interest , any back action of the projective measurement on the combined system will have a relatively small ( i.e. weak ) effect on the system of interest , compared to its natural evolution .it is also common to make a markov approximation , which assumes that any correlations between the system and the environment are damped on timescales shorter than any of the timescales present in the system of interest . by averaging over the environmental degrees of freedom ,it is possible to obtain an effective evolution for the system of interest , either in average form ( the master equation ) or in a form which provides a quantum state that is conditioned on the specific measurement record obtained for the single system ( the stochastic master equation , sme ) .the evolution of the conditional state represents the best estimate of the evolution of the single quantum system of interest and provides a quantum trajectory for that system ( often referred to as an ` unraveling ' of the master equation ) .the ability to generate a quantum trajectory from an experimental measurement record provides an opportunity to implement a particular form of measurement - based / incoherent feedback control called _ rapid purification_. in 2003 , jacobs proposed the use of quantum feedback control to modify the speed at which a completely mixed ( unknown ) quantum state could be purified .that is , feedback is used to modify the rate at which information about the quantum state is extracted from the system . in , jacobs showed that the use of measurement - based feedback could increase the average rate of purification for a single qubit by up to a factor of two .this control protocol is relevant for future experiments because existing experiments require that the system be initialized in a known pure state using a projective measurement before the quantum trajectory can be found .more general approaches using mixed initial states require information to be extracted gradually as the weak measurements are recorded .subsequent studies on rapid purification showed that the proposed protocol was optimal , in that it maximized the average rate of purification , but it did not minimize the time taken to reach a given level of purity .other work has generalized these results for different optimization conditions , n - level systems , practical implementation of the controls , shared entangled states , imperfections , inefficient detection , and mixed protocols . in particular , li et al . demonstrated that when the efficiency of the detector is lower than 50% , there is no predicted speed up in purification rate .this result is significant for the recent quantum trajectory experiments in superconducting microwave systems because the estimated detection efficiency is given as 40% in .the paper has two main aims : the extension of a numerical integration method to address the case of inefficient continuous weak measurements , and the application of this method to a specific example of quantum control for two coupled qubits in a mixed initial state .the paper provides an efficient numerical method to solve the stochastic master equation for a given measurement record or realization .it uses this method to demonstrate a two - qubit feedback control scheme for completely mixed initial states that is robust and amenable to implementation in current experimental systems .the integration method was discussed for efficient continuous weak measurements in ; here it is adapted to deal with inefficient detection and extended to include higher order corrections .the resultant method ensures the positiveness of the conditional quantum state ( as represented by the density matrix ) and can be simplified further , reducing the computational load associated with the solution of the sme without a commensurate effect on the accuracy of the conditioned state .the intention is to develop a quantum filter that could be used for online quantum state estimation and feedback control .for a quantum system , represented by a density matrix , coupled to a markovian environment , represented by a set of environmental operators  , the master equation in linblad form is given by , + \sum_{j=1}^{m } \left\ { v_{j } \rho v^{\dagger}_{j } -\frac{1}{2}\left(v^{\dagger}_{j } v_{j } \rho + \rho v^{\dagger}_{j } v_{j } \right)\right\}dt\ ] ] where is the hamiltonian of the system and is an infinitesimal time increment .if we select some of the environmental operators to provide a set of continuous weak measurements  , with efficiencies , the corresponding stochastic master equation for the conditioned state is then given by , \nonumber\\ & & + \sum_{j=1}^{m - m ' } \left\ { v_{j } \rho_c v^{\dagger}_{j } -\frac{1}{2}\left(v^{\dagger}_{j } v_{j } \rho_c + \rho_c v^{\dagger}_{j } v_{j } \right)\right\}dt  \nonumber\\ & & + \sum_{r=1}^{m ' } \left\ { l_{r } \rho_c l^{\dagger}_{r } -\frac{1}{2}\left(l^{\dagger}_{r } l_{r } \rho_c + \rho_c l^{\dagger}_{r } l_{r } \right)\right\}dt  \nonumber\\ & & + \sum_{r=1}^{m ' } \sqrt{\eta_r}\left(l_{r}\rho_c+\rho_c l^{\dagger}_{r}-\mathrm{tr}(l_{r}\rho_c+\rho_c l^{\dagger}_{r } ) \right)dw_{r}\nonumber\\\end{aligned}\ ] ] where we have taken to be a real wiener increment ( such that and ) .the measurement record for each of the measurement operators during a time step is given by , . the previous case considered in deals with the case where for all environmental operators .  for a given initial state , a system and a set of environmental operators ,the sme may be integrated using standard numerical stochastic integration methods , e.g. the euler - maruyama method ( which is weakly convergent to first order ) or the euler - milstein method ( strongly convergent to first order ) . for an sme of the form ( [ sme1 ] ) with commuting measurement operators ( = 0 $ ] for between and ) ,the euler - milstein increment for a finite time step and finite stochastic increments can be computed using  ,  \delta t+\sum_{j=1}^{m - m ' } \left\ { v_{j } \rho_c v^{\dagger}_{j } -\frac{1}{2}\left(v^{\dagger}_{j } v_{j } \rho_c + \rho_c v^{\dagger}_{j } v_{j } \right)\right\}\delta t  \nonumber\\ & & + \sum_{r=1}^{m ' } \left\ { l_{r } \rho_c l^{\dagger}_{r } -\frac{1}{2}\left(l^{\dagger}_{r } l_{r } \rho_c + \rho_c l^{\dagger}_{r } l_{r } \right)\right\}\delta t + \sum_{r=1}^{m ' }  \sqrt{\eta_r}\left(l_{r}\rho_c+\rho_c l^{\dagger}_{r}-\mathrm{tr}(l_{r}\rho_c+\rho_c l^{\dagger}_{r } ) \right)\delta w_{r }  \nonumber \\ & & + \sum_{r , s=1}^{m ' }  \frac{\sqrt{\eta_r \eta_s}}{2 } \left ( \begin{array}{l } l_r l_s \rho_c+\rho_c l^{\dagger}_{r } l^{\dagger}_{s}+l_{s}\rho_cl^{\dagger}_{r}+ l_{r}\rho_c l^{\dagger}_{s } \\ -\mathrm{tr}\left ( \begin{array}{l } l_r l_s \rho_c + \rho_c l^{\dagger}_{r } l^{\dagger}_{s}+l_{s}\rho_c l^{\dagger}_{r}+ l_{r}\rho_c l^{\dagger}_{s } \end{array}\right)\rho_c\\ -\mathrm{tr}(l_s\rho_c+\rho_c l^{\dagger}_{s } ) ( l_r\rho_c+\rho_c l^{\dagger}_{r})\\ -\mathrm{tr}(l_r\rho_c+\rho_c l^{\dagger}_{r})(l_s\rho_c+\rho_c l^{\dagger}_{s})\\ + 2\mathrm{tr}(l_r\rho_c+\rho_c l^{\dagger}_{r})\mathrm{tr}(l_s\rho_c+\rho_c l^{\dagger}_{s } ) \rho_c \end{array } \right)\left(\delta w_r\delta w_s-\delta_{r , s}\delta t\right)\end{aligned}\ ] ]  where is the kronecker delta symbol , and the s are independent gaussian variables with zero mean and a variance equal to .the euler - milstein increment is strongly convergent to first order in , but the stochastic increment is only half order in , so terms need to be retained .the form given in ( [ sme2 ] ) does guarantee that is hermitian , up to numerical rounding errors due to the accuracy of the floating point calculations , however , it is computationally expensive and the formula does not guarantee the positivity of the resultant density matrix .an alternative form for the increment , which ensures the positivity of can be derived from an alternative expansion  to second order in that provides a positive operator valued measure ( povm ) .the increment for a time step is then given by , , where    and where the operator is given by   however , we note that the integrated measurement records over the time step are given by , so the operator can be simplified to give    where the state update is still given by ( [ sme3 ] ) .to demonstrate the efficiency and the robustness of the proposed method , we consider a specific two - qubit example . a two qubit example has been selected because single qubit quantum control and state estimation has been studied extensively , and two - qubit quantum control offers the potential to manipulate the entanglement between the two qubits as well as the rate of purification .examples demonstrating that quantum feedback can be used to stabilise or control the level of entanglement in two qubit systems have been discussed for stochastic quantum jump processes and continuous unravelings of the master equation .we consider the case where each qubit has an intrinsic hamiltonian , where , and are the pauli matrices .this hamiltonian generates a rotation of the individual qubit bloch vector around the respective -axis , with angular frequency .we introduce a coupling between the two qubits that couples the -components of the bloch vectors .the total hamiltonian ( in the absence of controls ) is given by , where the tensor product is implied ( i.e. ) , is the identity matrix , and is the strength of the coupling between the two qubits .  