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In this paper, we consider impulse control problems involving conditional McKean–Vlasov jump diffusions, with the common noise coming from the σ -algebra generated by the first components of a Brownian motion and an independent compensated Poisson random measure. We first study the well-posedness of the conditional McKean–Vlasov stochastic differential equations (SDEs) with jumps. Then, we prove the associated Fokker–Planck stochastic partial differential equation (SPDE) with jumps. Next, we establish a verification theorem for impulse control problems involving conditional McKean–Vlasov jump diffusions. We obtain a Markovian system by combining the state equation with the associated Fokker–Planck SPDE for the conditional law of the state. Then we derive sufficient variational inequalities for a function to be the value function of the impulse control problem, and for an impulse control to be the optimal control. We illustrate our results by applying them to the study of an optimal stream of dividends under transaction costs. We obtain the solution explicitly by finding a function and an associated impulse control, which satisfy the verification theorem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study optimal control of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). To this end, we first prove a stochastic Fokker-Planck equation for the conditional law of the solution of such equations. Combining this equation with the original state equation, we obtain a Markovian system for the state and its conditional law. Furthermore, we apply this to formulate a Hamilton-Jacobi-Bellman equation for the optimal control of conditional McKean-Vlasov jump diffusions. Then we study the situation when the law is absolutely continuous with respect to Lebesgue measure. In that case the Fokker-Planck equation reduces to a stochastic partial differential equation for the Radon-Nikodym derivative of the conditional law. Finally we apply these results to solve explicitly the linear-quadratic optimal control problem of conditional stochastic McKean-Vlasov jump diffusions, and optimal consumption from a cash flow. [ABSTRACT FROM AUTHOR]

We study optimal control for mean-field forward–backward stochastic differential equations with payoff functionals of mean-field type. Sufficient and necessary optimality conditions in terms of a stochastic maximum principle are derived. As an illustration, we solve an optimal portfolio with mean-field risk minimization problem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the optimal stopping problem 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. The key is that we combine the conditional McKean–Vlasov equation with the associated stochastic Fokker–Planck partial integro-differential equation for the conditional law of the state. This leads to a Markovian system which can be handled by using a version of a Dynkin formula. Our verification result is illustrated by finding the optimal time to sell in a market with common noise and jumps. [ABSTRACT FROM AUTHOR]

Our purpose of this paper is to study stochastic control problems for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case. In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem. For infinite horizon, we derive sufficient and necessary maximum principles. As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system. [ABSTRACT FROM AUTHOR]

By a memory mean-field process we mean the solution X(·) of a stochastic mean-field equation involving not just the current state X(t) and its law L(X(t)) at time t,  but also the state values X(s) and its law L(X(s)) at some previous times s

The purpose of this paper is to study the following topics and the relation between them: (i) Optimal singular control of mean-field stochastic differential equations with memory; (ii) reected advanced mean-field backward stochastic differential equations; and (iii) optimal stopping of mean-field stochastic differential equations. More specifically, we do the following: (1) We prove the existence and uniqueness of the solutions of some reected advanced memory backward stochastic differential equations; (2) we give sufficient and necessary conditions for an optimal singular control of a memory mean-field stochastic differential equation (MMSDE) with partial information; and (3) we deduce a relation between the optimal singular control of an MMSDE and the optimal stopping of such processes. [ABSTRACT FROM AUTHOR]

We consider the problem of optimal control of a mean-field stochastic differential equation (SDE) under model uncertainty. The model uncertainty is represented by ambiguity about the law of the state X(t) at time t. For example, it could be the law of X(t) with respect to the given, underlying probability measure. This is the classical case when there is no model uncertainty. But it could also be the law with respect to some other probability measure or, more generally, any random measure on with total mass 1. We represent this model uncertainty control problem as a stochastic differential game of a mean-field related type SDE with two players. The control of one of the players, representing the uncertainty of the law of the state, is a measure-valued stochastic process and the control of the other player is a classical real-valued stochastic process u(t). This optimal control problem with respect to random probability processes in a non-Markovian setting is a new type of stochastic control problems that has not been studied before. By constructing a new Hilbert space of measures, we obtain a sufficient and a necessary maximum principles for Nash equilibria for such games in the general nonzero-sum case, and for saddle points in zero-sum games. As an application we find an explicit solution of the problem of optimal consumption under model uncertainty of a cash flow described by a mean-field related type SDE. [ABSTRACT FROM AUTHOR]

In this paper we study the linear mean-field backward stochastic differential equations (mean-field BSDE) of the form (0.1) d Y (t) = − [ α 1 (t) Y (t) + β 1 (t) Z (t) + ∫ R 0 η 1 (t , ζ) K (t , ζ) ν (d ζ) + α 2 (t) E [ Y (t) ] + β 2 (t) E [ Z (t) ] + ∫ R 0 η 2 (t , ζ) E [ K (t , ζ) ] ν (d ζ) + γ (t) ] d t + Z (t) d B (t) + ∫ R 0 K (t , ζ) N ̃ (d t , d ζ) , t ∈ 0 , T , Y (T) = ξ. where (Y , Z , K) is the unknown solution triplet, B is a Brownian motion, N ̃ is a compensated Poisson random measure, independent of B. We prove the existence and uniqueness of the solution triplet (Y , Z , K) of such systems. Then we give an explicit formula for the first component Y (t) by using partial Malliavin derivatives. To illustrate our result we apply them to study a mean-field recursive utility optimization problem in finance. [ABSTRACT FROM AUTHOR]