Karthyek R. A. Murthy

Quantifying Distributional Model Risk Via Optimal Transport

This paper deals with the problem of quantifying the impact of model misspecification when computing general expected values of interest. The methodology that we propose is applicable in great generality, in particular, we provide examples involving path dependent expectations of stochastic processes. Our approach consists in computing bounds for the expectation of interest regardless of the probability measure used, as long as the measure lies within a prescribed tolerance measured in terms of a flexible class of distances from a suitable baseline model. These distances, based on optimal transportation between probability measures, include Wassersteins distances as particular cases. The proposed methodology is well-suited for risk analysis, as we demonstrate with a number of applications. We also discuss how to estimate the tolerance region non-parametrically using Skorokhod-type embeddings in some of these applications.

Tail asymptotics for delay in a half-loaded GI/GI/2 queue with heavy-tailed job sizes

We obtain asymptotic bounds for the tail distribution of steady-state waiting time in a two-server queue where each server processes incoming jobs at a rate equal to the rate of their arrivals (that is, the half-loaded regime). The job sizes are taken to be regularly varying. When the incoming jobs have finite variance, there are basically two types of effects that dominate the tail asymptotics. While the quantitative distinction between these two manifests itself only in the slowly varying components, the two effects arise from qualitatively very different phenomena (arrival of one extremely big job or two big jobs). Then there is a phase transition that occurs when the incoming jobs have infinite variance. In that case, only one of these effects dominates the tail asymptotics; the one involving arrival of one extremely big job.

Exact simulation of multidimensional reflected Brownian motion

We present the first exact simulation method for multidimensional reflected Brownian motion (RBM). Exact simulation in this setting is challenging because of the presence of correlated local-time-like terms in the definition of RBM. We apply recently developed so-called

Incorporating views on marginal distributions in the calibration of risk models

\varepsilon-

Optimal rare event Monte Carlo for Markov modulated regularly varying random walks

strong simulation techniques (also known as Tolerance-Enforced Simulation) which allow us to provide a piece-wise linear approximation to RBM with

Robust Wasserstein Profile Inference and Applications to Machine Learning

\varepsilon

On distributionally robust extreme value analysis

(deterministic) error in uniform norm. A novel conditional acceptance/rejection step is then used to eliminate the error. In particular, we condition on a suitably designed information structure so that a feasible proposal distribution can be applied.

State-independent importance sampling for random walks with regularly varying increments

We apply entropy based ideas to portfolio optimization and options pricing. The known abstracted problem corresponds to finding a probability measure that minimizes relative entropy with respect to a specified measure while satisfying moment constraints on functions of underlying assets. We generalize this to also allow constraints on marginal distribution of functions of underlying assets. These are applied to Markowitz portfolio framework to incorporate fatter tails as well as to options pricing to incorporate implied risk neutral densities on liquid assets.

Optimal Transport Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes.

Most of the efficient rare event simulation methodology for heavy-tailed systems has concentrated on processes with stationary and independent increments. Motivated by applications such as insurance risk theory, in this paper we develop importance sampling estimators that are shown to achieve asymptotically vanishing relative error property (and hence are strongly efficient) for the estimation of large deviation probabilities in Markov modulated random walks that possess heavy-tailed increments. Exponential twisting based methods, which are effective in light-tailed settings, are inapplicable even in the simpler case of random walk involving i.i.d. heavy-tailed increments. In this paper we decompose the rare event of interest into a dominant and residual component, and simulate them independently using state-independent changes of measure that are both intuitive and easy to implement.