Optimal stopping with irregular reward functions

International audienceWe consider optimal stopping problems with finite horizon for one dimensional diffusions. We assume that the reward function is bounded and Borel-measurable, and we prove that the value function is continuous and can be characterized as the unique solution of a variational inequality in the sense of distributions

Connecting discrete and continuous lookback or hindsight options in exponential L\'evy models

Motivated by the pricing of lookback options in exponential L\'evy models, we study the difference between the continuous and discrete supremum of L\'evy processes. In particular, we extend the results of Broadie et al. (1999) to jump-diffusion models. We also derive bounds for general exponential L\'evy models.Comment: 31 p

When can the two-armed bandit algorithm be trusted?

We investigate the asymptotic behavior of one version of the so-called two-armed bandit algorithm. It is an example of stochastic approximation procedure whose associated ODE has both a repulsive and an attractive equilibrium, at which the procedure is noiseless. We show that if the gain parameter is constant or goes to 0 not too fast, the algorithm does fall in the noiseless repulsive equilibrium with positive probability, whereas it always converges to its natural attractive target when the gain parameter goes to zero at some appropriate rates depending on the parameters of the model. We also elucidate the behavior of the constant step algorithm when the step goes to 0. Finally, we highlight the connection between the algorithm and the Polya urn. An application to asset allocation is briefly described

A duality approach for the weak approximation of stochastic differential equations

In this article we develop a new methodology to prove weak approximation results for general stochastic differential equations. Instead of using a partial differential equation approach as is usually done for diffusions, the approach considered here uses the properties of the linear equation satisfied by the error process. This methodology seems to apply to a large class of processes and we present as an example the weak approximation of stochastic delay equations.Comment: Published at http://dx.doi.org/10.1214/105051606000000060 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Residual risks and hedging strategies in Markovian markets

19 pagesWe prove two explicit formulae for the quadratic residual risk and for the optimal hedging portfolio of a European contingent claim when the underlying stock prices are functions of a Markov proces