Kind Of Stochastic Control Problem Research Articles

Backward SDE representation for stochastic control problems with nondominated controlled intensity

We are interested in stochastic control problems coming from mathematical finance and, in particular, related to model uncertainty, where the uncertainty affects both volatility and intensity. This kind of stochastic control problem is associated to a fully nonlinear integro-partial differential equation, which has the peculiarity that the measure $(\lambda(a,\cdot))_{a}$ characterizing the jump part is not fixed but depends on a parameter $a$ which lives in a compact set $A$ of some Euclidean space $\mathbb{R}^{q}$. We do not assume that the family $(\lambda(a,\cdot))_{a}$ is dominated. Moreover, the diffusive part can be degenerate. Our aim is to give a BSDE representation, known as a nonlinear Feynman–Kac formula, for the value function associated with these control problems. For this reason, we introduce a class of backward stochastic differential equations with jumps and a partially constrained diffusive part. We look for the minimal solution to this family of BSDEs, for which we prove uniqueness and existence by means of a penalization argument. We then show that the minimal solution to our BSDE provides the unique viscosity solution to our fully nonlinear integro-partial differential equation.

Risk-Sinsitive Dynamic Portfolio Optimization with Partial Information on Infinite Time Horizon

We consider an optimal investment problem for a factor model treated by Bielecki and Pliska (Appl. Math. Optim. 39 337–360) as a risk-sensitive stochastic control problem, where the mean returns of individual securities are explicitly affected by economic factors defined as Gaussian processes. We relax the measurability condition assumed as Bielecki and Pliska for the investment strategies to select. Our investment strategies are supposed to be chosen without using information of factor processes but by using only past information of security prices. Then our problem is formulated as a kind of stochastic control problem with partial information. The case on a finite time horizon is discussed by Nagai (Stochastics in Finite and Infinite Dimension 321–340. Birkhauser, Boston). Here we discuss the problem on infinite time horizon.

Super Replication and Uncertain Volatility: European, American and Passport Options

Despite a large number of works, there is no unique way to model the stochastic volatility of an underlying. A new approach has been developed where one determines the (super)hedging strategies which are admissible for any model where the volatility is lying in a known interval without other restriction.Our first objective standardizes and to develops the existing results on European options in a common setting. A second objective is the study of American options. We characterize the price by a stochastic control problem with a control over volatility and over stopping times. A last objective is the study of European and American passport options. The prices are characterized by the same kind of stochastic control problem with moreover a control over all the trading strategies of the option's buyer.All those characterizations are valid for only very smooth payoff functions. We extend them to the practical case where the payoff functions are merely continuous. With an analytic point of view, the computation of those prices is given by the resolution of Hamilton-Jacobi-Bellman (HJB) equations (European case) or of a system of HJB equations (American case) with an initial condition.

Examples of optimal control for partially observable systems:comparison, classical, and martingale methods

The following kind of stochastic control problem is considerd: to minimize independent Wiener processes, X 0 Gaussian random variable independent of observed, with all causal functions u t bounded by unity admissible. By using recent comparison theorems for solutions of stochastic differential equations, it is possible to prove that the physically obvious law: is indeed optimal. The same result is established via the classical and the martingale methods of approach to stochastic control. Various generalizations of the above model are also discussed.