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

Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions

We establish new existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam. We consider the widely studied boundary conditions corresponding to clamped and hinged ends and many non-local boundary conditions, with a unified approach. Our method is to show that each boundary-value problem can be written as the same type of perturbed integral equation, in the space $C[0,1]$, involving a linear functional $\alpha[u]$ but, although we seek positive solutions, the functional is not assumed to be positive for all positive $u$. The results are new even for the classic boundary conditions of clamped or hinged ends when $\alpha[u]=0$, because we obtain sharp results for the existence of one positive solution; for multiple solutions we seek optimal values of some of the constants that occur in the theory, which allows us to impose weaker assumptions on the nonlinear term than in previous works. Our non-local boundary conditions contain multi-point problems as special cases and, for the first time in fourth-order problems, we allow coefficients of both signs

Symmetry of Nodal Solutions for Singularly Perturbed Elliptic Problems on a Ball

In [40], it was shown that the following singularly perturbed Dirichlet problem \ep^2 \Delta u - u+ |u|^{p-1} u=0, \ \mbox{in} \ \Om,\] \[ u=0 \ \mbox{on} \ \partial \Om has a nodal solution u_\ep which has the least energy among all nodal solutions. Moreover, it is shown that u_\ep has exactly one local maximum point P_1^\ep with a positive value and one local minimum point P_2^\ep with a negative value and, as \ep \to 0, \varphi (P_1^\ep, P_2^\ep) \to \max_{ (P_1, P_2) \in \Om \times \Om } \varphi (P_1, P_2), where \varphi (P_1, P_2)= \min (\frac{|P_1-P_2}{2}, d(P_1, \partial \Om), d(P_2, \partial \Om)). The following question naturally arises: where is the {\bf nodal surface} \{ u_\ep (x)=0 \}? In this paper, we give an answer in the case of the unit ball \Om=B_1 (0). In particular, we show that for \epsilon sufficiently small, P_1^\ep, P_2^\ep and the origin must lie on a line. Without loss of generality, we may assume that this line is the x_1-axis. Then u_\ep must be even in x_j, j=2, ..., N, and odd in x_1. As a consequence, we show that \{ u_\ep (x)=0 \} = \{ x \in B_1 (0) | x_1=0 \}. Our proof is divided into two steps: first, by using the method of moving planes, we show that P_1^\ep, P_2^\ep and the origin must lie on the x_1-axis and u_\ep must be even in x_j, j=2, ..., N. Then, using the Liapunov-Schmidt reduction method, we prove the uniqueness of u_\ep (which implies the odd symmetry of u_\ep in x_1). Similar results are also proved for the problem with Neumann boundary conditions

Inverse heat conduction problems by using particular solutions

Based on the method of fundamental solutions, we develop in this paper a new computational method to solve two-dimensional transient heat conduction inverse problems. The main idea is to use particular solutions as radial basis functions (PSRBF) for approximation of the solutions to the inverse heat conduction problems. The heat conduction equations are first analyzed in the Laplace transformed domain and the Durbin inversion method is then used to determine the solutions in the time domain. Least-square and singular value decomposition (SVD) techniques are adopted to solve the ill-conditioned linear system of algebraic equations obtained from the proposed PSRBF method. To demonstrate the effectiveness and simplicity of this approach, several numerical examples are given with satisfactory accuracy and stability.Peer reviewe