Abstract
We study a class of stochastic partial integral-differential equations with an asymmetrical non-local operator $\frac{1} {2}\Delta +a^{\alpha }\Delta ^{\frac{\alpha } {2}}+b\cdot \nabla $ and a distribution expressed as divergence of a measurable field. For $0<\alpha <2$, the existence and uniqueness of solution is proved by analytical method, and a probabilistic interpretation, similar to the Feynman-Kac formula, is presented for $ 0<\alpha <1$. The method of backward doubly stochastic differential equations is also extended in this work.
Highlights
We consider the following stochastic partial integral-differential equation in this article ←−dut(x) + [Aut(x) + ft(x, ut(x), ∇ut(x))] dt + ht(x, ut(x), ∇ut(x)) dBt+ divgt(x, ut(x), ∇ut(x)) dt = 0, (t, x) ∈ [0, T ] × Rd; uT (x) = Φ(x), x ∈ Rd. (1.1) For constants a > and < α2, the non-local operator
We study a class of stochastic partial integral-differential equations with an asymmetrical non-local operator and a distribution expressed as divergence of a measurable field
Inspired by a serial of works on dealing with the singular term divg, we consider a stochastic PIDE with non-local operator that is associated with a perturbed Lévy process (Xt)t≥0
Summary
We consider the following stochastic partial integral-differential equation in this article. This is a generalization of the result in [5], where only symmetric operator was considered. Inspired by a serial of works (see, for example, [5, 8, 10, 18]) on dealing with the singular term divg, we consider a stochastic PIDE with non-local operator that is associated with a perturbed Lévy process (Xt)t≥0. The last section is devoted to building the connection between stochastic PIDEs and BDSDEs
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