This thesis is concerned with developing MCMC methodology for sampling from given target distributions. In particular we are interested in situations where sampling is difficult using current techniques, and where we use methods which can be seen as exact approximations of preferable algorithms that for various reasons might be unavailable or impractical. We develop algorithms that attempt to reduce the autocorrelation seen in the output of MCMC samplers using the principle of local approximation, and we explore the performance of existing methods that apply in situations where the target density may not be evaluated at every point. This leads to guidelines that suggest how computational resources should best be allocated to yield low-variance estimators.

Dynamic models of forward looking agents, whose goal is to maximize expected in-tertemporal payoffs, are useful modelling frameworks in economics. With an exception of a small class of dynamic decision processes, the estimation of the primitives in these models is computationally burdensome due to the presence of the value functions that has no closed form. We follow a popular two-step approach which estimates the functions of interest rather than use direct numerical approximation. The first chapter, joint with Oliver Linton, considers a class of dynamic discrete choice models that contain observable continuously distributed state variables. Most papers on the estimation of dynamic discrete choice models assume that the observable state variables can only take finitely many values. We show that the extension to the infinite dimensional case leads to a well-posed inverse problem. We derive the distribution theory for the finite and the infinite dimensional parameters. Dynamic models with continuous choice can sometimes avoid the numerical issues related to the value function through the use of Euler's equation. The second chapter considers models with continuous choice that do not necessarily belong to the Euler class but frequently arise in applied problems. In this chapter, a class of minimum distance estimators is proposed, their distribution theory along with the infinite dimensional parameters of the decision models are derived. The third chapter demonstrates how the methodology developed for the discrete and continuous choice problems can be adapted to estimate a variety of other dynamic models. The final chapter discusses an important problem, and provides an example, where some well-known estimation procedures in the literature may fail to consistently estimate an identified model. The estimation methodologies I propose in the preceding chapters may not suffer from the problems of this kind.

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The main aim of this thesis is to examine stability properties of the solutions to stochastic differential equations (SDEs) driven by Levy noise. Using key tools such as Ito's formula for general semimartingales, Kunita's moment estimates for Levy-type stochastic integrals, and the exponential martingale inequality, we find conditions under which the solutions to the SDEs under consideration are stable in probability, almost surely and moment exponentially stable. In addition, stability properties of stochastic functional differential equations (SFDEs) driven by Levy noise are examined using Razumikhin type theorems. In the existing literature the problem of stochastic stabilization and destabilization of first order non-linear deterministic systems has been investigated when the system is perturbed with Brownian motion. These results are extended in this thesis to the case where the deterministic system is perturbed with Levy noise. We mainly focus on the stabilizing effects of the Levy noise in the system, prove the existence of sample Lyapunov exponents of the trivial solution of the stochastically perturbed system, and provide sufficient criteria under which the system is almost surely exponentially stable. From the results that are established the Levy noise plays a similar role to the Brownian motion in stabilizing dynamical systems. We also establish the variation of constants formula for linear SDEs driven by Levy noise. This is applied to study stochastic stabilization of ordinary functional differential equation systems perturbed with Levy noise.

Option prices in the Black-Scholes model can usually be expressed as solutions of partial differential equations (PDE). In general exponential Levy models an additional integral term has to be added and the prices can be expressed as solutions of partial integro-differential equations (PIDE). The sensitivity of a price function to changes in its arguments is given by its derivatives, in finance known as greeks. The greeks can be obtained as a solution to a PDE or PIDE which is obtained by differentiating the equation and side conditions of the price function. We call the method of simultaneously solving the equations for the price function and the greeks the dynamic partial (-integro) differential approach. So far this approach has been analysed for a few contracts in the Black-Scholes model and in a Markov Chain model. In this thesis, we extend the use of the dynamic approach in the Black-Scholes model and apply it to a financial market where the underlying stock prices are driven by Levy processes. We derive and solve systems of equations that determine the price and the greeks both for vanilla and for exotic options. In particular we are interested in options whose prices depend only on time and one state variable. Furthermore, we calculate sensitivities of option prices with respect to changes in the stochastic model of the underlying price process. Such sensitivities can again be expressed as solutions to PIDE. The occurring systems of PIDE are solved numerically via a finite difference approach and the results are compared with simulation and numerical integration methods to compute prices and sensitivities. We show that the dynamic approach in many cases outperforms its competitors. Finally, we investigate the smoothness of the price functions and give conditions for the existence of solutions of the PIDE.

In this thesis we use Markov chain Monte Carlo (MCMC) simulation to calibrate a two-factor arbitrage-free model for the term structure of interest rates which is proposed by Cairns (2004a) based on the positive-interest framework (Flesaker and Hughston, 1996). The model is a time-homogeneous model driven by latent state variables which follow a two-dimensional Ornstein-Uhlenbeck process. A number of MCMC algorithms are developed and employed for estimating both model parameters and latent variables where simulated data are used in the first place in order to validate the algorithms and ensure that they can result in reasonable and reliable estimates before using UK market data. Once the posterior estimates are obtained, we next investigate goodness of fit of the model and eventually assess the impact of parameter uncertainty on the forecasting of yield curves in which the achieved MCMC output can be used directly. Additionally, the developed algorithm is also applied for estimating the two-factor Vasicek term structure model for comparison. We conclude that our algorithms work reasonably well for estimating the Cairns term structure model. The model is then fitted to UK Strips data, and it found to produce reasonable fits for medium- and long-term yields, but we also conclude that some improvement may be required for the short-end of the yield curves.

We consider the numerical approximation of stochastic differential equations interpreted both in the It^o and Stratonovich sense and develop three stochastic time-integration techniques based on the deterministic exponential time differencing schemes. Two of the numerical schemes are suited for the simulations of It^o stochastic ordinary differential equations (SODEs) and they are referred to as the stochastic exponential time differencing schemes, SETD0 and SETD1. The third numerical scheme is a new numerical method we propose for the simulations of Stratonovich SODEs. We call this scheme, the Exponential Stratonovich Integrator (ESI). We investigate numerically the convergence of these three numerical methods, in addition to three standard approximation schemes and also compare the accuracy and efficiency of these schemes. The effect of small noise is also studied. We study the theoretical convergence of the stochastic exponential time differencing scheme (SETD0) for parabolic stochastic partial differential equations (SPDEs) with infinite-dimensional additive noise and one-dimensional multiplicative noise. We obtain a strong error temporal estimate of O(¢tµ + ²¢tµ + ²2¢t1=2) for SPDEs forced with a one-dimensional multiplicative noise and also obtain a strong error temporal estimate of O(¢tµ + ²2¢t) for SPDEs forced with an infinite-dimensional additive noise. We examine convergence for second-order and fourth-order SPDEs. We consider the effects of spatially correlated and uncorrelated noise on bifurcations for SPDEs. In particular, we study a fourth-order SPDE, the Swift-Hohenberg equation, and allow the control parameter to fluctuate. Numerical simulations show a shift in the pinning region with multiplicative noise.