in most of the cases considered here , we will fix the qubit rotation frequency to be a standard value , , which will be used to fix the size of the time steps used in the numerical integration , so that and is the number of steps per cycle .each of the qubits is measured along its bloch -axis and these measurements are inefficient , so that the environmental operators are and , where and are the measurement strengths , with efficiencies and respectively .we will also fix the qubit - qubit coupling and the measurement strengths to be relatively weak so that the individual qubit evolution is perturbed but not dominated by the coupling or the measurement interactions .we use , , and in all of the examples .the measurement efficiencies are better than those available to the experimental systems described in ; however , the values are not so close to 100% efficiency so as to be unrealistic .lower values for the measurement efficiency do reduce the level and robustness of the steady state entanglement found in the two - qubit control example below , but the results presented below are not critically dependent on the precise value of the measurement efficiency .the two ( local ) measurement records are and there are no extraneous unprobed environmental degrees of freedom , , for simplicity .to explore the accuracy of the proposed method , a separable initial pure state and a completely mixed ( unknown ) initial state are integrated over 50 oscillation periods using the euler - milstein increment with and steps per cycle .no controls are applied , and the qubits are allowed to rotate freely around their respective axes .figure 1a shows the average purity for both the pure state and the mixed state both states converge to an entangled mixed state with an average purity of around 0.85 .the integrated state found using 5000 steps per cycle is then compared against integrated states found using the euler - milstein method with longer integration steps ( i.e. fewer steps per cycle ) and compared against the proposed method with much longer time steps .the only modification to the methods described above is due to the fact that the accuracies of the resultant states are quite sensitive to the hamiltonian part of the integration step . because of this sensitivity, the hamiltonian evolution terms are expanded to at least second order in .the other second order terms in are much smaller and can be removed ) .the comparison uses the fidelity ,  , \right|^2\ ] ] between the estimated ( conditioned ) state and the conditioned state generated using 5000 steps / cycle , , as a metric .[ fig_1 ]  ( color online ) ( a ) average purity for initial pure state ( solid black line ) and initial completely mixed state ( dashed black line ) calculated using the euler - milstein method with 5000 steps per cycle ; ( b ) 1-average fidelity , where fidelity is calculated between an initial pure state with 5000 steps per cycle and an initial completely mixed state using euler - milstein method with 1000 steps ( blue - cross ) , 500 steps ( red - circle ) and 250 steps ( green - asterisk ) per cycle ; and , ( c ) 1-average fidelity , where fidelity is calculated between initial pure state with 5000 steps per cycle and an initial completely mixed state using the proposed method with 250 steps ( blue - crosses ) , 100 steps ( red - circles ) and 50 steps ( green - asterisks ) per cycle all values averaged over 1000 realizations.,title="fig : " ]  figure 1b shows the accuracy of the euler - milstein integration method for different number of steps per qubit cycle : 1000 steps / cycle , 500 steps / cycle and 250 steps / cycle , and gives the average value of , where the fidelity is averaged over a large number of realizations ( 1000 realizations , in this case ) .the integration errors reduce with more steps / cycle , as expected , but 250 steps / cycle are sufficient to produce a relatively high fidelity state .the corresponding calculations for the proposed method are shown in figure 1c , and also compared against the state generated using the euler - milstein method with 5000 steps / cycle . in figure 1c , the number of steps per cycle are 250 , 100 and 50 .the same accuracy can be achieved using the state update ( 4 ) with 4 - 5 times fewer integration steps than the standard euler - milstein method : 50 steps per cycle provides a conditioned state with high average fidelity .recent experiments have generated quantum trajectories using measurements that had been sampled around 60 times per qubit oscillation period , so measurement rates of around 50 - 60 measurements per cycle can be considered to be the current ` state - of - the - art ' for practical systems , albeit without feedback control . to move towards a practical implementation of quantum feedback control in such systems , the quantum trajectories and the corresponding controls would need to be generated in real time .  although it is not shown here , the numerical benefit is even more pronounced when the purity of the state approaches one .the euler - milstein method can give rise to unphysical conditioned states with purities significantly above one .this numerical instability is more with larger integration steps , and it can lead to unreliable results numerical integration . for the proposed method , the normalization contained in the povm - like update ( 4 ) dramatically reduces the likelihood of such unphysical conditioned states and the resultant purity is significantly more stable than that found using the euler - milstein method .in this section , we examine how the proposed numerical method can be used with quantum feedback control to affect the entanglement of the two qubits described by the hamiltonian ( [ h ] ) .the two measurement records are local and we restrict ourselves to local controls , for simplicity and to demonstrate that feedback control provides some advantages even without explicit control over the entangling interaction itself . in the one - qubit case ,the approach to quantum rapid purification has centered on the use of bloch rotations to rotate the conditioned qubit state towards the plane orthogonal to the measurement axis , or towards the measurement axis , or some combination of the two .these approaches have the advantage of being relatively robust and insensitive to small errors in these rotations . in the two qubit case ,we examine the use of local controls that rotate the bloch vectors for the individual qubits ( corresponding to a partial trace of the density matrix over the other qubit state ) towards one or other of the three main axes : i.e. rotations towards the , or axis for qubit 1 , and , or axis for qubit 2 .the controls applied are calculated at each time step by taking the estimated one qubit bloch vector and a target state ( aligned along one or other of the one qubit axes ) and then rotating the bloch vector onto the target state using a rotation based on rodriques rotation formula .the resultant rotations are similar to bang - bang controls , but once the bloch vectors are close to the target axes , the controls are relatively small .the controls are applied at the end of an integration time step , so the number of time steps reflects the number of control rotations that are required in every qubit cycle .we characterize the entanglement in terms of the concurrence and the negativity .two entanglement metrics are used because the ordering of mixed states in terms of their entanglement ( i.e. the entanglement of the state is greater than that of the state ) can be different for different entanglement measures , and two different metrics gives more confidence that the relative ordering of the resultant states is robust although , in some of the examples , the difference between the concurrence and the negativity is very small and only the concurrence values will be given .for a two - qubit mixed state density matrix the concurrence is given by where are the eigenvalues of the matrix in nondecreasing order and the negativity is given by twice the absolute sum of the negative eigenvalues of the partial trace of with respect to one of the individual systems .[ fig_2 ]  ( color online ) ( a ) average concurrence ( dashed lines ) and average negativity ( solid lines ) for the three control strategies for a completely mixed initial state , averaged over 500 realizations and 250 steps per cycle using the proposed integration scheme : two - qubit & controls ( blue - crosses ) , one - qubit controls ( red - circles ) , and no controls / free rotation about the qubit axes ( green - asterisks ) ; ( b ) concurrence ( dashed lines ) and negativity ( solid lines ) for one realization of the three control strategies in ( a).,title="fig : " ]  if no controls are applied to the system , and the qubits are allowed to rotate freely under the action of their respective hamiltonians , there is small build up of entanglement in the system because of the coupling term .figure 2a shows the average entanglement for a completely mixed initial state with no controls applied ( green lines with asterisks ) .the average concurrence is limited to around ( ) .figure 2b shows a single realization , where the entanglement varies significantly as a function of time ( between zero and ) .figure 2 also shows the average entanglement if controls are only applied one of the two qubits : one qubit bloch vector is rotated towards its axis ( , in this case ) and the other is allowed to rotate under the action of the hamiltonian .the result is a small improvement in the rate of generation of entanglement and a marginal improvement in the average entanglement value ( ) , and as with the no control case the entanglement for an individual realization is stochastic .controls that rotate one of the qubits towards one of the other axes , or , do not generate entanglement as long as the intrinsic hamiltonian rotations are significant .due to the symmetry of the example ( two identical qubits ) , there are six possible two - qubit controls available , rotating towards the following pairs of axes , , , , and . of these ,only simultaneously rotating towards the and axes generates a reasonable level of entanglement .the steady state entanglement for the controls is approximately ( ) with small fluctuations around this value , see figure 2b .