On Malliavin's differentiability of BSDEs with time delayed generators driven by Brownian motions and Poisson random measures
By Lukasz Delong, Peter Imkeller. 2010
87 citations recorded by the source corpus.
We investigate solutions of backward stochastic differential equations (BSDEs) with time delayed generators driven by Brownian motions and Poisson random measures, that constitute the two components of a Levy process. In these new types of equations, the generator can depend on the past values of a solution, by feeding them back into the dynamics with a time lag. For such time delayed BSDEs, we prove the existence and uniqueness of solutions provided we restrict on a sufficiently small time horizon or the generator possesses a sufficiently small Lipschitz constant. We study differentiability in the variational or Malliavin sense and derive equations that are satisfied by the Malliavin gradient processes. On the chosen stochastic basis this addresses smoothness both with respect to the continuous part of our Levy process in terms of the classical Malliavin derivative for Hilbert space valued random variables, as well as with respect to the pure jump component for which it takes the form of an increment quotient operator related to the Picard difference operator.
Subfields: Computational Theory and Mathematics, Finance, Modeling and Simulation
View Paper: https://www.semanticscholar.org/paper/0d6702b47083b38d219cce685b2013215041ed21
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In Computational Theory and Mathematics
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Citations in the corpus
- Applications of time-delayed backward stochastic differential equations to pricing, hedging and portfolio management in insurance and finance (2010, methodology): “The unique solution derived under Theorem 2.1 of Delong and Imkeller (2010a) and Theorem 2.1 of Delong (2011) is X = π = 0 and it is not clear at the moment how to derive a non-zero solution (if any).”