\(\mathbb{L}^{p}\) solutions of reflected backward stochastic differential equations with jumps. (English) Zbl 1454.60084
Summary: Given \(p\in (1, 2)\), we study \(\mathbb{L}^{p}\)-solutions of a reflected backward stochastic differential equation with jumps (RBSDEJ) whose generator \(g\) is Lipschitz continuous in \((y, z, u)\). Based on a general comparison theorem as well as the optimal stopping theory for uniformly integrable processes under jump filtration, we show that such a RBSDEJ with \(p\)-integrable parameters admits a unique \(\mathbb{L}^{p}\) solution via a fixed-point argument. The \(Y\)-component of the unique \(\mathbb{L}^{p}\) solution can be viewed as the Snell envelope of the reflecting obstacle \(\mathfrak{L}\) under \(g\)-evaluations, and the first time \(Y\) meets \(\mathfrak{L}\) is an optimal stopping time for maximizing the \(g\)-evaluation of reward \(\mathfrak{L}\).
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60F25 | \(L^p\)-limit theorems |
| 60J76 | Jump processes on general state spaces |
Keywords:
reflected backward stochastic differential equations with jumps; \(\mathbb{L}^{p}\) solutions; comparison theorem; optimal stopping; Snell envelope; Doob-Meyer decomposition; martingale representation theorem; fixed-point argument; \(g\)-evaluationsReferences:
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