Gradient formula for transition semigroup corresponding to stochastic equation driven by a system of independent Lévy processes

2020/06/16 by Alexei Kulik, Kulik, Alexei, Szymon Peszat +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #60G51 #60H07 #60H10 #60J75 #FOS: Mathematics #Nonlinear Differential Equations Analysis #Probability (math.PR) #Stochastic processes and financial applications #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2006.09133

openalex publication_date 2020/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (P_t) be the transition semigroup of the Markov family (X^x(t)) defined by SDE d X= b(X) dt + d Z, X(0)=x, where Z=(Z₁, …, Z_d)^* is a system of independent real-valued Lévy processes. Using the Malliavin calculus we establish the following gradient formula ∇ P_tf(x)= 𝔼 f(X^x(t)) Y(t,x), f∈ B_b(ℝ^d), where the random field Y does not depend on f. Sharp estimates on ∇ P_tf(x) when Z₁, … , Z_d are α-stable processes, α∈ (0,2), are also given.

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