Itô Stochastic Differential Equations with Strong Solutions

* A Brief Introduction to Stochastic Differential Equations *

Itô Stochastic Differential Equations with Strong Solutions

Itô stochastic differential equations with m Wiener processes take the form

$\begin{eqnarray} dX^j_t&=&a^j({\bf X}_t,t)~dt+\sum_{k=1}^mb^j_k({\bf X}_t,t)~dW^k_t,\nonumber \end{eqnarray}$

where j = 1,..., n. We assume that the coefficients $a^j({\bf X}_t,t)$ and $b^j_k({\bf X}_t,t)$ have regularity properties which guarantee that X^j_t are some fixed functions of the Wiener processes i.e. X^j_t = X_j(t, W¹_t,..., W^m_t), and that they are differentiable to high order. Such equations are said to have strong solutions. Other types of sde exist and we briefly comment on these equations below.

* Innovative Stochastic Algorithms *