Symmetric stochastic differential equations with nonsmooth coefficients

V. Mackevičius

Sbornik. Mathematics, Tome 44 (1983) no. 4, pp. 527-534

Résumé

The concept of a solution of a symmetric stochastic equation $$ X_t=x+\int^t_0\sigma(s,X_s)\circ dB_s+\int^t_0b(s,X_s)\,ds,\qquad t\geqslant0, $$ is generalized to the case when the coefficient $\sigma=\sigma(t,x)$, $(t,x)\in\mathbf R_+\times\mathbf R$, is continuous and continuously differentiable with respect to $t$, i.e., $\sigma\in C^{1,0}$. Here $B_t$, $t\geqslant0$, is a one-dimensional Brownian motion, and the stochastic integral is understood in the symmetric sense (in the sense of Stratonovich). Sufficient conditions are obtained for the existence and uniqueness of a solution, and the stability of a solution under perturbations of the coefficients is investigated. Bibliography: 14 titles.