University of California Santa Barbara

A well-behaved field theory living on a fixed background has a causality structure defined by the background metric. In holography, however, signals can travel through the bulk, and some bulk metrics would allow a signal to travel faster than the speed of light as seen on the boundary. These are called boundary causality violating metrics. Holographers usually work with a classical bulk metric, in which case they declare that boundary causality violating metrics are forbidden. However, in a full quantum gravity path integral, these metrics do contribute. The question is then: how to avoid causality violation in this context? In this talk I will give a prescription that achieves this.

We present linear-quadratic stochastic differential games on directed chains inspired by the directed chain stochastic differential equations introduced by Detering, Fouque, and Ichiba in a previous work. We solve explicitly for Nash equilibria with a finite number of players and we study more general finite-player games with a mixture of both directed chain interaction and mean field interaction. We investigate and compare the corresponding games in the limit when the number of players tends to infinity. 

The limit is characterized by Catalan functions and the dynamics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain, with or without the presence of mean field interaction.

Joint work with Yichen Feng and Tomoyuki Ichiba.

We consider calculation of VaR/TVaR capital requirements when the underlying economic scenarios are determined by simulatable risk factors. This problem involves computationally expensive nested simulation, since evaluating expected portfolio losses of an outer scenario (aka computing a conditional expectation) requires inner-level Monte Carlo. We introduce several inter-related machine learning techniques to speed up this computation, in particular by properly accounting for the simulation noise. Our main workhorse is an advanced Gaussian Process (GP) regression approach which uses nonparametric spatial modeling to efficiently learn the relationship between the stochastic factors defining scenarios and corresponding portfolio value. Leveraging this emulator, we develop sequential algorithms that adaptively allocate inner simulation budgets to target the quantile region. The GP framework also yields better uncertainty quantification for the resulting VaR/\TVaR estimators that reduces bias and variance compared to existing methods.  Time permitting, I will highlight further related applications of statistical emulation in risk management.
This is joint work with Jimmy Risk (Cal Poly Pomona).