Stochastic modelling and equilibrium in mathematical finance and statistical sequential analysis

The focus of this thesis are the equilibrium problem under derivative market imbalance, the sequential analysis problems for some time-inhomogeneous diffusions and multidimensional Wiener processes, and the first passage times of certain non-affine jump-diffusions. First, we investigate the impact of imbalanced derivative markets - markets in which not all agents hedge - on the underlying stock market. The availability of a closed-form representation for the equilibrium stock price in the context of a complete (imbalanced) market with terminal consumption allows us to study how this equilibrium outcome is affected by the risk aversion of agents and the degree of imbalance. In particular, it is shown that the derivative imbalance leads to significant changes in the equilibrium stock price process: volatility changes from constant to local, while risk premia increase or decrease depending on the replicated contingent claim, and become stochastic processes. Moreover, the model produces implied volatility smiles consistent with empirical observations. Secondly, we study the sequential hypothesis testing and quickest change-point (disorder) detection problem with linear delay penalty costs for certain observable time-inhomogeneous Gaussian diffusions and fractional Brownian motions. The method of proof consists of the reduction of the initial problems into the associated optimal stopping problems for onedimensional time-inhomogeneous diffusion processes and the analysis of the associated free boundary problems. We derive explicit estimates for the Bayesian risk functions and optimal stopping boundaries for the associated weighted likelihood ratios and obtain their exact rates of convergence under large time values. Thirdly, we study the quickest change-point detection problems for the correlated components of a multidimensional Wiener process changing their drift rates at certain random times. These problems seek to determine the times of alarm which are as close as possible to the unknown change-point (disorder) times at which some of the components have changed their drift rates. The optimal times of alarm are shown to be the first times at which the appropriate posterior probability processes exit certain regions restricted by the stopping boundaries. We characterize the value functions and optimal boundaries as unique solutions of the associated free boundary problems for partial differential equations. We provide estimates for the value functions and boundaries which are solutions to the appropriately constructed ordinary differential free boundary problems. Fourthly, we compute the Laplace transforms of the first times at which certain non-affine one-dimensional jump-diffusion processes exit connected regions restricted by two constant boundaries. The method of proof is based on the solution of the associated integro-differential boundary problems for the corresponding value functions. We derive analytic expressions for the Laplace transforms of the first exit times of the jump-diffusion processes driven by compound Poisson processes with multi-exponential jumps. The results are illustrated on the constructed non-affine pure jump analogues of the diffusion processes which represent closed-form solutions of the appropriate stochastic differential equations. Finally, we obtain closed-form expressions for the values of generalised Laplace transforms of the first times at which two-dimensional jump-diffusion processes exit from regions formed by constant boundaries. It is assumed that the processes form the models of stochastic volatility with independent driving Brownian motions and independent compound Poisson processes with exponentially distributed jumps. The proof is based on the solution to the equivalent boundary-value problems for partial integro-differential operators. We illustrate our results in the examples of Stein and Stein, Heston, and other jump analogues of stochastic volatility models.