Stochastic Integral Equations in Banach Spaces

Abstract

In this chapter, we first study the solutions of stochastic differential equations with non-Markovian Lipschitz condition and growth condition. In this case the drift and noise coefficients $a(t,Z)$ and $f(t,x,Z)$, being non-anticipating, depend on the path of the solution $Z$ $=(Z_t)_{t\in [0,T]}$. This is done by defining the equation on a probability space with $\Omega :=$ $D(\mathbb {R}_+,E)$, the space of càdlàg functions on $\mathbb {R}_+ \rightarrow E$, and with the $\sigma$-algebra generated by the cylinder sets of $D(\mathbb {R}_+,E)$, where $E$ is a separable Banach space.