Stopping Problems of Certain Multiplicative Functionals and Optimal Investment with Transaction Costs

Optimal stopping and impulse control problems with certain multiplicative functionals are considered. The stopping problems are solved by showing the unique existence of the solutions of relevant variational inequalities. However, since functions defining the multiplicative costs change the signs, some difficulties arise in solving the variational inequalities. Through gauge transformation we rewrite the variational inequalities in different forms with the obstacles which grow exponentially fast but with positive killing rates. Through the analysis of such variational inequalities we construct optimal stopping times for the problems. Then optimal strategies for impulse control problems on the infinite time horizon with multiplicative cost functionals are constructed from the solutions of the risk-sensitive variational inequalities of "ergodic type" as well. Application to optimal investment with fixed ratio transaction costs is also considered.     

Stopping of functionals with discontinuity at the boundary of an open set

We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set O. The stopping horizon is either random, equal to the first exit from the set O, or fixed (finite or infinite). The payoff function is continuous with a possible jump at the boundary of O. Using a generalization of the penalty method, we derive a numerical algorithm for approximation of the value function for general Feller-Markov processes and show existence of optimal or ε-optimal stopping times.     

Uniqueness of unbounded viscosity solutions for impulse control problem

We study here the impulse control problem in infinite as well as finite horizon. We allow the cost functionals and dynamics to be unbounded and hence the value function can possibly be unbounded. We prove that the value function is the unique viscosity solution in a suitable subclass of continuous functions, of the associated quasivariational inequality. Our uniqueness proof for the infinite horizon problem uses stopping time problem and for the finite horizon problem, comparison method. However, we assume proper growth conditions on the cost functionals and the dynamics.     

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.     

Approximation of optimal stopping problems

We consider optimal stopping of independent sequences. Assuming that the corresponding imbedded planar point processes converge to a Poisson process we introduce some additional conditions which allow to approximate the optimal stopping problem of the discrete time sequence by the optimal stopping of the limiting Poisson process. The optimal stopping of the involved Poisson processes is reduced to a differential equation for the critical curve which can be solved in several examples. We apply this method to obtain approximations for the stopping of iid sequences in the domain of max-stable laws with observation costs and with discount factors.     

Continuity Properties of Optimal Multiple Stopping Value

This article is concerned with the optimal multiple stopping problem for discrete time finite stage stochastic processes. We study lower semicontinuity and continuity properties of optimal stopping values with respect to the topology of convergence in distribution. Also, we formulate the multiple stopping version of the prophet inequality for the optimal stopping problem and apply the lower semicontinuity property of optimal stopping values to the prophet inequality for the optimal multiple stopping problem.     

Risk-sensitive stopping problems for continuous-time Markov chains

In this paper we consider stopping problems for continuous-time Markov chains under a general risk-sensitive optimization criterion for problems with finite and infinite time horizon. More precisely our aim is to maximize the certainty equivalent of the stopping reward minus cost over the time horizon. We derive optimality equations for the value functions and prove the existence of optimal stopping times. The exponential utility is treated as a special case. In contrast to risk-neutral stopping problems it may be optimal to stop between jumps of the Markov chain. We briefly discuss the influence of the risk sensitivity on the optimal stopping time and consider a special house selling problem as an example.     

A note on optimal stopping of diffusions with a two-sided optimal rule

We consider a class of optimal stopping problems of diffusions with a two-sided optimal rule. We propose an approach for finding and characterizing the solution. We establish that the optimal stopping rule can be associated with the unique fixed point of an auxiliary function. The results are illustrated with an explicit example.     

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.     

Existence and explicit determination of optimal stopping times

We consider large classes of continuous time optimal stopping problems for which we establish the existence and form of the optimal stopping times. These optimal times are then used to find approximate optimal solutions for a class of discrete time problems.     

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We generalize the framework of [18] for optimal stopping time problem to allow a certain restricted class of stopping times. By using classical results in probability theory on families of random variables indexed by a restricted family of stopping times, we prove the existence of an optimal time, givecharacterizations of the minimal and maximal optimal stopping times, and provide some local properties of the value function family, in concert with all special cases studied previously.     

