The symbol associated with the solution of a stochastic differential equation

Abstract

Let (Zt)t≥0 be an ℝn-valued Lévy process. We consider stochastic differential equations of the dXxt = Φ(Xtt-)dZt Xx0 = x, x ∈ ℝd, where Φ: ℝd → ℝd×n is Lipschitz continuous. We show that the infinitesimal generator of the solution process (Xxt)t≥0 is a pseudo-differential operator whose symbol p : ℝd × → ℝd → ℂ can be calculated by p(x, ξ) ≔ -limt↓0𝔼x(ei(Xσt-x)⊤ ξ - 1/t). For a large class of Feller processes many properties of the sample paths can be derived by analysing the symbol. It turns out that the process (Xxt)t≥0 is a Feller process if Φ is bounded and that the symbol is of the form p(x, ξ) = ψ(Φ⊤(x)ξ), where ψ is the characteristic exponent of the driving Lévy process. © 2010 Applied Probability Trust.