An optimal stopping problem for a geometric Brownian motion with poissonian jumps

This paper examines an optimal stopping problem for a geometric Brownian motion with random jumps. It is assumed that jumps occur according to a time-homogeneous Poisson process and the proportions of these sizes are independent and identically distributed nonpositive random variables. The objective is to find an optimal stopping time of maximizing the expected discounted terminal reward which is defined as a nondecreasing power function of the stopped state. By applying the “smooth pasting technique” [1,2], we derive almost explicitly an optimal stopping rule of a threshold type and the optimal value function of the initial state. That is, we express the critical state of the optimal stopping region and the optimal value function by formulae which include only given problem parameters except an unknown to be uniquely determined by a nonlinear equation.     

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.     

Conditional hitting time estimation in a nonlinear filtering model by the Brownian bridge method

We consider a model composed of a signal process X given by a classic stochastic differential equation and an observation process Y, which is supposed to be correlated to the signal process. We assume that process Y is observed from time 0 to s>0 at discrete times and aim to estimate, conditionally on these observations, the probability that the non-observed process X crosses a fixed barrier after a given time t>0. We formulate this problem as a usual nonlinear filtering problem and use optimal quantization and Monte Carlo simulations techniques to estimate the involved quantities.     

Optimal stopping in a fuzzy environment

The problem under consideration is that of optimally controlling and stopping either a deterministic or a stochastic system in a fuzzy environment. The optimal decision is the sequence of controls that maximizes the membership function of the intersection of the fuzzy constraints and a fuzzy goal. The fuzzy goal is a fuzzy set in the cartesian product of the state space with the set of possible stopping times. Dynamic programming is applied to yield a numerical solution. This approach yields an algorithm that corrects a result of Kacprzyk.     

Optimal stopping in a search for a vertex with full degree in a random graph

We consider the following on-line decision problem. The vertices of a realization of the random graph G(n,p) are being observed one by one by a selector. At time m, the selector examines the mth vertex and knows the graph induced by the m vertices that have already been examined. The selector’s aim is to choose the currently examined vertex maximizing the probability that this vertex has full degree, i.e. it is connected to all other vertices in the graph. An optimal algorithm for such a choice (in other words, optimal stopping time) is given. We show that it is of a threshold type and we find the threshold and its asymptotic estimation.     

Brownian Optimal Stopping and Random Walks

One way to compute the value function of an optimal stopping problem along Brownian paths consists of approximating Brownian motion by a random walk. We derive error estimates for this type of approximation under various assumptions on the distribution of the approximating random walk.     

Brownian Optimal Stopping and Random Walks

One way to compute the value function of an optimal stopping problem along Brownian paths consists of approximating Brownian motion by a random walk. We derive error estimates for this type of approximation under various assumptions on the distribution of the approximating random walk.     

Optimal stopping of switching diffusions with state dependent switching rates

This paper is concerned with a continuous-time and infinite-horizon optimal stopping problem in switching diffusion models. In contrast to the assumption commonly made in the literature that the regime-switching is modeled by an independent Markov chain, we consider in this paper the case of state-dependent regime-switching. The Hamilton–Jacobi–Bellman (HJB) equation associated with the optimal stopping problem is given by a system of coupled variational inequalities. By means of the dynamic programming (DP) principle, we prove that the value function is the unique viscosity solution of the HJB system. As an interesting application in mathematical finance, we examine the problem of pricing perpetual American put options with state-dependent regime-switching. A numerical procedure is developed based on the DP approach and an efficient discrete tree approximation of the continuous asset price process. Numerical results are reported.     

Monotone stopping rules forstochastic processes in a semimartingale representation with applications

A monotone stopping problem is considered for stochastic processes in a semimartingale representation. Such a representation allows a direct infinitesimal characterization of the optimal stopping time. Transformations of such processes are investigated, which leave the semimartingale property unchanged. One of these transformations is a change of tiltration which leads to the stopping problem with partial information. Findly an application is discussed.     

