On the existence of solutions of stochastic differential equations of hyperbolic type with coefficients depending on ”the past”. (Russian) Zbl 0531.60059

Kislenko, S. S.

Teor. Veroyatn. Mat. Stat. 28, 40-51 (1983).

The stochastic differential equation with random coefficients \[ \frac{\partial^ 2\xi(s,t)}{\partial s\partial t}=a(s,t,\xi(\cdot))+B_ 1(s,t,\xi(\cdot))\frac{\partial^ 2M(s,t)}{\partial s\partial t}+ \]
\[ B_ 2(s,t,\xi(\cdot))\frac{\partial M(s,t)}{\partial s}+B_ 3(s,t,\xi(\cdot))\frac{\partial M(s,t)}{\partial t} \] in the domain \(T=[0,1]\times [0,1]\) with the boundary conditions \(\xi(s,0)=p_ 1(s), \xi(0,t)=p_ 2(t), p_ i(0)=\xi_ 0\), \(i=1,2\), is studied. Here (M(z),\({\mathfrak F}'_ z\), \(z\in T)\) is a continuous (a.s.) square integrable martingale field with values in \(R^ m\) and the family (\({\mathfrak F}'_ z)\) satisfies Cairoli-Walsh conditions. The notion of weak solution is defined. Theorems of existence and uniqueness of a solution under Lipschitz type conditions and of existence (without uniqueness) under continuous type conditions are proved.

Reviewer: A.Yu.Veretennikov

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60G60	Random fields
60H20	Stochastic integral equations

Keywords:

Cairoli-Walsh conditions; weak solution; existence and uniqueness of a solution