Stochastic delay difference and differential equations: applications to financial markets

摘要

This thesis deals with the asymptotic behaviour of stochastic difference and functional differential equationsudof Itˆo type. Numerical methods which both minimise error and preserve asymptotic features of the underlying continuous equation are studied. The equations have a form which makes them suitable to model financial markets in which agents use past prices. The second chapter deals with the behaviour of moving average models of price formation. We show that the asset returns are positively and exponentially correlated, while the presence of feedback traders causes either excess volatility or a market bubble or crash. udThese results are robust to the presence of nonlinearities in the traders’ demand functions. In Chapters 3 andud4, we show that these phenomena persist if trading takes place continuously by modelling the returns usingudlinear and nonlinear stochastic functional differential equations (SFDEs). In the fifth chapter, we assumeudthat some traders base their demand on the difference between current returns and the maximum return overudseveral trading periods, leading to an analysis of stochastic difference equations with maximum functionals.udOnce again it is shown that prices either fluctuate or undergo a bubble or crash. In common with the earlierudchapters, the size of the largest fluctuations and the growth rate of the bubble or crash is determined. Theudlast three chapters are devoted to the discretisation of the SFDE presented in Chapter 4. Chapter 6 highlightsudproblems that standard numerical methods face in reproducing long–run features of the dynamics ofudthe general continuous–time model, while showing these standard methods work in some cases. Chapter 7uddevelops an alternative method for discretising the solution of the continuous time equation, and shows thatudit preserves the desired long–run behaviour. Chapter 8 demonstrates that this alternative method convergesudto the solution of the continuous equation, given sufficient computational effort.