Search

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]

This article studies singular mean field control problems and singular mean field two-players stochastic differential games. Both sufficient and necessary conditions for the optimal controls and for the Nash equilibrium are obtained. Under some assumptions the optimality conditions for singular mean-field control are reduced to a reflected Skorohod problem, whose solution is proved to exist uniquely. Motivations are given as optimal harvesting of stochastic mean-field systems, optimal irreversible investments under uncertainty and mean-field singular investment games. In particular, a simple singular mean-field investment game is studied, where the Nash equilibrium exists but is not unique. [ABSTRACT FROM PUBLISHER]

We combine stochastic control methods, white noise analysis, and Hida–Malliavin calculus applied to the Donsker delta functional to obtain explicit representations of semimartingale decompositions under enlargement of filtrations. Some of the expressions are more explicit than previously known. The results are illustrated by examples. [ABSTRACT FROM AUTHOR]

We study the problem of optimal control of a jump diffusion, that is, a process which is the solution of a stochastic differential equation driven by Lévy processes. It is required that the control process is adapted to a given subfiltration of the filtration generated by the underlying Lévy processes. We prove two maximum principles (one sufficient and one necessary) for this type of partial information control. The results are applied to a partial information mean-variance portfolio selection problem in finance. [ABSTRACT FROM AUTHOR]

We prove a sufficient maximum principle for the optimal control of systems described by a quasilinear stochastic heat equation. The result is applied to solve a problem of optimal harvesting from a system described by a stochastic reaction-diffusion equation. [ABSTRACT FROM AUTHOR]

A Meyer-Tanaka formula involving weighted local time is derived for fractional Brownian motion and geometric fractional Brownian motion. The formula is applied to the study of the stop-loss-start-gain (SLSG) portfolio in a fractional Black-Scholes market. As a consequence, we obtain a fractional version of the Carr-Jarrow decomposition of the European call and put option prices into their intrinsic and time values. [ABSTRACT FROM AUTHOR]

We study how the value function (minimal cost function) Vc of certain impulse control problems depends on the intervention cost c. We consider the case when the cost of interfering with an impulse control of size ζ∈R is given bywith c≥0,λ>0 constants, and we show (under some assumptions) that Vc is very sensitive (non-robust) to an increase in c near c=0 in the sense that The paper provides a general scheme to prove results of this type. In particular, we show that the scheme applies to several of the standard processes used in financial markets, i.e., Brownian motion, Geometric Brownian motion with jumps, and the Ornstein–Uhlenbeck process. [ABSTRACT FROM AUTHOR]

We find the chaos expansion of local time ℓT(H)(x,·) of fractional Brownian motion with Hurst coefficient H∈(0,1) at a point x∈Rd. As an application we show that when H0d<1 then ℓT(H)(x,·)∈L2(μ). Here μ denotes the probability law of B(H) and H0=max{H1,…,Hd}. In particular, we show that when d=1 then ℓT(H)(x,·)∈L2(μ) for all H∈(0,1). [ABSTRACT FROM AUTHOR]