The Elimination algorithm for the problem of optimal stopping

We present a new algorithm for solving the optimal stopping problem. The algorithm is based on the idea of elimination of states where stopping is nonoptimal and the corresponding correction of transition probabilities. The formal justification of this method is given by one of two presented theorems. The other theorem describes the situation when an aggregation of states is possible in the optimal stopping problem.     

Optimal stopping of Markov switching Lévy processes

We consider a finite time horizon optimal stopping of a regime-switching Lévy process. We prove that the value function of the optimal stopping problem can be characterized as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman variational inequalities.     

On the optimal renewal of bilinear functionals in linear Normed spaces

We study the problem of optimal renewal of bilinear functionals on the basis of optimal linear information in the general statement. We also represent some new results for special spaces of functions.     

On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions

We introduce a sequence of stopping times that allow us to study an analogue of a life-cycle decomposition for a continuous time Markov process, which is an extension of the well-known splitting technique of Nummelin to the continuous time case. As a consequence, we are able to give deterministic equivalents of additive functionals of the process and to state a generalisation of Chen’s inequality. We apply our results to the problem of non-parametric kernel estimation of the drift of multi-dimensional recurrent, but not necessarily ergodic, diffusion processes.     

A leavable bounded-velocity stochastic control problem

This paper studies bounded-velocity control of a Brownian motion when discretionary stopping, or ‘leaving’, is allowed. The goal is to choose a control law and a stopping time in order to minimize the expected sum of a running and a termination cost, when both costs increase as a function of distance from the origin. There are two versions of this problem: the fully observed case, in which the control multiplies a known gain, and the partially observed case, in which the gain is random and unknown. Without the extra feature of stopping, the fully observed problem originates with Beneš (Stochastic Process. Appl. 2 (1974) 127–140), who showed that the optimal control takes the ‘bang–bang’ form of pushing with maximum velocity toward the origin. We show here that this same control is optimal in the case of discretionary stopping; in the case of power-law costs, we solve the variational equation for the value function and explicitly determine the optimal stopping policy.We also discuss qualitative features of the solution for more general cost structures. When no discretionary stopping is allowed, the partially observed case has been solved by Beneš et al. (Stochastics Monographs, Vol. 5, Gordon & Breach, New York and London, pp. 121–156) and Karatzas and Ocone (Stochastic Anal. Appl. 11 (1993) 569–605). When stopping is allowed, we obtain lower bounds on the optimal stopping region using stopping regions of related, fully observed problems.     

The best choice problem for a union of two linear orders with common maximum

We find optimal stopping times for the secretary problem considered on partially ordered sets that are unions of two independent finite linear orders that have a common best element. We also determine the optimal probability of success, i.e. choosing the best element, and the asymptotic value of this probability and a certain threshold behavior of the optimal stopping times when the number of elements in our set tends to infinity.     

Optimal variance stopping with linear diffusions

We study the optimal stopping problem of maximizing the variance of an unkilled linear diffusion. Especially, we demonstrate how the problem can be solved as a convex two-player zero-sum game, and reveal quite surprising application of game theory by doing so. Our main result shows that an optimal solution can, in a general case, be found among stopping times that are mixtures of two hitting times. This and other revealed phenomena together with suggested solution methods could be helpful when facing more complex non-linear optimal stopping problems. The results are illustrated by a few examples.     

Conditions for stability and stabilization of systems of neutral type with nonmonotone Lyapunov functionals

We suggest new tests for the stability and uniform asymptotic stability of an equilibrium in systems of neutral type. By using these tests, we prove conditions for optimal stabilization and derive new estimates for perturbations that can be countered by a system closed by an optimal control. We show that, by using nonmonotone sign-indefinite functionals as Lyapunov functionals, one can obtain conditions for uniform asymptotic stability that do not contain the a priori requirement of stability of the difference operator and do not imply the boundedness of the right-hand side of the system. When studying the action of perturbations on the stabilized systems, these conditions permit one to obtain new estimates of perturbations preserving the stabilizing properties of optimal controls. The obtained estimates do not imply any constraint on the value of perturbations in some domains of the phase space that are defined when constructing an optimal stabilizing control. Some examples are considered.