Viscosity solutions methods are used to pass to the limit in some penalization problems for rst order and second order, degenerate parabolic, Hamilton-Jacobi-Bellman equations. This characterizes the limit of the value functions of singularly perturbed optimal control problems for nonlinear deterministic systems and controlled degenerate diiusions, respectively. The results cover also cases whe...

We consider impulse control problems in finite horizon for diffusions with decision lag and execution delay. The new feature is that our general framework deals with the important case when several consecutive orders may be decided before the effective execution of the first one. This is motivated by financial applications in the trading of illiquid assets such as hedge funds. We show that the ...

The adaptation is related to the natural filtration of the Brownian motion (Bt)t≤T . In 1990, Pardoux and Peng introduced the notion of nonlinear backward stochastic differential equation (BSDE), namely (1.1), and gave existence and uniqueness result in their founder paper [26]. Since then the interest in BSDEs has kept growing steadily and there have been several works on that subject. The mai...

This paper treats a finite time horizon optimal control problem in which the controlled state dynamics is governed by a general system of stochastic functional differential equations with a bounded memory. An infinite-dimensional HJB equation is derived using a Bellman-type dynamic programming principle. It is shown that the value function is the unique viscosity solution of the HJB equation.

We consider three applications of impulse control in financial mathematics, a cash management problem, optimal control of an exchange rate, and portfolio optimisation under transaction costs. We sketch the different ways of solving these problems with the help of quasi-variational inequalities. Further, some viscosity solution results are presented.

In a series of papers, we characterized the value function in optimal control as the unique viscosity solution of the corresponding Bellman equation that satisfies appropriate side conditions. The novelty of our results was that they applied to exit time problems with general nonnegative instantaneous costs, including cases where the instantaneous cost is not uniformly bounded below by positive...

This paper is a survey on some recent aspects and developments in stochastic control theory. We discuss the two main historical approaches, Bellman’s optimality principle and Pontryagin’s maximum principle, and their modern exposition with viscosity solutions and backward stochastic differential equations. Some original proofs are presented in a unifying context including degenerate singular co...

Let M be a smooth connected and complete manifold of dimension n, and ∆ be a smooth nonholonomic distribution of rank m ≤ n on M . We prove that, if there exists a smooth Riemannian metric on ∆ for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of ∆ on M . Moreover, in dimension three, the assumption of the absence of singular minimizin...

We consider an optimal control problem of linear stochastic integro-differential equation with conic constraints on the phase variable and the control of singular-regular type. Our setting includes consumption-investment problems for models of financial markets in the presence of proportional transaction costs where that the prices are geometric Lévy processes and the investor is allowed to tak...

We consider an optimal stopping problem for a Hilbert-space valued di usion. We prove that the value function of the problem is the unique viscosity solution of an obstacle problem for the associated parabolic partial di erential equation in the Hilbert space. The results are applied to investigate the pricing of American interest rate options in the lognormal Heath-Jarrow-Morton model of yield...