Optimal stopping problems by two or more decision makers: a survey

A review of the optimal stopping problem with more than a single decision maker (DM) is presented in this paper. We classify the existing literature according to the arrival of the offers, the utility of the DMs, the length of the sequence of offers, the nature of the game and the number of offers to be selected. We enumerate various definitions for this problem and describe some dynamic approaches. Fouad Ben Abdelaziz is on leave from the Institut Superieur de Gestion, University of Tunis, Tunisia e-mail: foued.benabdelaz@isg.run.tn.     

Optimal stopping, free boundary, and American option in a jump-diffusion model

This paper considers the American put option valuation in a jump-diffusion model and relates this optimal-stopping problem to a parabolic integro-differential free-boundary problem, with special attention to the behavior of the optimal-stopping boundary. We study the regularity of the American option value and obtain in particular a decomposition of the American put option price as the sum of its counterpart European price and the early exercise premium. Compared with the Black-Scholes (BS) [5] model, this premium has an additional term due to the presence of jumps. We prove the continuity of the free boundary and also give one estimate near maturity, generalizing a recent result of Barleset al. [3] for the BS model. Finally, we study the effect of the market price of jump risk and the intensity of jumps on the American put option price and its critical stock price.     

Optimal stopping problems with discontinous reward: Regularity of the value function and viscosity solutions

In this paper we show that the solution of a stochastic boundary value problem with additive noise and with a completely nonlinear drift is a Markov field if only if the boundary condition is an initial or a final type condition     

Optimal Selling of an Asset with Jumps Under Incomplete Information

ABSTRACT

We study the optimal liquidation strategy of an asset with price process satisfying a jump diffusion model with unknown jump intensity. It is assumed that the intensity takes one of two given values, and we have an initial estimate for the probability of both of them. As time goes by, by observing the price fluctuations, we can thus update our beliefs about the probabilities for the intensity distribution. We formulate an optimal stopping problem describing the optimal liquidation problem. It is shown that the optimal strategy is to liquidate the first time the point process falls below (goes above) a certain time-dependent boundary.     

An optimal sequential procedure for a multiple selling problem with independent observations

We consider a sequential problem of selling K identical assets over the finite time horizon with a fixed number of offers per time period and no recall of past offers. The objective is to find an optimal sequential procedure which maximizes the total expected revenue. In this paper, we derive an effective number of stoppings for an optimal sequential procedure for the selling problem with independent observations.     

Optimal stopping of strong Markov processes

We characterize the value function and the optimal stopping time for a large class of optimal stopping problems where the underlying process to be stopped is a fairly general Markov process. The main result is inspired by recent findings for Lévy processes obtained essentially via the Wiener–Hopf factorization. The main ingredient in our approach is the representation of the β$\beta$-excessive functions as expected suprema. A variety of examples is given.     

Optimal retirement strategy with a negative wealth constraint

This paper investigates an optimal consumption, portfolio, and retirement time choice problem of an individual with a negative wealth constraint. We obtain analytical results of the optimal consumption, investment, and retirement behaviors and discuss the effect of the negative wealth constraint on the optimal behaviors. We find that, as an individual can borrow more with better credit, she is more likely to retire at a higher wealth level, to consume more, and to invest more in risky assets.     

Mayer and optimal stopping stochastic control problems with discontinuous cost

We study two classes of stochastic control problems with semicontinuous cost: the Mayer problem and optimal stopping for controlled diffusions. The value functions are introduced via linear optimization problems on appropriate sets of probability measures. These sets of constraints are described deterministically with respect to the coefficient functions. Both the lower and upper semicontinuous cases are considered. The value function is shown to be a generalized viscosity solution of the associated HJB system, respectively, of some variational inequality. Dual formulations are given, as well as the relations between the primal and dual value functions. Under classical convexity assumptions, we prove the equivalence between the linearized Mayer problem and the standard weak control formulation. Counter-examples are given for the general framework.