A PDE approach to regularity of solutions to finite horizon optimal switching problems

We study optimal 2-switching and n-switching problems and the corresponding system of variational inequalities. We obtain results on the existence of viscosity solutions for the 2-switching problem for various setups when the cost of switching is non-deterministic. For the n-switching problem we obtain regularity results for the solutions of the variational inequalities. The solutions are C^1,1-regular away for the free boundaries of the action sets.     

Asymptotic analysis for a downside risk minimization problem under partial information

We give an analytic characterization of a large-time “downside risk” probability associated with an investor’s wealth. We assume that risky securities in our market model are affected by “hidden” economic factors, which evolve as a finite-state Markov chain. We formalize and prove a duality relation between downside risk minimization and the related risk-sensitive optimization. The proof is based on an analysis of an ergodic-type Hamilton–Jacobi–Bellman equation with large (exponentially growing) drift.     

Optimal stopping for partially observed piecewise-deterministic Markov processes

This paper deals with the optimal stopping problem under partial observation for piecewise-deterministic Markov processes. We first obtain a recursive formulation of the optimal filter process and derive the dynamic programming equation of the partially observed optimal stopping problem. Then, we propose a numerical method, based on the quantization of the discrete-time filter process and the inter-jump times, to approximate the value function and to compute an ?$?$-optimal stopping time. We prove the convergence of the algorithms and bound the rates of convergence.     

An optimal stopping problem for fragmentation processes

In this article we consider a toy example of an optimal stopping problem driven by fragmentation processes. We show that one can work with the concept of stopping lines to formulate the notion of an optimal stopping problem and moreover, to reduce it to a classical optimal stopping problem for a generalized Ornstein–Uhlenbeck process associated with Bertoin’s tagged fragment. We go on to solve the latter using a classical verification technique thanks to the application of aspects of the modern theory of integrated exponential Lévy processes.     

Optimal transmission of Gaussian signals through a feedback channel

Using the methodology and results of the theory of filtering of conditionally Gaussian processes, the optimal schemes of transmission of Gaussian signals through the noisy feedback channel are constructed under the new power conditions.     

Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps

We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). The financial position is given by an RCLL adapted process. We first state some properties of RBSDEs with jumps when the obstacle process is RCLL only. We then prove that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of optimal stopping times is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, we investigate robust optimal stopping problems related to the case with model ambiguity and their links with mixed control/optimal stopping game problems. We prove that, under some hypothesis, the value function is equal to the solution of an RBSDE. We then study the existence of saddle points when the obstacle is left-upper semi-continuous along stopping times.     

An analytic approach to purely nonlocal Bellman equations arising in models of stochastic control

Given a bounded domain Ω⊂Rd and two integro-differential operators L¹, L² of the form we study the fully nonlinear Bellman equation

(0.1)     

Construction of asymptotically optimal controls for control and game problems

Recently the connection between control and game problems and Backward Stochastic Differential Equations has been established. This allows us to use an approximation scheme for such equations in order to construct an ɛ-optimal control. Received: 13 November 1995 / Revised version: 11 February 1998     

On optimal stopping of inhomogeneous standard Markov processes

The connection between the optimal stopping problems for inhomogeneous standard Markov process and the corresponding homogeneous Markov process constructed in the extended state space is established. An excessive characterization of the value-function and the limit procedure for its construction in the problem of optimal stopping of an inhomogeneous standard Markov process is given. The form of -optimal (optimal) stopping times is also found.     

Finite approximation schemes for Lévy processes,and their application to optimal stopping problems

This paper proposes two related approximation schemes, based on a discrete grid on a finite time interval [0,T]$[0,T]$, and having a finite number of states, for a pure jump Lévy process _Lt$L_t$. The sequences of discrete processes converge to the original process, as the time interval becomes finer and the number of states grows larger, in various modes of weak and strong convergence, according to the way they are constructed. An important feature is that the filtrations generated at each stage by the approximations are sub-filtrations of the filtration generated by the continuous time Lévy process. This property is useful for applications of these results, especially to optimal stopping problems, as we illustrate with an application to American option pricing. The rates of convergence of the discrete approximations to the underlying continuous time process are assessed in terms of a “complexity” measure for the option pricing algorithm.     

Optimal expulsion and optimal confinement of a Brownian particle with a switching cost

We solve two stochastic control problems in which a player tries to minimize or maximize the exit time from an interval of a Brownian particle, by controlling its drift. The player can change from one drift to another but is subject to a switching cost. In each problem, the value function is written as the solution of a free boundary problem involving second order ordinary differential equations, in which the unknown boundaries are found by applying the principle of smooth fit. For both problems, we compute the value function, we exhibit the optimal strategy and we prove its generic uniqueness.     

