On the existence and uniqueness of solutions to stochastic differential equations of mixed Brownian and Poissonian sheet type.
Let \((W_ {z}\), \(z\in \mathbb R^ 2_ {+})\) be a Brownian sheet, and \(Y\) a Poisson random measure on \(\mathbb R^ 2_ {+}\) such that \(\mathbf E Y(A) = \Lambda (A)\) for Borel sets \(A\) in \(\mathbb R^ 2_ {+}\), the measure \(\Lambda \) having a density with respect to Lebesgue measure. Set \(N = Y-\Lambda \) and for \(z = (z_ 1,z_ 2) \in \mathbb R^ 2_ {+}\) denote by \(R_ {z}\) the rectangle \(] 0, z_ 1]\times ] 0,z_ 2]\). A stochastic differential equation in the plane \[ X_ {z} = X_ {0}+ \int _ {R_ {z}} f_ 1(\zeta ,X_ \zeta )\,d\zeta + \int _ {R_ {z}} f_ 2(\zeta ,X_ \zeta )\,dW_ \zeta + \int _ {R_ {z}} f_ 3(\zeta ,X_ \zeta )N(d\zeta )+ \int _ {R_ {z}} f_ 4(\zeta ,X_ \zeta )\Lambda (d\zeta ) \tag{1} \] is studied under the following hypothesis: \(f_ {i}: \mathbb R^ 2_ {+}\times \mathbb R\to \mathbb R\), \(1\leq i\leq 4\), are Borel functions satisfying \(| f_ {i}(\zeta ,u)-f_ {i}(\zeta ,v)| \leq \rho _ {i}(| u-v| )\) for all \(\zeta \in \mathbb R^ 2_ {+}\), \(u,v\in \mathbb R\), where \(\rho _ {i}: \mathbb R_ {+}\to \mathbb R_ {+}\) are concave nondecreasing functions such that \(\rho _ {i}(0)=0\), \(\rho _ {i}>0\) on \(\mathopen ]0,\infty \mathclose [\), and \(\int _ {0+} (\sum ^ 4_ {i=1}\rho ^ 2_ {i}(u))^ {-1} u\, du = \infty \). The method of successive approximations is used to show that there exists a unique solution to (1). Furthermore, equations with more general driving martingales are briefly treated.
two-parameter stochastic differential equations
scientific article; zbMATH DE number 2119519