Abstract
We provide necessary and sufficient first order geometric conditions for the stochastic invariance of a closed subset of $\mathbb{R} ^d$ with respect to a jump-diffusion under weak regularity assumptions on the coefficients. Our main result extends the recent characterization proved in Abi Jaber, Bouchard and Illand (2016) to jump-diffusions. We also derive an equivalent formulation in the semimartingale framework.
Highlights
We consider a weak solution to the following stochastic differential equation with jumps dXt = b(Xt)dt + σ(Xt)dWt + ρ(Xt−, z) (μ(dt, dz) − Fdt), X0 = x, (1.1)
That is: a filtered probability space (Ω, F, F = (F )t≥0, P) satisfying the usual conditions and supporting a d-dimensional Brownian motion W, a Poisson random measure μ on R+ × Rd with compensator dt ⊗ F, and a F-adapted process X with càdlàg sample paths such that (1.1) holds P-almost surely
We provide an equivalent formulation of the stochastic invariance with respect to semimartingales in Theorem 3.2
Summary
Throughout this paper, we assume that b : Rd → Rd, σ : Rd → Md and ρ : Rd × Rd → Rd are measurable, where Md denotes the space of d × d matrices. Note that a first order characterization for a smooth volatility matrix σ is given in [14], where the Stratonovich drift appears (see [9] for the diffusion case). Combining the techniques used in [1, 24], we derive for the first time in Theorem 2.2 below, a first order geometric characterization of the stochastic invariance with respect to (1.1) when the volatility matrix σ can fail to be differentiable. We provide an equivalent formulation of the stochastic invariance with respect to semimartingales in Theorem 3.2 This extends [1] to the jump-diffusion case. In the Appendix, we adapt to our setting some technical results, mainly from [1]
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