2010/05/25 by Łukasz Delong, Delong, Łukasz, Peter Imkeller +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Geometric Analysis and Curvature Flows #Mathematical Biology Tumor Growth #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1005.4702
openalex publication_date 2010/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate solutions of backward stochastic differential equations (BSDE) with time delayed generators driven by Brownian motions and Poisson random measures, that constitute the two components of a Levy process. In this new type 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 BSDE, we prove 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.