Bismut formula for Lions derivative of distribution-path dependent SDEs
By Jianhai Bao, Panpan Ren, Feng-Yu Wang. Journal of Differential Equations, 2020
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To characterize the regularity of distribution-path dependent SDEs in the initial distribution which varies as probability measure on the path space, we introduce the intrinsic and Lions derivatives for probability measures on Banach spaces, and prove the chain rule of the Lions derivative for the distribution of Banach-valued random variables. By using Malliavin calculus, we establish the Bismut type formula for the Lions derivatives of functional solutions to SDEs with distribution-path dependent drifts. When the noise term is also path dependent so that the Bismut formula is invalid, we establish the asymptotic Bismut formula. Both non-degenerate and degenerate noises are considered. The main results of this paper generalize and improve the corresponding ones derived recently in the literature for the classical SDEs with memory and McKean-Vlasov SDEs without memory.
Subfields: Economics and Econometrics, Finance, Management Science and Operations Research
View Paper: https://www.semanticscholar.org/paper/e695ae3cfebeffad2b0dba532e7becebd9fcd5d4
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Citations in the corpus
- Distribution dependent SDEs driven by fractional Brownian motions (2021, methodology): “, [5, 29]), when taking the special choices of b((1)), b((2)) and σ, the above model will reduce to distribution dependent stochastic Hamiltonian system with fractional noise.”