Path independence of the additive functionals for McKean–Vlasov stochastic differential equations with jumps

Abstract

In this paper, the path independent property of additive functionals of McKean–Vlasov stochastic differential equations with jumps is characterized by nonlinear partial integro-differential equations involving [Formula: see text]-derivatives with respect to probability measures introduced by Lions. Our result extends the recent work16 by Ren and Wang where their concerned McKean–Vlasov stochastic differential equations are driven by Brownian motions.

Path independence of the additive functionals for stochastic differential equations driven by G-Lévy processes

“…By the Itô formula for G-Itô-Lévy processes to V (t, Y t ), we have (12). And then one can apply (10) to (12) to get (11). That is, F s,t is path independent in the sense of (7).…”

“…This is just right[11, (3)] without the distribution of Y r for r ∈ [s, t]. So, in the case Definition 2.11 and Theorem 3.2 are Definition 2.1 and Theorem 3.2 in[11] without the distribution of Y r for r ∈ [s, t], respectively. Therefore, our result overlaps[11, Theorem 3.2] in some sense.…”

Stochastic averaging principle for multi-valued McKean–Vlasov stochastic differential equations

“…Let us mention some related works. Qiao and Wu [24] proved that the path independent property of additive functionals of MVSDEs with jumps can be characterized by nonlinear partial integro-differential equations involving L-derivatives with respect to probability measures introduced by Lions. Liu et al [18] established large and moderate deviation principles for MVSDEs with jumps by applying the weak convergence method.…”

Coupled McKean-Vlasov stochastic differential equations with jumps

“…Song [26] observed the exponential ergodicity of MVSDEs with jumps. Qiao and Wu [20] demonstrated the path independence of additive functionals for MVSDEs with jumps.…”