Properties and advances of probabilistic and statistical algorithms with applications in finance

Abstract

This thesis is concerned with the construction and enhancement of algorithms involving probability and statistics. The main motivation for these are problems that appear in finance and more generally in applied science. We consider three distinct areas, namely, credit risk modelling, numerics for McKean Vlasov stochastic differential equations and stochastic representations of Partial Differential Equations (PDEs), therefore the thesis is split into three parts. Firstly, we consider the problem of estimating a continuous time Markov chain (CTMC) generator from discrete time observations, which is essentially a missing data problem in statistics. These generators give rise to transition probabilities (in particular probabilities of default) over any time horizon, hence the estimation of such generators is a key problem in the world of banking, where the regulator requires banks to calculate risk over different time horizons. For this particular problem several algorithms have been proposed, however, through a combination of theoretical and numerical results we show the Expectation Maximisation (EM) algorithm to be the superior choice. Furthermore we derive closed form expressions for the associated Wald confidence intervals (error) estimated by the EM algorithm. Previous attempts to calculate such intervals relied on numerical schemes which were slower and less stable. We further provide a closed form expression (via the Delta method) to transfer these errors to the level of the transition probabilities, which are more intuitive. Although one can establish more precise mathematical results with the Markov assumption, there is empirical evidence suggesting this assumption is not valid. We finish this part by carrying out empirical research on non-Markov phenomena and propose a model to capture the so-called rating momentum. This model has many appealing features and is a natural extension to the Markov set up. The second part is based on McKean Vlasov Stochastic Differential Equations (MV-SDEs), these Stochastic Differential Equations (SDEs) arise from looking at the limit, as the number of weakly interacting particles (e.g. gas particles) tends to infinity. The resulting SDE has coefficients which can depend on its own law, making them theoretically more involved. Although MV-SDEs arise from statistical physics, there has been an explosion in interest recently to use MV-SDEs in models for economics. We firstly derive an explicit approximation scheme for MV-SDEs with one-sided Lipschitz growth in the drift. Such a condition was observed to be an issue for standard SDEs and required more sophisticated schemes. There are implicit and explicit schemes one can use and we develop both types in the setting of MV-SDEs. Another main issue for MVSDEs is, due to the dependency on their own law they are extremely expensive to simulate compared to standard SDEs, hence techniques to improve computational cost are in demand. The final result in this part is to develop an importance sampling algorithm for MV-SDEs, where our measure change is obtained through the theory of large deviation principles. Although importance sampling results for standard SDEs are reasonably well understood, there are several difficulties one must overcome to apply a good importance sampling change of measure in this setting. The importance sampling is used here as a variance reduction technique although our results hint that one may be able to use it to reduce propagation of chaos error as well. Finally we consider stochastic algorithms to solve PDEs. It is known one can achieve numerical advantages by using probabilistic methods to solve PDEs, through the so-called probabilistic domain decomposition method. The main result of this part is to present an unbiased stochastic representation for a first order PDE, based on the theory of branching diffusions and regime switching. This is a very interesting result since previously (Itô based) stochastic representations only applied to second order PDEs. There are multiple issues one must overcome in order to obtain an algorithm that is numerically stable and solves such a PDE. We conclude by showing the algorithm’s potential on a more general first order PDE.