Optimal stopping of conditional McKean-Vlasov jump diffusions

Abstract

We study the problem of optimal stopping of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). We obtain sufficient variational inequalities for a function to be the value function of such a problem and for a stopping time to be optimal. To achieve this, we combine the state equation for the conditional McKean-Vlasov equation with the associated stochastic Fokker-Planck equation for the conditional law of the solution of the state. This gives us a Markovian system which can be handled by using a version of the Dynkin formula. We illustrate our result by solving explicitly two optimal stopping problems for conditional McKean-Vlasov jump diffusions. More specifically, we first find the optimal time to sell in a market with common noise and jumps, and, next, we find the stopping time to quit a project whose state is modelled by a jump diffusion, when the performance functional involves the conditional mean of the state.

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“…A deeper look into SDew in which coefficients satisfy both Lipchitz continuity and linear growth property, there by implying uniqueness in existential time evolution satisfying the SDew, under a wide range of transformations at initial value constraints was made by [14]. The attributes of linear and logarithmic transformations derived by these authors have made their ways to applications in dynamic programming techniques with random spikes as in [15] and [16].…”