Criticality of viscous Hamilton–Jacobi equations and stochastic ergodic control

This paper deals with nonlinear additive eigenvalue problems for viscous Hamilton–Jacobi equations which appear in stochastic ergodic control. Certain qualitative properties of principal eigenvalues and associated eigenfunctions are studied. Such analysis plays a key role in studying the recurrence and transience of feedback diffusions for the corresponding stochastic control problems. Our results can be regarded as a nonlinear extension of the criticality theory for Schrödinger operators with decaying potentials.     

Particle filters with random resampling times

Particle filters are numerical methods for approximating the solution of the filtering problem which use systems of weighted particles that (typically) evolve according to the law of the signal process. These methods involve a corrective/resampling procedure which eliminates the particles that become redundant and multiplies the ones that contribute most to the resulting approximation. The correction is applied at instances in time called resampling/correction times. Practitioners normally use certain overall characteristics of the approximating system of particles (such as the effective sample size of the system) to determine when to correct the system. As a result, the resampling times are random. However, in the continuous time framework, all existing convergence results apply only to particle filters with deterministic correction times. In this paper, we analyse (continuous time) particle filters where resampling takes place at times that form a sequence of (predictable) stopping times. We prove that, under very general conditions imposed on the sequence of resampling times, the corresponding particle filters converge. The conditions are verified when the resampling times are chosen in accordance to the effective sample size of the system of particles, the coefficient of variation of the particles’ weights and, respectively, the (soft) maximum of the particles’ weights. We also deduce central-limit theorem type results for the approximating particle system with random resampling times.     

A BSDE approach to stochastic differential games with incomplete information

We consider a two-player zero-sum stochastic differential game in which one of the players has a private information on the game. Both players observe each other, so that the non-informed player can try to guess his missing information. Our aim is to quantify the amount of information the informed player has to reveal in order to play optimally: to do so, we show that the value function of this zero-sum game can be rewritten as a minimization problem over some martingale measures with a payoff given by the solution of a backward stochastic differential equation.     

A stochastic partially reversible investment problem on a finite time-horizon: Free-boundary analysis

We study a continuous-time, finite horizon, stochastic partially reversible investment problem for a firm producing a single good in a market with frictions. The production capacity is modeled as a one-dimensional, time-homogeneous, linear diffusion controlled by a bounded variation process which represents the cumulative investment–disinvestment strategy. We associate to the investment–disinvestment problem a zero-sum optimal stopping game and characterize its value function through a free-boundary problem with two moving boundaries. These are continuous, bounded and monotone curves that solve a system of non-linear integral equations of Volterra type. The optimal investment–disinvestment strategy is then shown to be a diffusion reflected at the two boundaries.     

Forgetting the initial distribution for Hidden Markov Models

The forgetting of the initial distribution for discrete Hidden Markov Models (HMM) is addressed: a new set of conditions is proposed, to establish the forgetting property of the filter, at a polynomial and geometric rate. Both a pathwise-type convergence of the total variation distance of the filter started from two different initial distributions, and a convergence in expectation are considered. The results are illustrated using different HMM of interest: the dynamic tobit model, the nonlinear state space model and the stochastic volatility model.     

De Finetti’s optimal dividends problem with an affine penalty function at ruin

In a Lévy insurance risk model, under the assumption that the tail of the Lévy measure is log-convex, we show that either a horizontal barrier strategy or the take-the-money-and-run strategy maximizes, among all admissible strategies, the dividend payments subject to an affine penalty function at ruin. As a key step for the proof, we prove that, under the aforementioned condition on the jump measure, the scale function of the spectrally negative Lévy process has a log-convex derivative.     

Lagrange approach to the optimal control of diffusions

A new approach to the optimal control of diffusion processes based on Lagrange functionals is presented. The method is conceptually and technically simpler than existing ones. A first class of functionals allows to obtain optimality conditions without any resort to stochastic calculus and functional analysis. A second class, which requires Ito's rule, allows to establish optimality in a larger class of problems. Calculations in these two methods are sometimes akin to those in minimum principles and in dynamic programming, but the thinking behind them is new. A few examples are worked out to illustrate the power and simplicity of this approach.Research performed at the Mathematisches Seminar der Universität Kiel with support provided by an Alexander von Humboldt Foundation fellowship.