79 research outputs found
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
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