( the maximum possible entanglement achievable in this system with and perfect 100% measurement efficiencies corresponds to ) .compared with the no control and one qubit control cases , this entanglement is relatively robust the stochastic fluctuations are small compared to the average entanglement .the entanglement for the two - qubit control case corresponds to a mixed state which is approximately described by with smaller and cross terms .the other combinations provide zero entanglement .it is the combined effect of the rotations about the -axis and the repeated rotations to the individual and axes that allows entanglement to be generated through the addition of the and terms in .note that rotating both qubits towards the axes ( i.e. controls ) generates zero entanglement , even though rotating one of the two qubits towards its axis does provide a small amount of additional entanglement .[ fig_3 ]  ( color online ) average concurrence ( dashed lines ) and average negativity ( solid lines ) for the two - qubit controls rotating towards the ( blue - crosses ) , ( red - circles ) and ( green - asterisks ) axes as functions of the hamiltonian rotation ( angular ) frequency , after 50 cycles ( ) averaged over 500 realizations and 250 steps per cycle using the proposed integration scheme.,title="fig : " ]  if we were to remove the hamiltonian evolution ( rotation around the respective axes ) the cases , and should be identical , and this is indeed the case . however , in these cases with no hamiltonian evolution the local controls generate the same level of entanglement as the rotating case ( ) .any combination that includes rotations towards one of the individual axes generates no entanglement .if rotating the individual bloch vectors towards the or axes generates a reasonable level of entanglement in the absence of hamiltonian rotations , and zero entanglement when , this could provide a mechanism to control the level of steady state entanglement in the system .varying the rate of hamiltonian rotation allows the build up of entanglement to be moderated without changing the control protocol see figure 3 .the sme provides a best ( bayesian ) estimate of the quantum trajectory of the individual system _ given _ the particular continuous weak measurement record or realization , .the potential problem is that the calculations required to construct this estimate could be impractical to perform in real time , or preferably quicker than real time to allow feedback controls to be applied in real time .there are a number of ways in which a state estimation filter can be modified to improve the computational efficiency . in classical state estimation , it is common to introduce approximations into the state update ( measurement ) and state prediction ( system dynamics ) processes .the rationale for these approximations is that a state estimate does not necessarily have to be optimal to be useful . in the case of quantum feedback control ,the requirement is that the estimated state is sufficiently close to optimal for the quantum filter / sme to be stable under the action of the resultant controls ( that is , deviations between the optimal and the approximate state estimate vanish in the long time limit ) .standard approaches that could be used to simplify the sme include : ( i ) removal of small terms , ( ii ) use of fixed or filtered values to replace the rapidly fluctuating dynamical variables , ( iii ) analytic approximations , and ( iv ) discrete values to replace continuous variables ( analog - to - digital conversion ) .using the state update ( [ sme3 ] ) , there are a number of benefits in terms of computational efficiency compared to the standard euler - milstein method .the first is that the measurement record can be used directly in ( [ mn2 ] ) and the majority of the operations applied to the density matrix are constant from time step to time step ( assuming that the measurement / environment operators do not change in time ) .this reduces the number of calculations required to find the new conditioned state by reducing the number of operations that require quantities to be calculated from . the only operation requiring properties of to be calculatedis in the normalization step , i.e. division by the trace of the numerator in ( [ sme3 ] ) .in fact , because the positiveness of the density matrix is ensured by the modified update step , the resultant quantum filter is also more stable numerically than the standard euler - milstein method , leading to the use of larger time steps seen in section [ sec : sec4 ] .in addition to these numerical benefits , we now consider two approximations to reduce the computational demands even further for the specific control case considered here .the next step is to identify the smallest terms in the operator to truncate the calculations ( i.e. removal of small terms ) . as already stated , the hamiltonian evolution is dominant , for , and the expansion of the hamiltonian terms needs to be at least second order in . of the other terms , the and terms are two orders of magnitude smaller than the other terms , andthey can be removed without a significant effect on the fidelity of the final estimated state .the resultant approximate state update is then given by ,    where   the effect of these approximations is to limit the accuracy of the integration and the resultant conditioned state .for 50 steps per cycle , the fidelity of the conditioned state is around 99.5% . for 20 steps per cycle ,the fidelity of the conditioned state is around 98% ; however , the entanglement of the estimated state is robust to the use of the approximate operator ( [ mn3 ] ) as long as the number of steps is greater than 20 per cycle . at 10 steps per cycle, there is a significant reduction in the entanglement of the estimated state .[ fig_4 ]  ( color online ) ( a ) average concurrence for the two - qubit controls rotating towards the for using the approximate update operator ( [ mn3 ] ) with 500 steps / cycle ( black - solid line ) , 50 steps / cycle ( blue - crosses ) , 20 steps / cycle ( red - circles ) , 10 steps / cycle ( green - asterisks ) and 5 steps / cycle ( purple - diamonds ) ; ( b ) 1-average fidelity values corresponding to the concurrence values in ( a ) all values averaged over 500 realizations with fidelities compared against solutions using the proposed integration scheme with 500 steps per cycle.,title="fig : " ]  [ fig_5 ]  ( color online ) ( a ) average concurrence for the two - qubit controls rotating towards the for using a full measurement record with 500 steps / cycle ( black - solid line ) , and 50 steps / cycle with a measurement record truncated to 6 bits per measurement ( blue - crosses ) , 4 bits per measurement ( red - circles ) , 3 bits per measurement ( green - asterisks ) , and 2 bits per measurement ( purple - diamonds ) ; ( b ) 1-average fidelity values corresponding to the concurrence values in ( a ) all values averaged over 500 realizations with fidelities compared against solutions using the proposed integration scheme with 500 steps per cycle.,title="fig : " ]  the next approximation that we consider is the precision of the measurement record itself .any numerical method using an experimental measurement requires that the analog measurements be digitized to a finite number of bits . the ability to digitize an analog measurement is limited by the speed of the analog - to - digital converters used in the experimental set - up , and any saving in terms of the number of bits used to store the measured values will have a knock on effect on the efficiency of the processing .conventionally , the number of bits used for an experimental value is either 8 , 16 or possibly 32 . here, we consider far fewer bits per measurement ( sample ) value .figure 5 shows the average concurrence and fidelity for different numbers of bits using the approximate operator ( [ mn3 ] ) : 6 bits , 4 bits , 3 bits and 2 bits . for each number of bits ,the signal is discretized between a set of thresholds ( bits , discrete values requires threshold values ) where the thresholds are uniformly distributed between .the measurement signals contain a significant gaussian noise term , so this range corresponds roughly to 3 standard deviations . for 4 or more bits per measurement, the accuracy of the resultant state is unaffected by the reduced number of bits , with the average fidelity approaching 99% , and the entanglement of