Advances in Optimal Decision Making under Uncertainty

We consider a class of stochastic control problems where the state process is a probability measure-valued process satisfying an additional martingale condition on its dynamics, called measure-valued martingales (MVMs). Such problems notably arise when considering model-independent derivatives pricing. We establish the ‘classical results’ of stochastic control for these problems: specifically, we show that the value function for the problem can be characterised as the unique solution to a Hamilton-Jacobi-Bellman equation in the sense of viscosity solutions. In order to obtain our results, we exploit structural properties of the MVM processes; in particular, we present an appropriate version of Itô’s formula for controlled MVMs and an existence result for an SDE featuring MVMs. The talk is based on joint work with Alex Cox, Martin Larsson and Sara Svaluto-Ferro.

In this talk we present a reduced form framework under model uncertainty. In particular we introduce a sublinear conditional operator with respect to a family of possibly nondominated probability measures in presence of a single and multiple ordered default times. Moreover, we use this operator for the valuation of credit portfolio derivatives under model uncertainty. As an application we present a Feynman-Kac formula for the computation of this operator in the G-setting.

This study concerns the estimation of the growth-optimal portfolio. The growth-optimal portfolio, or the numeraire portfolio, maximises the expected logarithmic growth rate and plays a fundamental role in portfolio choice and asset pricing. Efficient estimates of the growth-optimal portfolio under the CAPM and multi-fund models are derived. The accuracy of estimation is greater, the larger the variances of the returns. A measure of investors’ economic loss caused by information loss due to missing data, or unavailable observations is provided. In particular, the economic loss is larger, the larger the investment universe or the factors affecting asset returns. The estimate has a Bayesian interpretation. A shrinkage method targets maximal growth with the least amount of deviation of the growth rates. Joint work with Kostas Kardaras and Hyeng Keun Koo

The objective of this talk is to see how Malliavin calculus can be applied to stochastic volatility modeling. This technique allows us to recover, unify, and generalize several previous results in the literature on stochastic volatility modeling related to the vanilla, the forward, and the VIX implied volatility surfaces. It can be applied to local, stochastic, and also to rough volatilities (driven by a fractional Brownian motion) leading to simple and explicit results.