Functional Integral Approach to the Solution of a System of Stochastic Differential Equations

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EPJ Web of Conferences 173, 02003 (2018) https://doi.org/10.1051/epjconf/201817302003 Mathematical Modeling and Computational Physics 2017

Edik Ayryan1,2,, Alexander Egorov3,, Dmitri Kulyabov1,2,, Victor Malyutin3,, and 2,4, Leonid Sevastianov † 1Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna, Russia 2Peoples’ Friendship University of Russia (RUDN University), Moscow, Russia 3Institute of Mathematics, National Academy of Sciences of Belarus, Minsk, Belarus 4Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Russia

Abstract. A new method for the evaluation of the characteristics of the solution of a system of stochastic differential equations is presented. This method is based on the representation of a probability density function p through a functional integral. The functional integral representation is obtained by means of the Onsager-Machlup functional technique for a special case when the diffusion matrix for the SDE system defines a Rie- mannian space with zero curvature.

1 Introduction Stochastic differential equations (SDE) are very popular in physics, chemistry, biology and etc. [1, 2]. There are many publications devoted to different aspects of SDE theory [1–3]. Except for some very special cases, SDE cannot be solved analytically and numerical methods are needed. The most com- mon approach to the numerical solution of SDE is based on the discretization of a time interval and numerical simulation of SDE solutions at discrete times [4–7]. In this work a new method for evaluating of the characteristics of a SDE solution is presented. This method is based on the representation of the probability density function (PDF) p through a functional integral. The Onsager-Machlup functional technique is used to get the functional integral representation [8–11]. For one dimension case this approach was considered in [12]. In this work this approach is given for a system of SDE. Here we consider the Onsager-Machlup functional technique only for the flat space when the diffusion matrix for a SDE system defines a Riemannian space with zero curvature. In section 2 the representation of PDF p through a functional integral is discussed. In section 3 the method of approximate evaluation of functional integrals is presented. Using the minimal action principle [13], we can distinguish among all trajectories the classical trajectory for which the action S takes the extremal value. The classical trajectory is found as the solution of the multidimensional Euler-Lagrange equation. Further, we use the expansion of the action with respect to the classical

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© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). EPJ Web of Conferences 173, 02003 (2018) https://doi.org/10.1051/epjconf/201817302003 Mathematical Modeling and Computational Physics 2017

trajectory to compute the integral. In section 4 approximate and exact values for the expectations of some functionals of solution of speciﬁc system of SDE are presented.

2 Representation of characteristics through functional integral

We consider the SDE system (the Ito interpretation)

d dx1(t) = a1( x, t) dt + j=1 g1 j( x, t) dw j(t), ... (1)  dx (t) = a ( x, t) dt + d g ( x, t) dw (t),  d d j=1 dj j   where the solution x(t) satisﬁes the initial condition x(t0) = x0, w (t) is a d-dimensional Wiener process. By means of the Onsager-Machlup functional technique the PDF can be represented in a form of the functional integral

t p( x, t, x0, t0) = D[ x] exp L0( x(τ), x˙(τ)) dτ , (2) − t0 N 1 d N − 1 D[ x] = lim dxkj , xkN = xk(t), N d/2 →∞ j= k= j= (2π∆t) det G( x j 1, t j 1) 1 1 1 − − d 1 1 L ( x(τ), x˙(τ)) = G− ( x(τ),τ) [˙x A ( x(τ),τ)][x ˙ A ( x(τ),τ)], 0 2 kj × k − k j − j k, j=1 1 d ∂ Ak( x(τ),τ) = ak( x(τ),τ) gij( x(τ),τ) gkj( x(τ),τ), − 2 ∂xi i, j=1 where G denotes a matrix of elements

