On the existence of a strong solution of an Ito stochastic differential equation

I. V. Fedorenko

Teoriâ veroâtnostej i ee primeneniâ, Tome 29 (1984) no. 1, pp. 120-123

Résumé

It is shown that the scalar stochastic differential equation $$ x_t=x_0+\int_0^t A(s,x_s)\,ds+\int_0^t B(s,x_s)\,dw_s,\qquad 0\le t\le T, $$ has at least one strong solution under the following conditions: a) scalar functions $A(t,x)$ and $B(t,x)$ are continuous in both $t$, $x$ for $0\le t\le T$, $-\infty<x<\infty$; b) $B(t,x)$ satisfies a local Lipschitz conditions in $x$; c) $|A(t,x)|+ |B(t,x)|\le L(1+|x|)$ for some constant $L$ and all $t$, $x$; d) $\mathbf Mx_0^2<\infty$.