Abstract
We consider some stochastic difference partial differential equations of the form
du(x,t,c)=L(x,t,D)u(x,t,c)dt+M(x,t,D)u(x,t−a,c)dw(t), where
L(x,t,D) is a linear uniformly elliptic partial differential operator of the second order,
M(x,t,D) is a linear partial differential operator of the first order, and w(t) is a Weiner process. The existence and uniqueness
of the solution of suitable mixed problems are studied for the
considered equation. Some properties are also studied. A more
general stochastic problem is considered in a Hilbert space and
the results concerning stochastic partial differential equations
are obtained as applications.