the estimated state is robust for 3 or more bits per measurement .similar calculations using 20 integration steps per cycle show similarly robust results , with an average fidelity of around .numerical calculations using a single bit per measurement have been demonstrated elsewhere , but this requires significantly higher measurement frequencies than the results demonstrated here .in this paper , we have discussed a method for the efficient integration of the stochastic master equation for a quantum system under the action of continuous weak measurements , and we have demonstrated that it offers significant advantages in terms of accuracy and stability over the euler - milstein integration method for finite time steps .we have used an example with two coupled qubits subject to inefficient local measurements and local controls to show that the proposed method can be used with incoherent ( measurement - based ) quantum feedback control .the fidelity of the estimated ( conditioned ) quantum state and the resultant quantum trajectory are very good , even when the number of integration steps is very low ( or the time steps are very long ) . in addition , the entanglement between the two qubits is remarkably robust for very low numbers of time steps , and with a number of additional simplifying approximations for the state update process and for the digitization of the measurement signal .using as few as 3 or 4 bits per measurement and 50 measurements per qubit period would allow the construction of a conditioned state that is sufficiently accurate ( fidelity , ) to allow feedback control to be used to generate states with a concurrence very close to the maximum obtainable .  the ability to use local measurements ,an efficient state estimation process , and local quantum feedback controls to manipulate the entanglement between two coupled qubits offers a range of experimental realizations .the required continuous measurement frequencies are within the scope of recent experimental developments and have been demonstrated through the calculation of quantum trajectories from continuous weak measurements . the main obstacles to demonstrating such feedback control of entanglement are the efficiency of the local measurement interactions and the the ability to calculate the updates in real time .the results given above are not critically dependent on the efficiency of the measurement process , but lower efficiency measurements limit the purity and the entanglement found in the final conditioned state . the ability to use fairly coarse approximations in the update step and limited bit digital measurements should assist greatly when implementing the state estimation process and the feedback controls for hardware and/or software reconfigurable processors .the authors would like to thank the newton institute ( university of cambridge ) for their hospitality , and the organizers of the workshop on quantum control engineering ( july - 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Abstract: we discuss an efficient numerical scheme for the recursive filtering of diffusive quantum stochastic master equations . we show that the resultant quantum trajectory is robust and may be used for feedback based on inefficient measurements . the proposed numerical scheme is amenable to approximation , which can be used to further reduce the computational burden associated with calculating quantum trajectories and may allow real - time quantum filtering . we provide a two - qubit example where feedback control of entanglement may be within the scope of current experimental systems .


complex scattering of waves has captured the interest of physicists for a long time . for instance , the problem of coherent multiple scattering of waves , which has long been of great importance in optics , has seen a revived interest in relation to the phenomenon of localization .the present article fits in the general topic of statistical theory of complex wave - interference phenomena " .in particular , we shall study the statistical fluctuations of transmission and reflection of waves , which are of considerable interest in mesoscopic physics .complexity in wave scattering may derive from :  * the chaotic nature of the underlying classical dynamics , as in microwave cavities and quantum dots , or * the randomness of the scattering potentials in a disordered medium , as a disordered conductor , or a disordered waveguide carrying classical waves ( electromagnetic , elastic , etc . ) , on which we shall concentrate here .why do statistics on the results of a scattering process ?the point is that the interference pattern resulting from the coherent multiple scattering of waves from the systems described above is so complex ( a small variation in some external parameter changes it completely ) that only a statistical treatment is meaningful .we shall find a recurrent theme in our presentation : the statistical regularity of the behavior , which involves a relatively small number of relevant physical parameters , while the rest of details serves as mere scaffolding " .this feature was captured in the past following a maximum - entropy approach , within the powerful , non - perturbative , framework known as random - matrix theory : shannon s information entropy is maximized , subject to the symmetries and constraints that are physically relevant . later , generalized central - limit theorems ( clt ) have been demonstrated . herewe shall revisit past and recent efforts towards discovering universal features in the statistical scattering of waves in disordered waveguides .the paper is organized as follows .we first indicate the various physical regimes to be encountered in the problem of disordered conductors .we then mention how the statistical regularities in the problem have been captured in the past within a maximum - entropy approach : we briefly mention the transfer - matrix method that was used and the random - matrix theory model that was constructed , giving rise to a diffusion equation in transfer - matrix space .we then indicate a clt that was proved , thus showing that the maximum - entropy approach captures the universal features found in the clt .we then describe a microscopic potential model that was developed recently , which gives rise to a further generalization of the clt and thus to a limiting macroscopic statistics .we then give our conclusions .the fundamental physical process occurring in the system under study , the disordered waveguide shown schematically in fig . [ q1d](a ) , is coherent multiple scattering of waves .the length of the disordered section of the waveguide is , is its width ( the system being assumed two - dimensional ) and is the total number of running modes supported by the waveguide ; the mean - free - path ( mfp ) is denoted by .the system might as well be a disordered conductor with the same dimensions .\(a ) ( b )  in the transport of waves through the system one encounters various physical regimes , indicated in fig . [ q1d](b ) : i ) the _ ballistic _ regime , for ; ii ) the _ diffusive _ , or _ metallic _ regime , for ; iii ) the _ insulating _ regime , for .the aim is to calculate the transport properties of waves through this system .one important quantity is , of course the conductance of the disordered quasi - one - dimensional ( q-1d ) system , which is given by landauer s formula : where denote the transverse ( running ) modes : .landauer s relation allows calculating the conductance a transport quantity from the scattering properties of the system . in microwave systems one can actually measure the individual transmission coefficients , as well as , experimentally .the main approaches to the problem have been : i ) perturbation theory in the disordered potential ; ii ) supersymmetry methods , giving rise to a non - linear sigma model and iii ) random - matrix theory ( rmt ) models of the scattering matrix , or the transfer matrix of the system .here we shall concentrate on these latter non - perturbative models .one has observed remarkable _ statistical regularities _, in the sense that the probability distribution for various macroscopic quantities involves a relatively small number of relevant physical parameters ( essentially the _ mean free path _ ) . within a random - matrix theory scheme ,this feature was captured in the past following a _ maximum - entropy approach _ , which we now describe .  to a waveguide of length assign the transfer matrix ( see fig .[ lbb ] ) , which has the property that acting on the wave amplitudes on the left gives the amplitudes on the right : for every configuration of disorder we have one transfer matrix .if we assign a probability density to our transfer matrices , what results is a _ random - matrix theory of transfer matrices_.  to the waveguide of length we now add a building block " ( bb ) of thickness which is much shorter than , but _ still contains many weak scatterers _( the so called dense - weak - scattering limit ( dwsl ) ) , as shown schematically in fig .[ lbb ] .the transfer matrix associated with the bb will be considered to be statistically independent from ; we write its probability density as .the transfer matrices and are combined as to obtain the total transfer matrix of the combined system , while the probability density of the latter , , is obtained from the individual ones by the convolution " where is the invariant measure associated with the group of transfer matrices .