Sandra Cerrai (eds.)354042136X, 9783540421368
The main objective of this monograph is the study of a class of stochastic differential systems having unbounded coefficients, both in finite and in infinite dimension. We focus our attention on the regularity properties of the solutions and hence on the smoothing effect of the corresponding transition semigroups in the space of bounded and uniformly continuous functions. As an application of these results, we study the associated Kolmogorov equations, the large-time behaviour of the solutions and some stochastic optimal control problems together with the corresponding Hamilton- Jacobi-Bellman equations. In the literature there exists a large number of works (mostly in finite dimen sion) dealing with these arguments in the case of bounded Lipschitz-continuous coefficients and some of them concern the case of coefficients having linear growth. Few papers concern the case of non-Lipschitz coefficients, but they are mainly re lated to the study of the existence and the uniqueness of solutions for the stochastic system. Actually, the study of any further properties of those systems, such as their regularizing properties or their ergodicity, seems not to be developed widely enough. With these notes we try to cover this gap. |
Table of contents :
Introduction….Pages 1-18
Kolmogorov equations in Rd with unbounded coefficients….Pages 21-63
Asymptotic behaviour of solutions….Pages 65-80
Analyticity of the semigroup in a degenerate case….Pages 81-101
Smooth dependence on data for the SPDE: the Lipschitz case….Pages 105-141
Kolmogorov equations in Hilbert spaces….Pages 143-170
Smooth dependence on data for the SPDE: the non-Lipschitz case (I)….Pages 171-203
Smooth dependence on data for the SPDE: the non-Lipschitz case (II)….Pages 205-220
Ergodicity….Pages 221-235
Hamilton- Jacobi-Bellman equations in Hilbert spaces….Pages 237-279
Application to stochastic optimal control problems….Pages 281-300 |
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