On approximate solution of stochastic differential equations with retarded argument

T. A. Zamanov

Teoriâ veroâtnostej i ee primeneniâ, Tome 15 (1970) no. 1, pp. 145-148

Résumé

In a separable Hilbert space the stochastic differential equation $$ dx(t)=\{Ax(t)+K[t,x(t),x(t-\tau)]\}\,dt+\int_\Lambda F[t,\beta,x(t),x(t-\tau)]w(dt\times d\beta) $$ with the initial condition $$ x(t)=\varphi(t),\quad-\tau\le t\le0 $$ is given. Here: $\Lambda$ is a measurable space with a measure $\nu(d\beta)$ on the $\sigma$-algebra of measurable sets; $w(dt\times d\beta)$ is a Wiener stochastic measure on $[0,l]\times\Lambda$, satisfying the conditions 1–3; $A$ is a negative determined self-adjoint operator with a dense domain; the operators $K$ and $F$ satisfy the conditions \begin{gather*} \|K[t,x,u]-K[t,y,v]\|^2\le N[\|x-y\|^2+\|u-v\|^2], \\ \int_\Lambda\|F[t,\beta,x(t),x(t-\tau)]-F[t,\beta,y(t),y(t-\tau)]\|^2\nu(d\beta)\le N[\|x(t)-y(t)\|^2+ \\ +\|x(t-\tau)-y(t-\tau)\|^2]. \end{gather*} In the paper, convergence of Galërkin's approximations is proved.