The Lévy–Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups

Abstract

Ito's construction of Markovian solutions to stochastic equations driven by a Lévy noise is extended to nonlinear distribution dependent integrands aiming at the effective construction of linear and nonlinear Markov semigroups and the corresponding processes with a given pseudo-differential generator. It is shown that a conditionally positive integro-differential operator (of the Lévy-Khintchine type) with variable coefficients (diffusion, drift and Lévy measure) depending Lipschitz continuously on its parameters (position and/or its distribution) generates a linear or nonlinear Markov semigroup, where the measures are metricized by the Wasserstein-Kantorovich metrics. This is a non-trivial but natural extension to general Markov processes of a long known fact for ordinary diffusions.

Integration by parts formula and applications to equations with jumps

“…We refer to [4,[14][15][16][17][18] for the links between Markov processes and pseudo-differential operators.…”

Continuously Monitored Barrier Options Under Markov Processes

“…Example 2.5 More generally, one may specify the price process S by directly prescribing its generator to act on sufficiently regular functions f as where is given in () and where for every , is a (Lévy) measure with support in (−1, ∞) such that The discounted process {e −γ t S t } t ≥ 0 is a local martingale. Sufficient conditions on σ and ν to guarantee the existence of a Feller process S corresponding to this generator were established in Kolokoltsov (2009, theorem 1.1).…”

3.1 Nonlinear Lévy and Nonlinear Feller Processes: an Analytic Introduction

“…which implies the first estimate in (25) first for small times, which is then extended to all finite times by the iteration. The second estimate in (25) follows from (8).…”