On the existence and uniqueness of solutions to stochastic equations in infinite dimension with integral-Lipschitz coefficients.
Let \(H\) be a separable Hilbert space, \(A\) an infinitesimal generator of a \(C_0\)-semigroup in \(H\), \(x\in H\) and \(B\) a cylindrical Hilbert space-valued Brownian motion. The authors study the stochastic differential equation (SDE) \[ dX_t= (AX_t+ b(t, X_t))\,dt+ \sigma(t, X_t)\,dW_t,\;t\geq 0,\;X_0= x \] for non-Lipschitz coefficients. In a first theorem they prove that under the so-called ``integral-Lipschitz assumption'' \[ | b(t,x)- b(t,x')|^2+ |\sigma(t,x)- \sigma(t,x')|^2\leq \beta_t\rho(| x- x'|^2) \] (\(\beta\in L^2_{\text{loc}}(R_+)\), \(\rho: R_+\to R_+\) increasing, continuous and concave such that \(\int_{0+}\rho(u)^{-1} \,du= +\infty\)) the equation possesses a unique mild solution. For this the authors borrow the method of \textit{J.-Y. Chemin} and \textit{N. Lerner} [J. Differ. Equations 121, 314--328 (1995; Zbl 0878.35089)] in order to simplify the method of successive approximation used by \textit{T. Yamada} in finite dimension [J. Math. Kyoto Univ. 21, 501--515 (1981; Zbl 0484.60053)] and translate it into the infinite-dimensional framework. The main result of the paper concerns mild existence and uniqueness under the even still weaker assumptions on \(b\) that \[ | b(t, x)- b(t, x')|\leq \beta^1_t \rho(| x- x'|),\quad \beta^1\in L^1_{\text{loc}}(R_+); \] the previous assumption on \(\sigma\) is preserved. The price of this generalization is that \(A\) is now supposed to be \(m\)-dissipative. In this case, the authors' proof of existence by successive approximation (using the Yosida approximation of \(A\)) is new, even for the finite-dimensional case. Their uniqueness result projected to the 1-dimensional case is slightly better than the well-known result of \textit{T. Yamada} and \textit{S. Watanabe} [ibid. 11, 155--167 (1971; Zbl 0236.60037)]. Finally, the last section of the paper is devoted to backward SDEs (in finite dimension) with an integral-Lipschitz assumption with respect to the variable \(y\).
stochastic differential equation in infinite dimension
backward stochastic differential equation
scientific article; zbMATH DE number 1998382