d Gij( x(τ),τ) = gik( x(τ),τ)g jk( x(τ),τ). k=1

3 Evaluation of functional integrals

The quantity L0( x(τ), x˙(τ)) in (2) can be considered as the Lagrangian of the system, and the quantity t S = L0( x(τ), x˙(τ)) dτ can be considered as its action. Using the minimal action principle [13], t0 we can distinguish among all trajectories the classical trajectory xcl for which the action S takes the extremal value. The classical trajectory is found as the solution of the multidimensional Euler- Lagrange equation d ∂L ∂L 0 0 = 0, dt ∂x˙ − ∂x  1 1  .  . (3)   d ∂L0 ∂L0  = 0. dt ∂x˙d − ∂xd    2 EPJ Web of Conferences 173, 02003 (2018) https://doi.org/10.1051/epjconf/201817302003 Mathematical Modeling and Computational Physics 2017 trajectory to compute the integral. In section 4 approximate and exact values for the expectations of Further, to compute the integral, we can use the decomposition of the action with respect to the some functionals of solution of specific system of SDE are presented. classical trajectory xcl 1 S [ x(τ)] S [ x (τ)] + δ2S [ x (τ)]. ≈ cl 2 cl 2 Representation of characteristics through functional integral 2 The second order variation δ S [ xcl(τ)] can be written in the form We consider the SDE system (the Ito interpretation) t d 2 δ S [ xcl(τ)] = δxiΛijδx j dτ, d t0 i, j=1 dx1(t) = a1( x, t) dt + j=1 g1 j( x, t) dw j(t), ... (1)  where x = xcl + δx,  d dxd(t) = ad( x, t) dt + = gdj( x, t) dw j(t),  j 1 ∂2L ∂2L d d ∂2L d ∂2L d  Λij = + .  ∂xi∂x j ∂xi∂x˙ j dt − dt ∂x˙i∂x j − dt ∂x˙i∂x˙ j dt where the solution x(t) satisfies the initial condition x(t0) = x0, w (t) is a d-dimensional Wiener process. xcl xcl xcl xcl By means of the Onsager-Machlup functional technique the PDF can be represented in a form of After these transformations the formula (2) is written in the form the functional integral t d t 1 p( x, t, x0, t0) = exp S [ xcl(τ)] D[δ x] exp δxiΛijδx j dτ . p( x, t, x , t ) = D[ x] exp L ( x(τ), x˙(τ)) dτ , (2) − − 2 0 0 − 0 t0 i, j=1 t0 By evaluating the functional integral from the exponential function with the quadratic form in N 1 d N − 1 exponent we obtain result for the PDF p. D[ x] = lim dxkj , xkN = xk(t), N d/2 →∞ j= k= j= (2π∆t) det G( x j 1, t j 1) 1 1 1 − − 4 Numerical results d 1 1 L ( x(τ), x˙(τ)) = G− ( x(τ),τ) [˙x A ( x(τ),τ)][x ˙ A ( x(τ),τ)], In this section we consider an application of the proposed method for the evaluation of the expectation 0 2 kj × k − k j − j k, j=1 of some functionals of the SDE system solutions d 1 ∂ dx1(t) = (a1 x1 + c1 √x1 x2 + b1) dt + σ1 √x1 dw1(t), Ak( x(τ),τ) = ak( x(τ),τ) gij( x(τ),τ) gkj( x(τ),τ), (4) − 2 ∂xi  i, j=1 dx2(t) = (a2 x2 + c2 √x1 x2 + b2) dt + σ2 √x2 dw2(t),  where G denotes a matrix of elements  x (t ) = x , x (t ) = x .  1 0 10 2 0 20 d The approximate and exact values of the expectation of some functionals of the SDE system Gij( x(τ),τ) = gik( x(τ),τ)g jk( x(τ),τ). solution (4) for concrete coefficients and initial conditions are presented in Tables 1 and 2. The form k=1 of the functionals is chosen in such way that we can calculate the exact values of the expectations. To compute the function xcl approximately we use the partitioning of the interval [t0, t] into n parts.

3 Evaluation of functional integrals Table 1. Approximate and exact values for expectations for a1 = a2 = 2, c1 = c2 = 2,σ1 = σ2 = 1, − b1 = b2 = 1/4, t0 = 0, t = 1, x10 = x20 = 1/4 The quantity L0( x(τ), x˙(τ)) in (2) can be considered as the Lagrangian of the system, and the quantity t E[ x ] E[ x ] E[(x x )] S = L0( x(τ), x˙(τ)) dτ can be considered as its action. Using the minimal action principle [13], √ 1 √ 2 1 2 t0 − Approximate values n = 40 0.4651 0.4651 0.00003 we can distinguish among all trajectories the classical trajectory xcl for which the action S takes the extremal value. The classical trajectory is found as the solution of the multidimensional Euler- Approximate values n = 120 0.4837 0.4837 0.00002 Lagrange equation Exact values 0.5 0.5 0 d ∂L ∂L 0 0 = 0, dt ∂x˙ − ∂x 5 Conclusion  1 1  .  . (3) A new method for evaluating of the characteristics of a SDE solution is proposed. From results   d ∂L0 ∂L0 presented in Tables 1 and 2 one can see that the increase of the parameter n leads to the decreasing  = 0. ﬀ dt ∂x˙d − ∂xd di erences between exact and approximate values.    3 EPJ Web of Conferences 173, 02003 (2018) https://doi.org/10.1051/epjconf/201817302003 Mathematical Modeling and Computational Physics 2017

Table 2. Approximate and exact values for expectations for a1 = a2 = 2, c1 = c2 = 2,σ1 = 9,σ2 = 10, − b1 = 81/4, b2 = 100/4, t0 = 0, t = 1, x10 = x20 = 4

E[√x ] E[√x ] E[(x x )] 1 2 1 − 2 Approximate values n = 40 6.037 10.72 11.17 − Approximate values n = 120 5.962 11.51 13.53 − Exact values 5.806 13.19 15.17 −

Acknowledgements This work was ﬁnancially supported by the Ministry of Education and Science of the Russian Fed- eration (the Agreement number 02.a03.21.0008) and by the Belarussian Republic’s Fund for Basic Research (project F16D-002).

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