( [ convolution ] ) has the structure of the _ smoluchowski equation in brownian motion theory _ .expecting the _ results to be largely independent of the details of the bb _ , the distribution for the bb was modelled , in ref . , by maximizing the _ shannon entropy _ = - \int p(m ' ) \ln p(m ' ) dm ' \ ; , \ ] ] subject to the constraints  being the elastic mean - free - path ( mfp ) .the only physical information conveyed by is the mfp .  following the procedure developed in the theory of brownian motion, we can convert the integral equation ( [ convolution ] ) into a differential equation , with the result : \;\ ; . \label{diff eqn dmpk}\end{aligned}\ ] ] this is a _ diffusion equation in transfer - matrix space _, known as the dmpk equation ( after dorokhov and mello , pereyra and kumar ) , which governs the evolution " with the length of the waveguide of the probability distribution of our transfer matrices .this equation has been written in terms of new variables as follows : i ) denotes the length in units of the mfp ; ii ) ( ) are radial " variables in terms of which the conductance can be written as ; iii ) we have used the jacobian .finally , eq . ( [ diff eqn dmpk ] ) has to be solved with the initial condition .the quantity ( ) denotes the universality class of random - matrix theory .the distribution of the conductance is known to evolve from a gaussian ( deep in the diffusive , metallic regime ) to a log - normal distribution ( deep in the localized , insulating regime ) . although can not be easily obtained algebraically from the above expressions , various approximationsshow that this behavior is well described by the dmpk equation ( [ diff eqn dmpk ] ) . in the crossover regime , found the main statistical properties of arising from the dmpk equation using a monte carlo calculation .the results are shown in fig .[ dmpk vs anderson ] , where is plotted ( solid lines ) for both and 2 for different values of .the results of extensive q-1d tight - binding - model calculations ( symbols ) , carried out for bulk disorder , are also shown .we see that the agreement between the two types of results is excellent .in contrast , the results for surface disorder and , also shown in the figure for comparison , are not described properly by the dmpk equation .it was shown later that a limiting distribution for arises when the individual , microscopic , scattering units are combined in the so called _ dense - weak - scattering limit _ ( dwsl ) and within a particular class of models .the dwsl corresponds to a large density of weak scatterers , with a fixed mfp . in the particular model that was studied ,the individual scattering units were defined through their transfer matrices and an isotropic " distribution of their phases was assumed .the limiting distribution that arises depends only on and is insensitive to other details of the microscopic distribution : it thus constitutes a _ generalized central - limit theorem _ ( clt ) .the result turns out to be identical to the dmpk equation ( [ diff eqn dmpk ] ) found in the maximum - entropy model described above .we can thus say that _ the maximum - entropy model selects the limiting distribution _, in the sense of the dwsl , within a class of models for the transfer matrices of the individual scattering units .  a class of limiting distributions wider than that of ref . was studied by one of the present authors ( pam ) and s. tomsovic in ref . , in which the isotropy assumption of ref . was relaxed to a large extent . in ref . the dwsl plays again an essential role and the result is a more general clt than that of ref . . the evolution with described by a generalized diffusion equation , in which the diffusion coefficients are the inverse mfp s for the various scattering processes that may occur in the problem .when the various mfp s can be represented by a single one , one encounters the dmpk equation that was described above .thus the model of ref . appears as a possible candidate to study the influence of the specific scattering properties of the various modes , which seem to be relevant , for instance , for the problem of waveguides with surface disorder , where dmpk does not give a proper description .we first present a general way of expressing the expectation value of an observable when we add a bb to an already existing waveguide of length , as shown in fig .the transfer matrix of the two pieces is combined as in eq .( [ m = mm ] ) , which can also be written as where we have expressed the transfer matrix of the bb as consider now a function of the transfer matrix , whose statistical properties we want to study : it might be , for instance , the conductance studied earlier , the transmission coefficient , or any other quantity of physical interest .its average for the enlarged piece can be written in terms of that for the original one as to simplify the notation , we have indicated symbolically with dots the relevant summations over channel and block indices ( see ref . for more details ) .it is clear that we now need an expression for the various moments of the quantities associated with the bb .recently , such an expression has been obtained from a potential model , which we now outline .we construct the bb as a sequence of random -potential slices , such that as shown schematically in fig .[ random_slices ] .  the -th -slice potential , defined as has matrix elements with respect to channels given by {ab } = { ( u_r)}_{ab}\delta ( x - x_r),\ ] ] in terms of which we specify the statistical model .the potentials , , are assumed to be _ statistically independent _ , _ identically distributed , with zero average _ and , for simplicity , _ zero odd moments _ , so that where , etc ., denote the second , fourth , etc . , cumulants of . from these expressionswe can calculate , in the dwsl , the various moments of needed in eq .( [ < > l+dl ] ) .one finds that the first moment vanishes , the second moment behaves linearly with and higher moments behave as higher powers thereof .also , the very important result emerges that _ the dependence on the cumulants of the potential higher than the second drops out in the dwsl_. the _ diffusion coefficients _ , or inverse mfp s , defined as depend only upon the second cumulants of the potential .they are energy dependent and also length dependent .  finally , we take the first term on the r.h.s . of eq . ([ < > l+dl ] ) to the l.h.s . ,divide both sides by and take the limit .the result is the fokker - planck equation : the fact that cumulants of the potential higher than the second are irrelevant in the end signals the existence of a _generalized clt _ : once the mfp s are specified , the limiting equation ( [ general_diff_eqn ] ) is _ universal _ , i.e. , independent of other details of the microscopic statistics .one of the main difficulties in solving eq .( [ general_diff_eqn ] ) , both analytically and numerically , is that it involves averages of different quantities on the l.h.s . and on the r.h.s .so far , that equation has been solved analytically for the one - open - channel case ( ) and a restricted number of observables " only : the results thus obtained are in excellent agreement with microscopic calculations .numerically , we have found no direct " way of solving eq .( [ general_diff_eqn ] ) .recently , a numerical algorithm which was called random walk in transfer - matrix space " has been implemented , in which a bb is constructed with the property given in eq .( [ general_diff_coeff ] ) , and then combined with successive bb s to construct a waveguide of finite length .the results have been compared with those arising from microscopic calculations , in which the entities that are combined are , literally , individual potential slices . for situations in which we have bulk disorder ,the comparison is excellent , even for quantities which are not described properly by dmpk . for surface disorder ,preliminary results indicate a reasonable agreement for the observables examined so far .in this paper we have first revisited some earlier results which seem to indicate that a maximum - entropy approach to the problem of transport in disordered waveguides works well when there is a central - limit theorem ( clt ) behind the scenes " .recently , we have shown that a clt arises in a model consisting of a random distribution of -potential slices .the parameters on which the result depends are the mfp s which depend on the variances of the potential - slices matrix elements and the correlation coefficient of these matrix elements .other details of the potential distribution are washed out " in the dense - weak - scattering limit ( dwsl ) .the result is expressed in terms of a generalized diffusion equation for the evolution with length of expectation values of physical observables .numerical results based on the random walk in transfer - matrix space " method indicate an excellent agreement with microscopic calculations with bulk disorder , and a reasonable one for the problem with surface disorder .p.a.m . and m.y .acknowledge financial support by conacyt , mexico , the former through grant no .l.s.f . and j.j.s .have been supported by the spanish mcyt ( ref .bfm2003 - 01167 ) and the eu integrated project `` molecular imaging '' ( eu contract lshg - ct-2003 - 503259 ) .

