Existence of weak solutions to stochastic differential equations in the plane with continuous coefficients

Abstract:

Let $B$ be a $2$-parameter Brownian motion on ${\mathbf {R}}_ + ^2$. Consider the nonMarkovian stochastic differential system in $2$-parameter \[ \left \{ {\begin {array}{*{20}{c}} {dX(z) = \alpha (z,X)\;dB(z) + \beta (z,X)\;dz} \hfill & {{\text {for}}\;z \in {\mathbf {R}}_ + ^2,} \hfill \\ {x(z) = \xi } \hfill & {{\text {for}}\;z \in \partial {\mathbf {R}}_ + ^2,} \hfill \\ \end {array} } \right .\] i.e., \[ \left \{ {\begin {array}{*{20}{c}} {X(z) = X(0) + \int _{{R_z}} {\alpha (\zeta ,X)\;dB(\zeta ) + \int _{{R_z}} {\beta (\zeta ,X)\;d\zeta } } } \hfill & {{\text {for}}\;z \in {\mathbf {R}}_ + ^2,} \hfill \\ {x(0) = \xi ,} \hfill & {} \hfill \\ \end {array} } \right .\] where ${R_z} = [0,s] \times [0,t]$ for $z = (s,t) \in {\mathbf {R}}_ + ^2$. An existence theorem for weak solutions of the system is proved in this paper. Under the assumption that $\alpha$ and $\beta$ satisfy a continuity condition and a growth condition and ${\mathbf {E}}[{\xi ^6}] < \infty$, it is shown that there exist a $2$-parameter stochastic process $X$ and a $2$-parameter Brownian motion $B$ on some probability space satisfying the stochastic integral equation above, with $X(0)$ having the same probability distribution as $\xi$.