Optimal stopping for extremal processes

For an extremal process (Z_t)_t the optimal stopping problem for X_t = f(Z_t)?g(t) gives the continuous time analogue of the optimal stopping problem for max{Y₁,…,Y_k}?c_k where Y₁, Y₂,… are i.i.d. For the continuous time problem we derive optimal stopping times in explicit form and also show that the optimal stopping boundary is the limit of the optimal stopping boundaries for suitably standardized discrete problems.     

Finite dimensional approximation in infinite dimensional mathematical programming

We consider the problem of approximating an optimal solution to a separable, doubly infinite mathematical program (P) with lower staircase structure by solutions to the programs (P(N)) obtained by truncating after the firstN variables andN constraints of (P). Viewing the surplus vector variable associated with theNth constraint as a state, and assuming that all feasible states are eventually reachable from any feasible state, we show that the efficient set of all solutions optimal to all possible feasible surplus states for (P(N)) converges to the set of optimal solutions to (P). A tie-breaking algorithm which selects a nearest-point efficient solution for (P(N)) is shown (for convex programs) to converge to an optimal solution to (P). A stopping rule is provided for discovering a value ofN sufficiently large to guarantee any prespecified level of accuracy. The theory is illustrated by an application to production planning.The work of Robert L. Smith was partially supported by the National Science Foundation under Grant ECS-8700836.     

Generalized Wald equations in discrete time

Let S = {S_n, n ? 1} be a martingale. Expectations of m^th order quantities associated with S are related by two forms of Wald-type identity, called Generalized Wald equations. The previously known sufficient conditions for the validity of Wald equations are shown to be of a set of three equivalent conditions, each of which is necessary as well as sufficient for the validity of both types of Generalized Wald Equation.     

Stopping criteria, forward and backward errors for perturbed asynchronous linear fixed point methods in finite precision

^** Email: pierre.spiteri{at}enseeiht.fr This paper deals with perturbed linear fixed point methods inthe presence of round-off errors. Successive approximationsas well as the more general asynchronous iterations are treated.Forward and backward error estimates are presented, and theseare used to propose theoretical stopping criteria for thesemethods. In the case of asynchronous iterations, macro-iterationsare used as a tool in order to obtain estimates.     

Rapidity distributions of net protons from AGS to LHC energy regions

Taking the conservation of baryon number into account in a non-uniform flow model, the rapidity distribution of the net protons at the LHC is predicted. The energy dependence of the rapidity distribution, baryon stopping and collective flow from BNL/AGS to CERN/LHC are systematically investigated.     

Optimal stopping of a Brownian bridge with an unknown pinning point

The problem of stopping a Brownian bridge with an unknown pinning point to maximise the expected value at the stopping time is studied. A few general properties, such as continuity and various bounds of the value function, are established. However, structural properties of the optimal stopping region are shown to crucially depend on the prior, and we provide a general condition for a one-sided stopping region. Moreover, a detailed analysis is conducted in the cases of the two-point and the mixed Gaussian priors, revealing a rich structure present in the problem.     

Pricing of American lookback spread options

We find the closed form formula for the price of the perpetual American lookback spread option, whose payoff is the difference of the running maximum and minimum prices of a single asset. We solve an optimal stopping problem related to both maximum and minimum. We show that the spread option is equivalent to some fixed strike options on some domains, find the exact form of the optimal stopping region, and obtain the solution of the resulting partial differential equations. The value function is not differentiable. However, we prove the verification theorem due to the monotonicity of the maximum and minimum processes.     

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.