Abstract: the statistical theory of certain complex wave interference phenomena , like the statistical fluctuations of transmission and reflection of waves , is of considerable interest in many fields of physics . in this article we shall be mainly interested in those situations where the complexity derives from the quenched randomness of scattering potentials , as in the case of disordered conductors , or , more in general , disordered waveguides .  in studies performed in such systems one has found remarkable _ statistical regularities _ , in the sense that the probability distribution for various macroscopic quantities involves a rather small number of relevant physical parameters , while the rest of the microscopic details serves as mere scaffolding " . we shall review past work in which this feature was captured following a maximum - entropy approach , as well as later studies in which the existence of a limiting distribution , in the sense of a generalized central - limit theorem , has been actually demonstrated . we then describe a microscopic potential model that was developed recently , which gives rise to a further generalization of the central - limit theorem and thus to a limiting macroscopic statistics .   disordered waveguides , quantum transport , random processes 05.60.gg , 73.23.-b , 05.40.-a , 84.40.az


early statistical models of financial markets assumed a gaussian distribution . however , there is evidence that stock price returns are not gauss distributed and that the distributions diminish slowly in the high end ( fat tails ) .a better statistical description is provided by a model where the logarithm of the stock price is a one dimensional compound poisson process with jumps being asymptotically power - law distributed ( stable lvy process ) .that model , however , does not eliminate discrepancies between experimental facts and the theory and drawbacks of the theory .  the exponents of the power laws ( tail indices ) in distributions appear to be outside of the range of validity predicted by the theory .furthermore models in the literature assume that jumps in the price are independent of waiting times between jumps ( no price - temporal coupling ) .fat tails can be explained by models basing on operator stable lvy processes . herethe price of the stock is an exponential from a sum of a large number of projections of many - dimensional jumps whose distributions are invariant under auto - convolution .this implies that the distributions are , in the high end , mixtures of power laws with different exponents , each one describing a walk in one of the dimensions .we briefly describe development of models that account for a price - temporal coupling ( coupled random walks ) below . an analytical model that yields probability distributions of price jumps under an `` ad hoc '' ansatz about the form of price - temporal coupling . derives a long time scaling limit for the jump probability distribution in coupled random walks for an anomalous fractional diffusion . obtains results for master equations of coupled random walks in the long time scaling limit and show their connection to fractional derivative calculus . considers one - dimensional continuous - time coupled random walks ( weierstrass flights ) and derives the long time scaling limit of the mean squared displacement of the walker as a function of two parameters and which are fractional dimensions that represent the flight and the waiting .an extension of weierstrass flights to a model where the walker , after random waiting times , moves with arbitrary finite velocities to new positions is formulated in and formulas are derived for the sojourn probability density and for likelihoods of extreme events .finally construct contingency tables between logarithmic price returns and waiting times for stocks on the new york stock exchange and for united states dollar japanese yen exchange rates and reject the hypothesis of the independence of returns on waiting times at a high confidence level .we formulate a multi - dimensional random walk model that recognises that the jumps are not independent of the waiting times .the model has a two - level structure with the higher level being a walk with random fluctuations preceded by random waiting times and coupled to those and the lower level describing the fluctuations as sums of random numbers of random jumps with a jump intensity ( lvy density ) .the levels of the model have a subordinated time horizon that will be explained in the next section .our model works with two arbitrary functions , viz the lvy density with a scaling index and a waiting time probability density function ( pdf ) with a scaling index and generalises previous models proposed in the literature in this subject matter .in this section we formulate an assumption about the price evolution of the stock and we recollect some definitions from the theory of stochastic processes . in section [ sec : thestockprice ] we recapitulate some known facts concerned with limit distributions of sums of independent random vectors . in sections[ sec : scal ] and [ sec : scalrot ] we obtain exact relations for fourier transforms of jump intensities and for fourier transforms of distributions of integer powers of one dimensional projections of jumps that are sums of large numbers of independent random vectors . in section [ eq : jointpdfs ]we derive the fourier transform of a joint probability density of an event that the stochastic process attains given values at given times and , in addition to that , that it attains a given maximal value in some predefined time interval . in section ( [ sec : optionprice ] ) we define an option on the stock , define a portfolio that is composed of stocks and of options on stocks , define a random variable ( deviation of portfolio ) that describes the temporal change of that portfolio in a small time span diminished by a compound rate of interest earned on the portfolio in that time span . andwe derive conditions on the deviation of the portfolio to have a zero mean .let be the logarithm of the price of the stock ( log - price ) at time .we assume that the log - price has a drift . in a small time interval the log - price may or may not change with respect to this drift .the probability of that change depends on a presumed waiting time pdf .this means that : where is a real number that will be specified later , the parameters and ( volatility ) are assumed to be constant as a function of time and of the stock price .we assume that the random vector ( fluctuation ) is infinitely divisible ( see e.g. for the precise definition ) .this implies that the logarithm of the fourier transform ( log - characteristic function ) of it has a unique representation in terms of the lvy - khintchin formula .this means that : = \imath \vec{a } \cdot \vec{k } - \frac{1}{2 } q(\vec{k},t ) + \int_{\vec{x}\ne 0 } \left ( e^{\imath \vec{k } \cdot \vec{x } }  - 1 - \frac{\imath \vec{k } \cdot \vec{x}}{1 + ||\vec{x}||^2 } \right )  \phi(\vec{x}|t ) d\vec{x}\ ] ] where is a constant , is a quadratic form that describes gaussian fluctuations and the integral represents a compound poisson process with some rate and with a jump pdf . here we do not use the measure - theory notation but instead we replace the measure by riemann integrals with some probability densities .we denote  ] where is a normalisation constant of the waiting time pdf ( ) for some .the large and large scaling limit of the of the joint jump pdf in a coupled random walk reads where is the probability density in ( termed as the jump intensity ) , and the matrix is an exponent of the operator stable law ( stable index ) corresponding to the jump random variable and is a normalisation constant of the jump pdf .the eigenvalues of satisfy \ge 1/2 ] and the eigenvalues = 1/2 ] are simple ( not degenerate ) .the scaling limit of the conditional jump pdf is : this follows in a straightforward manner by integrating over in ( [ eq : scalinglimitth ] ) .the jump intensity satisfies a functional equation : where is the auto - convolution defined as a limit of -times auto - convolutions for some rational such that . since the random process is infinitely divisible the definition is correct .the corollary follows from the self - similarity of the random walk .this means that there exists a sequence of linear operators and a sequence of real numbers and a sequence of integer numbers such that for some independent , identically distributed random variables we have : on the other hand if we take the sequence of rational numbers we have now if we take large enough we can write for a certain subsequence . therefore the random variable has the same pdf except that the argument of the pdf is linearly transformed .the fourier transform (\vec{k}) ] where .then and the function can has an asymptotic expansion :   in the next section we are going to analyse the langevin equation ( [ eq : stockpricefluct ] ) for the log - price of the stock driven by operator stable fluctuations . for this purposewe need the time dependence of the conditional jump pdf on the waiting time for small waiting times and we need the probability distribution of the power of a scalar product for ( the conditional -marginal pdf ) .we have :  the conditional jump pdf depends on time as follows : this is a straightforward conclusion from ( [ eq : scalinglimitcor ] ) and from the fact that  = e^{\mbox{tr}} ] for every linear operator .the fourier transform of the conditional -marginal probability density function (k) ] takes following form : where & \quad\mbox{if is odd }  \label{eq : kerneldefia}\\ 2 e^{\imath \frac{\pi}{2 n } }  \int_0^\infty d\xi e^{-\mathfrak{k } \xi^n }  \cos\left ( \cos(\pi \frac{(n-1)}{2 n } ) \mathfrak{l } \xi\right ) & \quad\mbox{if is even }  \end{array } \right .