Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise

By Sergio Albeverio, V. Mandrekar, B. Rüdiger. 2009

80 citations recorded by the source corpus.

Existence and uniqueness of the mild solutions for stochastic differential equations for Hilbert valued stochastic processes are discussed, with the multiplicative noise term given by an integral with respect to a general compensated Poisson random measure. Parts of the results allow for coefficients which can depend on the entire past path of the solution process. In the Markov case Yosida approximations are also discussed, as well as continuous dependence on initial data, and coefficients. The case of coefficients that besides the dependence on the solution process have also an additional random dependence is also included in our treatment. All results are proven for processes with values in separable Hilbert spaces. Differentiable dependence on the initial condition is proven by adapting a method of S. Cerrai.

Subfields: Applied Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/678f112e604b36616b11775bebd289eeb6cb1f6d

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Citations in the corpus

Exponential ergodicity and regularity for equations with Lévy noise (2011, background): “, see the papers [4, 2, 28, 26, 20, 32, 41], the book [27], and the references therein.”
Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise (2008, background): “The following well-posedness result in H2(T ) is quite simple to prove and it essentially relies only on the isometric formula for stochastic integrals with respect to compensated Poisson measures (see also [1, 24] for similar results).”
LOCAL WELL‐POSEDNESS OF MUSIELA’S SPDE WITH LÉVY NOISE (2007, unknown): “…Compilation C© 2010 Wiley Periodicals, Inc. 341 and Pistone 1976 and Gyöngy 1982 for the variational approach, Lescot and Röckner 2004 for an analytic approach based on generalized Mehler semigroups, and Albeverio, Mandrekar, and Rüdiger 2009; Peszat and Zabczyk 2006 for the “mild” approach).”