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The object of study in this thesis is a number of different models of branching Levy processes in inhomogeneous breeding potential. We employ some widely-used spine techniques to investigate various features of these models for their subsequent comparison. The thesis is divided into 5 chapters. In the first chapter we introduce the general framework for branching Markov processes within which we are going to present all our results. In the second chapter we consider a branching Brownian motion in the potential β|·|p, β> 0, p ≥0. We give a new proof of the result about the critical value of p for the explosion time of the population. The main advantage of the new proof is that it can be easily generalised to other models. The third chapter is devoted to continuous-time branching random walks in the potential β|·|p, β> 0, p ≥0. We give results about the explosion time and the right most particle behaviour comparing them with the known results for the branching Brownian motion. In the fourth chapter we look at general branching Levy processes in the potential β|·|p, β> 0, p ≥0. Subject to certain assumptions we prove some results about the explosion time and the rightmost particle. We exhibit how the corresponding results for the branching Brownian motion and and the branching random walk fit into the general structure. The last chapter considers a branching Brownian motion with branching taking place at the origin on the local time scale. We present some results about the population dynamics and the right most particle behaviour. We also prove the Strong Law of Large Numbers for this model.

In this work we develop a Markov Chain Jump-Diffusion (MCJD) model, where we have a financial market in which there are several possible states. Asset prices in the market follow a generalised geometric Brownian motion, with drift and volatility depending on the state of the market. So for example, one state may represent a bull market where drifts are high, whilst another state may represent a bear market where where drifts are low. The state the market is in is governed by a continuous time Markov chain. We add to this diffusion process jumps in the asset price which occur when the market changes state, and the jump sizes are dependent on the states the market is transiting to and transiting from. We also allow the market to transit to the same state, which corresponds to a jump in the asset price with no change to the drift or volatility. We will develop conditions of no arbitrage in such a market, and methods for pricing derivatives of assets whose prices follow MCJD processes. We will also consider Term-Structure models where the short rate (or forward rate) follows an MCJD process.

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