Existence and pathwise uniqueness of solutions for stochastic differential equations with respect to martingales in the plane
A stochastic differential equation \[ X_{z} = Z_{(s,0)} + Z_{(0,t)} - Z_{(0,0)} + \int _{[0,z]} \alpha (v,X_{v}) dM_{v} + \int _{[0,z]}\beta (v,X_{v}) dA_{v}, \tag{1} \] \(z = (s,t) \in \mathbb R^{2}_{+}\), for a two-parameter stochastic process \(X\) is studied. It is assumed that \(M= (M_{z}, z\in \mathbb R^{2}_{+})\) is a continuous square integrable martingale, \([M]^1 = [M]^2 = \langle M \rangle \), \(A\) is a continuous increasing process, \(M = A = 0\) on \(\partial \mathbb R^{2}_{+}\), and \(Z\) is an adapted continuous process on the boundary \(\partial \mathbb R^{2}_{+}\). In a previous paper [ibid. 62, No. 2, 263-276 (1996; Zbl 0863.60059)], the author and \textit{M. Zheng} proved existence theorems for the problem (1), which are now amended with pathwise uniqueness results of Yamada-Watanabe type [see \textit{T. Yamada} and \textit{S. Watanabe}, J. Math. Kyoto Univ. 11, 155-167 (1971; Zbl 0236.60037)]. Namely, one of the two main theorems of the paper reads as follows: Let \(\rho , k: [0,\infty [\to [0, \infty [\) be strictly increasing concave functions, \(\rho (0) = k(0) = 0\), such that \(|\alpha (v,x)-\alpha (v,y)|\leq \rho (|x-y|)\), \(|\beta (v,x)-\beta (v,y)|\leq k(|x-y|)\) for all \(v\in \mathbb R^{2}_{+}\), \(x,y\in \mathbb R\). Suppose that \(\int _{0+} \frac {u}{k^{2}(u) + \rho ^{2}(u)} du = +\infty\). Let there exist a nonnegative continuous function \(B\) on \(\mathbb R^{2}_{+}\) which is a distribution function of a measure dominating the random measure generated by \(A\) and \(\langle M\rangle \), \(d\langle M\rangle _{v} + dA_{v} \leq dB_{v}\). Then pathwise uniqueness holds for the equation (1).
stochastic differential equations in the plane
scientific article; zbMATH DE number 1792543
scientific article