​In this talk, I will present some new integration by parts (IBP) formulae for the marginal law at a given time maturity of killed diffusions as well as a class of stochastic volatility models with unbounded drift. Relying on a perturbation argument for Markov processes, our formulae are based on a simple Markov chain evolving on a random time grid for which we develop a tailor-made Malliavin calculus. Though such formulae could be further analyzed to study fine properties of the associated densities, our main motivation lies in their numerical approximation. Indeed, we show that an unbiased Monte Carlo path simulation method directly stems from our formulae so that it can be used in order to numerically compute with optimal complexity option prices as well as their sensitivities with respect to the initial values, the so-called Greeks, namely the Delta and Vega, for a large class of non-smooth European payoff. Numerical results are proposed to illustrate the efficiency of the method.

 

 

 

 

We investigate generalized Langevin dynamics in the sense of Baudoin-Hairer-Teichmann and its convergence to a Gibbs measure of a loss function. We derive a pathwise decay formula for entropy as in Karatzas-Schachermayer-Tschiderer and obtain in this generalized context similar results to Hu-Ren-Siska-Szpruch, which are applied to training neural networks in machine learning.
(joint work with Robert Crowell, Christa Cuchiero, Yuuki Ida and Yuri Imamura)