Analytic Approximation of Transition Probabilities

A transition probability is essentially a likelihood of ’something random’ transitioning from one state of being to another. Though, more formally, for all intents and purposes, the ’something random’ is a sequence of random events, which is a stochastic process. There are many stochastic processes that are valuable to understand. Examples can be found in a multitude of topics, from finance to biology and physics. Many naturally occurring stochastic processes cannot be explicitly defined. However, there are many stochastic models that describe their behaviour very well. Once a stochastic model is established, transition probabilities for the underlying stochastic process can also be described, which in turn allows some predictability. Estimations of transition probabilities are often limited in terms of convergence. The Euler-Maruyama (E-M) method for instance, is a numerical method for approximating Stochastic differential equations that represent the behaviour of stochastic processes. This means that it effectively approximates transition probabilities between time-steps of a process. The E-M method is weakly convergent with order 1, which entails that the error of the approximation decreases linearly with the size of the time-step. In this study, the aim is to beat the aforementioned linear decrease in error of approximation via an analytic approximation using a combination of the E-M method and the Laplace method for some well known stochastic models. The two methods work neatly in conjunction, however, a correctional function is necessary for the Laplace method to work due to the nature of the problem. The resulting approach shows astonishingly good results with room for further improvements. (Less)