\label{eq : kerneldefiia}\end{aligned}\ ] ] and . for small values of conditional -marginal pdf reads : where and is defined in appendix a the proof of the proposition is in appendix a.  the mean value of the power of the fluctuation term in ( [ eq : stockpricefluct ] ) conditioned on the waiting time is infinite if is even and and is zero if is odd . = \left\ { \begin{array}{ll } \infty   & \quad\mbox{for } \ ; n \in 2\mathbb{n } \quad\mbox{and }\\ 0     & \quad\mbox{for } \ ; n \in \mathbb{n } \diagdown 2\mathbb{n }  \end{array } \right .\label{eq : infinitemean}\ ] ] the first statement follows from ( see ( [ eq : smallkvalnmargpdf ] ) ) and the second statement follows from the assumption that the conditional jump pdf and hence the jump intensity are symmetric in ( see beginning of section [ sec : thestockprice ] ) .  the power of the fluctuation term in ( [ eq : stockpricefluct ] )has a form : where is a random number that depends neither on time nor on .we have : ( z ) = } \label{eq : nmarginaltimescali}\\ & &  \mathcal{f}^{-1}_{k } \left [ \tilde{\varpi } \left ( ( \delta t)^{\beta \underline{\underline{e}}^{t } }  \left ( \mathfrak{g } k^{1/n } \vec{\sigma } \right ) \right ) \right ] ( z ) =  \mathcal{f}^{-1}_{k } \left [  \tilde{\varpi}^{(\delta t)^\beta } \left ( \mathfrak{g } k^{1/n } \vec{\sigma } \right )  \right](z)= \label{eq : nmarginaltimescalii}\\ & & \mathcal{f}^{-1}_{k } \left [  1 - ( \delta t)^\beta ( \tilde{\varpi } - 1 )  \left ( \mathfrak{g } k^{1/n } \vec{\sigma } \right )  + o\left((\delta t)^{2 \beta}\right ) \right](z)= \label{eq : nmarginaltimescaliii}\\ & & \delta(z - z_0 ) -  ( \delta t)^\beta \mathcal{f}^{-1}_{k } \left [  ( \tilde{\varpi } - 1 )  \left ( \mathfrak{g } k^{1/n } \vec{\sigma } \right )  \right](z ) + o\left((\delta t)^{2 \beta}\right ) \label{eq : nmarginaltimescaliv}\end{aligned}\ ] ] for some .( [ eq : nmarginaltimescaliv ] ) is equivalent to ( [ eq : nmarginaltimescal ] ) for .the random variable conforms to a distribution whose characteristic function is .the concrete dependence of on time depends on the form of the matrix ( stable index ) .we investigate that dependence in next sections .now we are going to investigate the functional form of the jump intensity in the limit of large jumps .it is more convenient to analyse the fourier transform for small values of .this function is a solution of equation ( [ eq : scalinglimitcorfourtransf ] ) and is determined in a unique way via the stable index . since , however , on the grounds of the jordan decomposition theorem , every matrix has a unique representation as a block diagonal matrix where every block is of the form : where is a real eigenvalue of in the first case , and in the second case where is a complex conjugate pair of eigenvalues of the set of all possible jump intensities is narrowed down to few classes of solutions only , each one corresponding to a particular jordan decomposition of the matrix . in the following we investigate these classes of solutions as a function of .we proceed as follows :  1 .take a jordan decomposition of , 2 .find the fourier transform of the jump intensity by solving the functional equation ( [ eq : scalinglimitcorfourtransf ] ) 3 .find , from ( [ eq : fourtrnmarginalprop ] ) the fourier transform of the conditional -marginal pdf ( see table [ tab : summarydef ] ) as a function and of for small and small .  in the following sections we seek for the solutions of the equation ( [ eq : isexpdef ] ) for different types of blocks in the jordan representation .the names of the sections refer to the type of jordan decompositions of .let where is a dimensional identity matrix and is a constant . herethe mapping : changes the length of only .we therefore assume the jump intensity to depend on the length of the vector only . substituting that assumption into ( [ eq : scalinglimitcorfourtransf ] ) and taking we get :    and where .the conditional -marginal pdf ( see table [ tab : summarydef ] ) from ( [ eq : fourtrnmarginalprop ] ) reads : where the kernel is defined in ( [ eq : kerneldefii ] ) and has a series expansion in given in ( [ eq : kernelexpansioniv ] ) . from ( [ eq : smallkvalnmargpdf ] ) we get the -marginal pdf for even and for small values of .we have : where and the constant is defined in appendix a.   take and let : the trace = \mu^{-1} ] .we denote by a two dimensional rotation by an angle .the mapping : changes the length of by a factor rotates by an angle .we seek solutions of ( [ eq : scalinglimitcorfourtransf ] ) in the form : where the function is assumed to depend on the length and on the angle . inserting ( [ eq : opstablesolansatz ] ) into ( [ eq : scalinglimitcorfourtransf ] ) we obtain : now we assume that the function factorizes : inserting ( [ eq : scalrotansatz ] ) into ( [ eq : eq4upsilon ] ) and taking with we get :    after separating variables and assuming that the jump intensity is even in we obtain in the following form :  \right ] \label{eq : fourjumintsoliia}\ ] ] where for are some numbers that depend on . we find only solutions for for and leave the generic case for future work .we check that ( [ eq : fourjumintsoliia ] ) is indeed a solution of ( [ eq : eq4upsilon ] ) for under following assumptions : this leads to a following solution for the jump intensity in the fourier domain : \label{eq : fourjumintsoliib}\ ] ] where for small the conditional -marginal pdf reads : \right]\ ] ] where where the number is defined in ( [ eq : defg ] ) .we summarise the results for the fourier transforms of the jump intensity and of the conditional -marginal pdf in both cases of pure scaling and of scaling & rotation :  where is defined in ( [ eq : defg ] ) and the results for the conditional -marginals hold for small only .the purpose of this section is to derive probability distributions of financial instruments like barrier european style options with multiple exercise times and american style options ( exotic options ) as a function of of the conditional -marginal pdf .we derive a joint probability distribution of log - prices of the stock at certain times and of the maximal log - price in some time interval .we fix values of time and we analyse the joint cumulative distribution function related to the log - prices for of the stock and to the maximal value of the log - price .we set , we define and we define integers such that . from the definition of the joint cumulative distribution function and from the the sde ( [ eq : stockpricefluct ] ) we have : where denotes a union of conditions , the integration region is an infinite region in bounded by hyperplanes of dimension and is the conditional probability density of the random variable .the fourier transform of that probability density is given in ( [ eq : summaryofresults ] ) .now the procedure consists of following three steps .  *derive the joint probability density function by differentiating the cumulative density function , * derive the joint characteristic function by taking fourier transforms of all random variables and expressing the result through fourier transforms (k) i \le i_j} ] equal to in order to ensure that the deviation variable exists in the limit .our approach differs here from the approach in financial mathematics where one assumes in the first place that the deviations have no drift ( there are no arbitrage opportunities on the market ) and one imposes conditions ( semi - martingale condition ) on the high end of the jump intensity in order to ensure this assumption .we can not make such an assumption for operator stable lvy processes .the processes that we are using are sums of a large number of of independent , identically distributed jumps where the distribution of the latter diminish according to a power law and not to an exponential in the high end .the processes are invariant under auto - convolutions ( [ eq : isexpdef ] ) ( self - similar ) and in our opinion it is much more appealing from the physical point of view ) ] to use them in financial modelling rather than to use the abstract notion of the metric space of lvy processes with right continuous and left bounded paths endowed with a shorokhod topology if there is no experimental justification that financial time series in question are martingales .our approach is not equivalent with theory in chapters * 20.4.3 * * 20.4.5 * pages 1416 1428 in .that theory assumes that the log - characteristic function of the fluctuations that drive is analytic ( is power expandable ) and derives a partial differential equation ( fokker - planck equation ) for the price of the option as a function of the price of the stock and of time and solves this equation subject to initial conditions for the price of the stock at maturity .operator stable processes are not power expandable. it might be possible to expand the log - characteristic function in a series of fractional powers of using the fractional calculus and in particular the fractional taylor expansion ( formula ( 4.11 ) page 89 in ) and develop a fractional fokker - planck equation for the price of the option on the stock . in futurework we will pursue investigations in this context in order to reconcile our results with that from .  