Short Communications
On approximate solution of stochastic differential equations with retarded argument
T. A. Zamanov Baku
Abstract:
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.
Received: 19.08.1968