A discretionary stopping problem in stochastic control: An application in credit exposure control

The dissertation studies a discretionary stopping problem in stochastic impulse control with a quantile-based performance criterion and applies it to the stochastic credit exposure control in the context of over-the-counter derivatives transactions. Problems combining stochastic control and optimal stopping arise in various business applications in which the system dynamics involve discrete actions. Such problems have been studied and dealt with in the existing literature. The main contribution of this work to the current literature is that it employs a new performance criterion in the form of the q-th quantile of the maximum of a diffusion process, rather than, the expected or long-run average value of a performance measure related to the state of the system. This new performance criterion would provide more information about extreme events and serve as a more robust measurement for stochastic optimization problems in decision marking under uncertainty. A generic framework is developed to find optimal strategies for a single stopping problem, the twice stopping problem in the form of simultaneous and sequential decisions, and an extension to the multiple stopping problem. This framework is then applied to solve the optimal mark-to-market timing problem arising in collateralization for stochastic credit exposure control in the context of over-the-counter derivatives transactions. Collateralization has been widely used in practice for mitigating counterparty credit risk by reducing stochastic credit exposure in over-the-counter derivatives markets. However, the relevant decisions are often made in an ad-hoc manner, without reference to an analytical framework. Very little academic research has addressed the quantitative analysis of mark-to-market timing. Another goal of this research is to fill this theoretical gap and propose a method for finding optimal timing of mark-to-market in collateral agreements to minimize potential future exposure of credit portfolio. With explicit consideration of the stochastic pathwise property of the underlying one-dimensional Ito diffusion, a quasi-analytic framework is built to model the single, twice and multiple mark-to-market timing strategies. Numerical methods are then employed to solve this stochastic dynamic optimization problem towards outlining optimal decision strategies.