equation ( [ eq : stockpricefluct ] ) defines the price of a stock in terms of an exponential from a lvy process .thus a small change of the stock price reads : -1 \right )  \label{eq : stockpricechangei } \\ & = & s_t \left ( \alpha ( \delta t)^\beta + \sum_{m=1}^{\infty }  \frac { \left(\vec{\sigma } \cdot \delta \vec{l}_t\right)^m}{m ! }\right ) \label{eq : stockpricechangeii}\end{aligned}\ ] ] where in ( [ eq : stockpricechangeii ] ) we expanded the exponential in a taylor series and retained all terms of the expansion since all these terms are proportional to ( [ eq : nmarginaltimescal ] ) for large values of .  from ( [ eq : stockpricechangei ] ) and ( [ eq : stockpricechangeii ] ) we see that the price of the stock is not a martingale .it is also not possible to construct a martingale from the price of the stock by subtracting the sum of means  ] of the portfolio ( [ eq : portfoliodef ] ) .we expand a change of the price of an option on the stock in a taylor series in powers of the change of the stock price : where the partial derivatives an are assumed to be independent of time and they will be determined later .  from ( [ eq : stockpricechangei ] ) and ( [ eq : stockpricechangeii ] ) we compute the powers of the changes of the stock price and we retain only those terms that are proportional to .we have  -1 \right)^n \quad\mbox{for }  \label{eq : stocknchange}\ ] ] where we have neglected all terms on the grounds of ( [ eq : nmarginaltimescal ] ) .we insert ( [ eq : optionchange ] ) and ( [ eq : stocknchange ] ) into ( [ eq : devdefinition ] ) and we get the deviation random variable : note that all terms on the right hand side of ( [ eq : deviationresult ] ) are proportional to .now we derive the fourier transform of the probability density function of conditioned on the value of the price of the stock at time ( conditional deviation characteristic function ) .we have : = } \label{eq : characfctdevi}\\ & &  \exp ( \imath k d_t ( \delta t)^\beta ) \int_{-\infty}^\infty dz  \nu^{(1)}(z | \delta t ) \exp\left ( \imath k \sum_{n=2}^\infty \frac{1}{n ! }  \frac{\partial^n c}{\partial s^n }  s_t^n \left ( \exp\left(z\right ) - 1 \right)^n \right ) = \label{eq : characfctdevii}\\ & & \exp ( \imath k d_t ( \delta t)^\beta ) \int_{-\infty}^\infty d\lambda  \tilde{\nu}^{(1)}(\lambda | \delta t ) \mathfrak{m}(k , \lambda ) = \label{eq : characfctdeviii}\\ & & \exp ( \imath k d_t ( \delta t)^\beta ) \int_{-\infty}^\infty d\lambda  \tilde{\varpi}((\delta t)^{\beta \underline{\underline{e}}^{t } }  \frac{\vec{\sigma}}{|\vec{\sigma}|}(\lambda ) ) \mathfrak{m}(k , \lambda )  \label{eq : characfctdeviv}\end{aligned}\ ] ] where and and in ( [ eq : characfctdeviv ] ) we made use of ( [ eq : fourtrnmarginalprop ] ) and of ( [ eq : kerneldefi ] ) . in further calculations we make use of the following proposition .the integral where is a real function such that the only root of is and tends , for small values of , to a following delta function : where is a number that satisfies ( [ eq : deltaconsteval ] ) , is the inverse function to in some vicinity of and is a positive real number . the proof is given in .  inserting ( [ eq : limlogdelta ] ) into ( [ eq : characfctdeviv ] )we get the conditional deviation characteristic function in the limit of small .we have : where and we check by direct substitution that for and the class of functions : with and and ensures that the drift of deviations =  \left .\frac{\partial \chi_{\mathfrak{d}_t\left|s_t\right.}(k)}{\partial k } \right|_{k=0 } = 0 \label{eq : nodrifti}\ ] ] is zero .the equation ( [ eq : nodrifti ] ) yields a differential equation for the option as a function of the price of the stock and of time viz note that the first term enclosed by parentheses in ( [ eq : differentialequationiii ] ) corresponds to the classical black & scholes equation and the second term depends via ( [ eq : optionpricingsol ] ) on all partial derivatives . in future workwe will analyse the uniqueness of the solution of the option pricing problem and we will derive conditions for the solution to have : a given initial value at maturity .we have applied the technique of characteristic functions to the problem of pricing an option on a stock that is driven by operator stable fluctuations .we have shown that it is possible to construct an option on a stock as a function of the price of the stock in such a way that deviations of the portfolio are compensated by the compound interest earned on the portfolio . in future workwe will price exotic options with different exercise times by computing the unconditional , joint characteristic function of the deviations of the portfolio related to values of stock price at these exercise times and by using initial conditions on the value of the portfolio at these exercise times .testing of the theory in monte carlo simulations and fitting the theory to market data will also be pursued in future work .we prove formula ( [ eq : fourtrnmarginalprop ] ) for the conditional -marginal probability function :   in ( [ eq : nmarginali ] ) we conditioned on the fluctuation and we integrated over . in ( [ eq : nmarginalii ] ) we assumed that the time is small enough that at most one jump occurs during it and therefore we replaced the conditional fluctuation pdf by the conditional jump pdf and in ( [ eq : nmarginaliii ] ) we used ( [ eq : scalinglimitcor ] ) . in ( [ eq : nmarginaliv ] )we replaced the jump pdf through an integral over from the fourier transform and we changed the order of integration integrating first over and then over . in ( [ eq : nmarginaliva ] )we defined , we introduced an orthogonal operator such that and we substituted for . in ( [ eq : nmarginalivb ] ) we performed the integrations over and in the remaining integral over we substituted for . in ( [ eq : nmarginalivc ] ) we eliminated the delta functions and transformed the -dimensional integral over into a one - dimensional integral over . in ( [ eq : nmarginalivd ] ) we defined a function ( kernel ) as : & \quad\mbox{if is odd } \\ 2 e^{\imath \frac{\pi}{2 n } }  \int_0^\infty d\xi e^{-\mathfrak{k } \xi^n } \cos\left ( \cos(\pi \frac{(n-1)}{2 n } ) \mathfrak{l } \xi\right ) & \quad\mbox{if is even }  \end{array } \right .\label{eq : kerneldefii}\end{aligned}\ ] ] and we evaluated it by rotating the range of integration in the complex plane by an angle ( wick rotation ) . in doing that we applied the cauchy theorem to a contour composed of an interval ] and  ] .the integrals over arches are bounded from above by  ] in which the function is monotone and thus invertible .we also denoted . in ( [ eq : poldeltafctproofiii ] ) we used the fact that and thus and we approximated the integrand by its large values .in ( [ eq : poldeltafctproofv ] ) we rotated the integration line by in the anti - clockwise direction making use of the fact that the integral over a quarter of a circle for  ] and the second one over  ] such that $ ] .in ( [ eq : poldeltafctproofvi ] ) we substituted and in ( [ eq : poldeltafctproofvii ] ) we performed the integral .now we take , small and and we evaluate the integral ( [ eq : poldeltafctproofix ] ) for this value . we fix and let .then and we see that . furthermore from ( [ eq : poldeltafctproofi ] ) we have we conclude that and the number can be evaluated from * q.e.d.*.       repetowicz p , richmond p , modelling share price evolution as a continuous time random walk ( ctrw ) with non - independent price changes and waiting times , physica a , in press , corrected proof , available online 1 september 2004  masoliver j , the ctrw in finance : direct and inverse problems , preprint cond - mat/0308017 ; masoliver j , montero m , continuous time random walk model for financial distributions , phys rev .e * 67 * , 021112 ( 2003 )

Abstract: we model the price of a stock via a langvin equation with multi - dimensional fluctuations coupled both in the price in time . we generalise previous models in that we assume that the fluctuations conditioned on the time step are assumed to be compound poisson processes with operator stable jump intensities . we derive exact relations for fourier transforms of the jump intensity in case of different scaling indices of the process . we express the fourier transform of the joint probability density of the process to attain given values at several different times and to attain a given maximal value in a given time period through fourier transforms of the jump intensity . then we consider a portfolio composed of stocks and of options on stocks and we derive the fourier transform of a random variable ( deviation of the portfolio ) that is defined as a small temporal change of the portfolio diminished by the the compound interest earned . we derive a functional equation for the price of the option on the stock as a function of the stock subject to the condition that the deviation of the portfolio has a zero mean = 0 $ ] therefore the option pricing problem may have a solution .