Diffusion processes provide a natural way of modelling a variety of physical and economic phenomena. It is often the case that one is unable to observe a diffusion process directly, and must instead rely on noisy observations that are discretely spaced in time. Given these discrete, noisy observations, one is faced with the task of inferring properties of the underlying diffusion process. For example, one might be interested in inferring the current state of the process given observations up to the present time (this is known as the filtering problem). Alternatively, one might wish to infer parameters governing the time evolution the diffusion process. In general, one cannot apply Bayes’ theorem directly, since the transition density of a general nonlinear diffusion is not computationally tractable. In this thesis, we investigate a novel method of simplifying the problem. The stochastic differential equation that describes the diffusion process is replaced with a simpler ordinary differential equation, which has a random driving noise that approximates Brownian motion. We show how one can exploit this approximation to improve on standard methods for inferring properties of nonlinear diffusion processes.

We are interested in a nonlinear filtering problem motivated by an information-based approach for modelling the dynamic evolution of a portfolio of credit risky securities. We solve this problem by `change of measure method\\\' and show the existence of the density of the unnormalized conditional distribution which is a solution to the Zakai equation. Zakai equation is a linear SPDE which, in general, cannot be solved analytically. We apply Galerkin method to solve it numerically and show the convergence of Galerkin approximation in mean square. Lastly, we design an adaptive Galerkin filter with a basis of Hermite polynomials and we present numerical examples to illustrate the effectiveness of the proposed method. The work is closely related to the paper Frey and Schmidt (2010).

We are interested in a nonlinear filtering problem motivated by an information-based approach for modelling the dynamic evolution of a portfolio of credit risky securities. We solve this problem by `change of measure method\\\'' and show the existence of the density of the unnormalized conditional distribution which is a solution to the Zakai equation. Zakai equation is a linear SPDE which, in general, cannot be solved analytically. We apply Galerkin method to solve it numerically and show the convergence of Galerkin approximation in mean square. Lastly, we design an adaptive Galerkin filter with a basis of Hermite polynomials and we present numerical examples to illustrate the effectiveness of the proposed method. The work is closely related to the paper Frey and Schmidt (2010).

Recent debiasing techniques are incorporated into the Ensemble Kalman-Bucy Filter (EnKBF). Specifically, a novel double randomization is applied. The EnKBF is a Monte Carlo (MC) method that approximates the Kalman-Bucy Filter (KBF), which in turn can be seen as the continuous-time version of the celebrated discrete-time Kalman Filter (KF). The KF is a method that combines sequential observations with an underlying dynamics model to predict the state of the quantity of interest. Our interest in the EnKBF comes from its relevance in high dimensions, where it overcomes the curse of dimensionality and outperforms other standard methods like the Particle Filter. We will consider debiasing techniques (also termed unbiased estimators) in order to improve the error-to-cost rate. Unbiased estimators are variance reduction techniques that produce unbiased and finite variance estimators. Applications of the EnKBF are numerous, from atmospheric sciences, numerical weather prediction, finance, machine learning, among others. Thus, improving the EnKBF is of interest. Numerical tests are done in order to evaluate the cost and the error-to-cost rate of the algorithm, where we consider Ornstein-Uhlenbeck processes. Specifically, a numerical comparison with the Multilevel Ensemble Kalman-Bucy Filter (MLEnKBF) is made using two different unbiased estimators, the coupled sum and the single term estimators. Additionally, we test two variants of the EnKBF, the Vanilla EnKBF, and the Deterministic EnKBF. We find that the error-to-cost rate is virtually the same, although the cost of the unbiased EnKBF is much higher.