Algorithmic Solution of Stochastic Differential Equations

This brief note presents an algorithm to solve ordinary stochastic differential equations (SDEs). The algorithm is based on the joint solution of a system of two partial differential equations and provides strong solutions for finite-dimensional systems of SDEs driven by standard Wiener processes and with adapted initial data. Several examples illustrate its use.


Introduction
In general, it is hard to find explicit expression for closed-form solutions for stochastic differential equations (SDEs), see [1][2][3][4][5][8][9][10][11][12]16] to name just a few monographs.In none of the aforementioned monographs one can find a general algorithm to find such solutions.Nowadays, sophisticated computer packages provide powerful tools to seek for solutions, e.g., MAPLE, MATHEMATICA, MATLAB, etc.Our aim is to provide a general algorithm to find explicit expression for the strong solution of initial value problems related to systems of finite-dimensional ordinary SDEs driven by independent Wiener processes W j on the filtered probability space (Ω, F, (F t ) t≥0 , P) which is completed with respect to all P-null sets.The drift f : [0, T ] × R d → R d and diffusion coefficients g j : [0, T ] × R d → R d are supposed to be Borel-measurable functions such that the strong solution of the IVP (1) exists (a.s.).The initial value X 0 in (1) is completely independent of the increments of all Wiener processes W j for t ≥ 0. The algorithm is applicable to both Itô-and Stratonovich-interpreted SDEs (1).It is mainly based on the fundamental theorem of stochastic calculus which originates from Itô's works [6,7].The algorithm can be extended to SDEs integrated in the sense of α-calculus, see [13,14].For the readers, the readership is supposed to be familiar with the basic concepts of stochastic calculus and real analysis (e.g., for an excellent survey, see [15]).
The paper is organized in 3 sections as follows.After this short introduction, Section 2 presents the algorithm.In Section 3 we discuss its application to solve numerous examples where the solution is and is not known from the literature.An appendix states the fundamental theorem of stochastic calculus, also known as Itô formula, and a theorem on the existence of unique strong solutions of linear SDEs.

The Linear PDE-Based Algorithm
Suppose the SDE with sufficiently smooth coefficients f , g j is solvable (in the strong sense).Define the partial differential operators L j by where j = 1, 2, ..., m, 0 ≤ t ≤ T and x ∈ R d .In passing, note that L 0 is called the infinitesimal generator of SDE (1) and occurs in several fundamental theorems of stochastic analysis (cf.Itô formula in appendix).

The algorithm for Itô SDEs
[1] Find the general solution F = F (t, x) of the PDE-problem with appropriate constants a 0 , a 1 ∈ R.
[2] Find the general solution F = F (t, x) of the PDE-problems with appropriate constants b j 0 , b j 1 ∈ R.
[3] Consider all common solutions F = F (t, x) of steps [1] and [2].Check whether there are invertible solutions F with respect to x.If there are no common, invertible solutions then the algorithm stops and one knows that there are no invertible transformations F to linear SDEs (6) which form common solutions of linear PDE-problems ( 4) and ( 5), otherwise one continues with step [4].
[4] Solve the linear SDE (or look up its solution in literature) with U (0) = F (0, X 0 ) and coefficients a i and b j i from steps [1] and [2].
[5] Let F −1 be the inverse of invertible common solution with respect to x from step [3].Define the final solution of the algorithm by for 0 ≤ t ≤ T .Note that the linear system (6) has always a strong solution (as stated by Theorem 2 in Section 3.3).The easiest choices are when all b j 0 = 0 or all b j 1 = 0.One is also tempted to seek for solutions with a 0 = b j 0 = 0 or a 1 = b j 1 = 0.These are the cases of pure homogeneous or non-state-dependent equations for U , respectively.However, there is no guarantee that one of this special choices works in a closed form (series solutions can be observed).If the algorithm stops with no found solution then one might also try to find common solutions to nonlinear PDE-problems -which is a much harder task to solve (however, not impossible in certain special cases).

