Stochastic Fokker-Planck PIDE for conditional McKean-Vlasov jump diffusions and applications to optimal control

Abstract

We study optimal control of McKean-Vlasov (mean-field) stochastic differential equations with jumps.-First we prove a Fokker-Planck equation for the law of the state.-Then we study the situation when the law is absolute continuous with respect to Lebesgue measure. In that case the Fokker-Planck equation reduces to a deterministic integro-differential equation for the Radon-Nikodym derivative of the law. Combining this equation with the original state equation, we obtain a Markovian system for the state and its law.-We apply this to formulate a Hamilton-Jacobi-Bellman (HJB) equation for the optimal control of McKean-Vlasov stochastic equations with jumps. Then we derive a Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation for the Nash equilibrium of McKean-Vlasov jump diffusion games.-Finally we apply these results to solve explicitly the following problems: (i) Linear-quadratic optimal control of stochastic McKean-Vlasov equations with jumps (ii) Optimal consumption from a cash flow modelled as a stochastic McKean-Vlasov equation with jumps.

Optimal bailout strategies resulting from the drift controlled supercooled Stefan problem

“…From an economic point of view, our findings indicate a high sensitivity on the parameter γ in (7). As shown in Figure 5b, for a certain value of γ and in a regime where α triggers jump discontinuities in the uncontrolled regime, the optimal control strategy switches from not avoiding a jump to avoiding a jump.…”

“…As a matter of fact, we deal with subprobability densities here describing the marginal distributions of the absorbed process X = X t 1 {τ >t} . We note that the total mass of these subprobability measures as well as the underlying dynamics (7) can exhibit jumps if Λ is discontinuous, and that these jumps emerge endogenously from the feedback mechanism. This is in contrast to some other recent papers where jumps are exogenously given.…”

“…L T − (β) = δ). Writing X(β) for the solution process associated with the minimal solution Λ(β), analogously defined as in ( 5) but now for (7), we thus minimise the following objective function…”

“…As we show numerically in Section 3, the optimal loss L T − as a fuction of γ is monotone decreasing, so that for large enough γ, the threshold δ becomes small enough to avoid systemic events. We shall analyse the objective function (10) together with the dynamics (7), which is a mean-field control problem with a singular interaction through hitting the boundary, and its relation to existing literature. As we show in Section 2, in particular Theorem 2.7, optimisation of the McKean-Vlasov equation ( 7) yields the same result as first optimising in the N -particle system and then passing to the limit.…”

“…This is in contrast to some other recent papers where jumps are exogenously given. For instance, the very recent article [7] considers the control of McKean-Vlasov dynamics with jumps and associated HJB-PIDEs, while in [8] a stochastic maximum principle is derived to anaylse a mean-field game with random jump time penalty and interaction on the control.…”