Propagation of chaos and conditional McKean-Vlasov SDEs with regime-switching

Abstract

This work concerns the optimal control problem for McKean-Vlasov SDEs. In order to characterize the value function, we develop the viscosity solution theory for Hamilton-Jacobi-Bellman (HJB) equations on the Wasserstein space using Mortensen's derivative. In particular, a comparison principle for viscosity solution is established. Our approach is based on Borwein-Preiss variational principle to overcome the loss of compactness for bounded sets in the Wasserstein space.

Well‐posedness of diffusion–aggregation equations with bounded kernels and their mean‐field approximations

“…Applying the superposition principle [17, Theorem 4.1], we find a weak solution to $$$$ \mathrm{d}{\tilde{Y}}_t^{\epsilon }=-\left({k}^{\epsilon}\ast {\rho}_t^{\epsilon}\right)\left({\tilde{Y}}_t^{\epsilon}\right)\mathrm{d}t+\sigma \mathrm{d}{B}_t^1,\kern6.05pt {\tilde{Y}}_0^N\sim {\rho}_0,\kern0.30em t\in \left[0,T\right], $$$$ with $$$ \mathrm{Law}\left({\tilde{Y}}_t^{\epsilon}\right)={\tilde{\rho}}_t^{\epsilon}\mathrm{d}x $$$. Since strong uniqueness holds [33, Theorem 4.10] for the above SDE, we have $$$ {\tilde{Y}}^{\epsilon }={Y}^{\epsilon } $$$. By the Yamada–Watanabe theorem [53, Chapter 5, Proposition 3.20], this implies uniqueness in law, and therefore, $$$$ {\tilde{\rho}}_t^{\epsilon}\mathrm{d}x={\rho}_t^{\epsilon}\mathrm{d}x,\kern0.30em t\in \left[0,T\right], $$$$…”

“…A drawback of the modulated free energy approach is that it requires the existence of an entropy solution on the particle level (microscopic level), see [29, Proposition 4.2], which is nontrivial outside a setting on the torus. Further results on propagation of chaos were proven for general $$$ {L}^p $$$‐interaction force kernels $$$ k $$$ for first‐ and second‐order systems on the torus [32] and on the whole space $$$ {\mathrm{\mathbb{R}}}^d $$$ [33–35]. For instance, [35] provides optimal bounds on the relative entropy of order $$$ \mathcal{O}\left({k}^2/{N}^2\right) $$$ by exploiting the BBGKY‐hierarchy combined with delicate estimates on the error of iterations.…”

Coupling by Change of Measure for Conditional McKean-Vlasov SDEs and Applications

“…When there exists a common noise in the mean field particle system, the limit equation of a single particle becomes a conditional distribution dependent stochastic differential equation, which is called conditional McKean-Vlasov stochastic differential equation and the conditional distribution of the solution with respect to the common noise as a probability measure-valued process solves stochastic nonlinear Fokker-Planck-Kolmogorov equation. Compared with the McKean-Vlasov SDEs, there are fewer results on conditional McKean-Vlasov ones, one can refer to [1,3,4,5,10,2,13,7,16,17,21] for well-posedness, [13,7,12] for the study of stochastic nonlinear Fokker-Planck-Kolmogorov equation and [2,6,8,12,18,1,16,17] for conditional propagation of chaos.…”

Conditional McKean-Vlasov SDEs with jumps and Markovian regime-switching: wellposedness, propagation of chaos, averaging principle

“…[19] also showed the law of large number of a meanfield interaction system with Markovian regime-switching, which deduces the conditional distribution L(Y t |F 0 t ) instead of the distribution L(Y t ) to be used in the presence of regime-switching. Later, [22] generalized the work [19] to conditional MKV SDEs driven by Brownian motion with state-dependent regime-switching. Comprehensive study on hybrid switching diffusion processes can be found in [27] and the references therein.…”