In this thesis we will focus on a number of issues related to the modelling and simulation of stochastic processes. In particular, we will focus on three areas, namely dynamics, delay problems, and Forward-Backward problems and their efficient numerical methods.
First, we will relate the dynamics of numerical methods for SDEs to the dynamics of the solution of a Stochastic Differential Equation (SDE). This will give important insights into the behaviour of numerical methods for solAdng SDEs from a distribution viewpoint.
Secondly, since Stochastic Delay Differential Equations (SDDEs) can be considered as a generalisation of both Delay Differential Equations (DDEs) and SDEs, we relate the dynamics of numerical SDDEs with that of DDEs. By investigating these two dynamical behaviours, we will obtain deeper insights to the dynamics of numerical SDDEs itself.
Finally, since one of the
most appealing features of Backward Stochastic Differential Equations (BSDEs) and Forward-Backward Stochastic Differential Equations (FBSDEs) is that they can be applied to finance problems and give deep insights into them. However, the availability of numerical methods for solving these problems is still very limited. In this thesis we propose the Waveform-Relaxation method for solving FBSSDEs problems and demonstrate their performance by windowing on a number of simulations.