The exit problem for diffusions with time-periodic drift and stochastic resonance

Physical notions of stochastic resonance for potential diffusions in periodically changing double-well potentials such as the spectral power amplification have proved to be defective. They are not robust for the passage to their effective dynamics: continuous-time finite-state Markov chains describing the rough features of transitions between different domains of attraction of metastable points. In the framework of one-dimensional diffusions moving in periodically changing double-well potentials we design a new notion of stochastic resonance which refines Freidlin's concept of quasi-periodic motion. It is based on exact exponential rates for the transition probabilities between the domains of attraction which are robust with respect to the reduced Markov chains. The quality of periodic tuning is measured by the probability for transition during fixed time windows depending on a time scale parameter. Maximizing it in this parameter produces the stochastic resonance points.Comment: Published at http://dx.doi.org/10.1214/105051604000000530 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Utility maximization in incomplete markets

We consider the problem of utility maximization for small traders on incomplete financial markets. As opposed to most of the papers dealing with this subject, the investors' trading strategies we allow underly constraints described by closed, but not necessarily convex, sets. The final wealths obtained by trading under these constraints are identified as stochastic processes which usually are supermartingales, and even martingales for particular strategies. These strategies are seen to be optimal, and the corresponding value functions determined simply by the initial values of the supermartingales. We separately treat the cases of exponential, power and logarithmic utility.Comment: Published at http://dx.doi.org/10.1214/105051605000000188 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Malliavin's calculus in insider models: Additional utility and free lunches

We consider simple models of financial markets with regular traders and insiders possessing some extra information hidden in a random variable which is accessible to the regular trader only at the end of the trading interval. The problems we focus on are the calculation of the additional utility of the insider and a study of his free lunch possibilities. The information drift, i.e. the drift to eliminate in order to preserve the martingale property in the insider's filtration, turns out to be the crucial quantity needed to answer these questions. It is most elegantly described by the logarithmic Malliavin trace of the conditional laws of the insider information with respect to the filtration of the regular trader. Several examples are given to illustrate additional utility and free lunch possibilities. In particular, if the insider has advance knowledge of the maximal stock price process, given by a regular diffusion, arbitrage opportunities exist