The algorithm for Stratonovich SDEs
The linear PDE-based algorithm for Stratonovich based SDEs (1) is very similar to that of Itô SDEs.One only has to exchange the generator L 0 by the 1st order partial differential operator in step [1].

Examples
With a series of fairly simple examples, we shall demonstrate the systematic applicability of our algorithm.

Linear equations: geometric Brownian motion
Most commonly cited example in mathematical finance is that of geometric Brownian motion satisfying the Itô equation with σ = 0. Let us go through the steps [1] − [5] of the algorithm.The simplest choice is to transform to the Wiener process dX(t) = dW (t) itself.For this purpose, in step [1] we solve One finds that its solution is given by with real constant F 0 = 0, since In step [2], one needs to solve One easily checks that F (t, x) defined as above represents a common solution of steps [1] and [2].Moreover, F (t, x) is invertible with respect to x and its inverse is where F 0 is any real constant.Thus, we can successfully proceed through step [3].
Step [4] brings up the trivial solution X(t) = W (t) of the transformed equation dX(t) = dW (t).Finally, the final answer is provided by step [5] and is given by with the choice F 0 = X(0) to match the initial data correctly.

Bernoulli-type and logistic equations
Consider stochastic Bernoulli equations which are interpreted in Itô sense.The most popular representative of these equations is the case with n = 2, also called the logistic growth model with growth rate r > 0 and environmental capacity K > 0. These equations have numerous applications in mathematical biology (e.g., to model the growth of populations).For simplicity, we suppose that X(0) > 0 (i.e., we seek only for physically meaningful solutions and additionally avoid technical difficulties of mathematical stopping procedures).To carry out the procedure of the algorithm, we begin with step [1] solving Suppose that n = 1.By choosing the parameters one recognizes that the (deterministically known) Leibniz transform represents a solution of step [1].
Step [2] asks for the calculation of hence we may choose the constants b 1 0 = 0 and b 1 1 = σ(1 − n).Clearly, F (x) = x 1−n is a common invertible solution of both step [1] and [2].Thus, step [3] provides the solution F (x) = x 1−n with its inverse Step [4] takes into consideration the linear Itô equation which can be solved by Theorem 2 from the appendix.Its solution is where with t ≥ 0. Finally, step [5] leads to the solution of ( 10)

A nonlinear example (pure Stratonovich diffusions)
Pure Stratonovich diffusions are governed by Stratonovich SDEs with differentiable coefficients σ(x) ≥ 0. For our analysis, we suggest to transfer this equation to its equivalent Itô SDE (see [2]) Now, we apply our algorithm.For this purpose, it is interesting to observe that the transform has vanishing operator image for L 0 .By chain rule, we find that Moreover, one encounters Therefore, steps [1], [2] and [3] with constants a 0 = a 1 = b 1 1 = 0 and b 1 0 = 1 can be successfully conducted, and its common solution is of the form (13) with existing inverse F −1 (invertible since σ ≥ 0).The trivial solution of step [4] is U (t) = W (t) with U (0) = F (X(0)).Eventually, step [5] yields the solution As a fairly simple sub-example, one can treat with X(0) ≥ 0, which is also known as Bessel-type diffusion with dimension parameter m > 0. Here, . Therefore, the algorithm gives the positive solution An extension of the algorithm to a nonlinear-PDE-based version is conceivable and subject to future research.For example, Equation (4) in step [1] and (5) in step [2] can be substituted by nonlinear systems of PDEs L 0 F = G 0 (t, x, F ), L j F = G j (t, x, F ) with appropriate nonlinear functions G 0 and G j , respectively.However, to find explicit expressions for common solutions F of m + 1-dimensional systems of nonlinear PDEs such as (17) (i.e.G 0 or G j nonlinear in F , j = 1, 2, ..., m) seems to be an extremely difficult field of research.
c 2010 by the author; licensee MDPI, Basel, Switzerland.This article is an Open Access article distributed under the terms and conditions of the Creative Commons Attribution license http://creativecommons.org/licenses/by/3.0/.