Malliavin calculus for marked binomial processes and applications. (English) Zbl 1511.60079
Summary: We develop stochastic analysis tools for marked binomial processes (MBP) that are the discrete analogues of the marked Poisson processes. They include in particular: (i) the statement of a chaos decomposition for square-integrable functionals of MBP, (ii) the design of a tailor-made Malliavin calculus of variations, (iii) the statement of the analogues of Stroock’s, Clark’s and Mehler’s formulas. We provide our formalism with two applications: (App1) studying the (compound) Poisson approximation of MBP functional by combining it with the Chen-Stein method and (App2) solving an optimal hedging problem in the trinomial model.
MSC:
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
| 60F05 | Central limit and other weak theorems |
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
| 60J74 | Jump processes on discrete state spaces |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
Keywords:
chaos expansion; Chen-Stein method; Malliavin calculus; optimal hedging; Poisson limit theorems; trinomial market modelReferences:
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