Numerical Integration of Stochastic Differential Equations

Using stochastic differential equations we can successfully model systems that function in the presence of random perturbations. Such systems are among the basic objects of modern control theory. However, the very importance acquired by stochastic differential equations lies, to a large extent, in the strong connections they have with the equations of mathematical physics. It is well known that problems in mathematical physics involve ‘damned dimensions’, often leading to severe difficulties in solving boundary value problems. A way out is provided by stochastic equations, the solutions of which often come about as characteristics.

Let (Ω, F, P) be a probability space, let F _t , t ₀ ≤ t ≤ t ₀ + T, be a nonincreasing family of σ-subalgebras of F, and let (w _r(t), F _t ), r = 1,...,q, be independent Wiener processes. Consider the system of stochastic differential equations in the sense of Itô where X, a, σ_r are vectors of dimension n.

In the numerical integration formulas used in the Taylor-type expansion of solutions of systems of stochastic equations (see §2) the repeated Itô integrals appeared, where i ₁,…, i _j take values from the set {0,1,…, q}, and dw ₀(θ _r) is understood to mean dθ _r .

As already mentioned in the Introduction, in cases when the modeling of solutions is intended for the application of Monte-Carlo methods we can refrain from mean-square approximations and use approximations that are in may respect simpler: weak approximations of solutions.

Numerical methods for Wiener integrals are expounded in the books [13], [14], and [45] (see also the references in these books).