Given a compensated Poisson process $(X_t)_{t \geq 0}$ based on $(\Omega, \mathcal{F}, \mathbb{P})$, the Wiener-Poisson isomorphism $\mathcal{W} : \mathfrak{F}_+(L^2 (\mathbb{R}_+)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$ is constructed. We restrict the isomorphism to $\mathfrak{F}_+(L^2 [0,1])$ and prove some novel properties of the Poisson exponentials $\mathcal{E}(f) := \mathcal{W}(e(f))$. A new proof of the result $\Lambda_t + A_t + A^{\dagger}_t = \mathcal{W}^{-1}\widehat{X_t} \mathcal{W}$ is also given. The analogous results for $\mathfrak{F}_+(L^2 (\mathbb{R}_+))$ are briefly mentioned. The concept of a compensated Poisson process over $\mathbb{R}_+$ is generalised to any measure space $(M, \mathcal{M}, \mu)$ as an isometry $I : L^2(M, \mathcal{M}, \mu) \to L^2 (\Omega,\mathcal{F}, \mathbb{P})$ satisfying certain properties. For such a generalised Poisson process we recall the construction of the generalised Wiener-Poisson isomorphism, $\mathcal{W}_I : \mathfrak{F}_+(L^2(M)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$, using Charlier polynomials. Two alternative constructions of $\mathcal{W}_I$ are also provided, the first using exponential vectors and then deducing the connection with Charlier polynomials, and the second using the theory of reproducing kernel Hilbert spaces. Given any measure space $(M, \mathcal{M}, \mu)$, we construct a canonical generalised Poisson process $I : L^2 (M, \mathcal{M}, \mu) \to L^2(\Delta, \mathcal{B}, \mathbb{P})$, where $\Delta$ is the maximal ideal space, with $\mathcal{B}$ the completion of its Borel $\sigma$-field with respect to $\mathbb{P}$, of a $C^*$-algebra $\mathcal{A} \subseteq \mathfrak{B}(\mathfrak{F}_+(L^2(M)))$. The Gelfand transform $\mathcal{A} \to \mathfrak{B}(L^2(\Delta))$ is unitarily implemented by the Wiener-Poisson isomorphism $\mathcal{W}_I: \mathfrak{F}_+(L^2(M)) \to L^2(\Delta)$. This construction only uses operator algebra theory and makes no a priori use of Poisson measures. A new Fock space proof of the quantum Ito formula for $(\Lambda_t + A_t + A^{\dagger}_t)_{0 \leq t \leq 1}$ is given. If $(F_{\ \! \! t})_{0 \leq t \leq 1}$ is a real, bounded, predictable process with respect to a compensated Poisson process $(X_t)_{0 \leq t \leq 1}$, we show that if $M_t = \int_0^t F_s dX_s$, then on $\mathsf{E}_{\mathrm{lb}} := \mathrm{linsp} \{ e(f) : f \in L^2_{\mathrm{lb}}[0,1] \}$, $\mathcal{W}^{-1} \widehat{M_t} \mathcal{W} = \int_0^t \mathcal{W}^{-1} \widehat{F_s} \mathcal{W} (d\Lambda_s + dA_s + dA^{\dagger}_s),$ and that $(\mathcal{W}^{-1} \widehat{M_t} \mathcal{W})_{0 \leq t \leq 1}$ is an essentially self-adjoint quantum semimartingale. We prove, using the classical Ito formula, that if $(J_t)_{0 \leq t \leq 1}$ is a regular self-adjoint quantum semimartingale, then $(\mathcal{W} \widehat{M_t} \mathcal{W}^{-1} + J_t)_{0 \leq t \leq 1}$ is an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula, and hence the quantum Ito formula. The equivalent result for the sum of a Brownian and Poisson martingale, provided that the sum is essentially self-adjoint with core $\mathsf{E}_{\mathrm{lb}}$, is also proved.

Model checking linear-time properties expressed in first-order logic has non-elementary complexity, and thus various restricted logical languages are employed. In the first part of this dissertation we consider two such restricted specification logics on words: linear temporal logic (LTL) and two-variable first-order logic (FO2). LTL is more expressive but FO2 can be more succinct, and hence it is not clear which should be easier to verify. We take a comprehensive look at the issue, giving a comparison of verification problems for FO2, LTL, and various sublogics thereof across a wide range of models. In particular, we look at unary temporal logic (UTL), a subset of LTL that is expressively equivalent to FO2. We give three logic-to-automata translations which can be used to give upper bounds for FO2 and UTL and various sublogics. We apply these to get new bounds for model checking both non-deterministic systems (hierarchical and recursive state machines, games) and for probabilistic systems (Markov chains, recursive Markov chains, and Markov decision processes). Our results give a unified approach to understanding the behaviour of FO2, LTL, and their sublogics. We further consider the problem of computing maximal probabilities for interval Markov chains (and recursive interval Markov chains, stochastic context-free grammars) to satisfy LTL specifications. Using again our automata constructions we describe an expectation-maximisation algorithm to solve this problem in practice. Our algorithm can be seen as a variant of the classical Baum-Welch algorithm on hidden Markov models. We also introduce a publicly available on-line tool Tulip to perform such analysis. Finally, we investigate the extension of our techniques from words to trees. We show that the parallel between the complexity of FO2 satisfiability on general and on restricted structures breaks down as we move from words to trees, since trees allow one to encode alternating exponential time computation.

This thesis consists of two parts. The first part (Chapters 2-4) focuses on the problem of inverting the signature of a path of bounded variation, and we present three results here. First, we give an explicit inversion formula for any axis path in terms of its signature. Second, we show that for relatively smooth paths, the derivative at the end point can be approximated arbitrarily closely by its signature sequence, and we provide explicit error estimates. As an application, we give an effective inversion procedure for piecewise linear paths. Finally, we prove a uniform estimate for the signatures of paths of bounded variations, and obtain a reconstruction theorem via that uniform estimate. Although this general reconstruction theorem is not computationally efficient, the techniques involved in deriving the uniform estimate are useful in other situations, and we also give an application in the case of expected signatures for Brownian motion. The second part (Chapter 5) deals with rough paths. After introducing proper backgrounds, we extend the uniform estimate above to the context of rough paths, and show how it can lead to simple proofs of distance bounds for Gaussian iterated integrals.

We study a natural generalization of the Λ-coalescent to a spatial continuum. We introduce the process, which is known as the Segregated Λ-coalescent, via its connections to the (non-spatial) Λ-coalescent and the Spatial Λ-Fleming-Viot process. The main new results contained in this thesis are as follows. The Segregated Λ-coalescent has a non-trivial construction which we present here in terms of stochastic flows. We describe the qualitative behaviour of the Segregated Λ-coalescent and compare it to the behaviour of the Λ-coalescent, showing in particular that the Segregated Λ-coalescent has an extra phase transition which is directly related to the introduction of space. We finish with some results concerning the rate at which the Segregated Λ-coalescent comes down from infinity.

We develop the theory of chaos spaces and chaos matrices. A chaos space is a Hilbert space with a fixed, countably-infinite, direct-sum decomposition. A chaos matrix between two chaos spaces is a doubly-infinite matrix of bounded operators which respects this decomposition. We study operators represented by such matrices, particularly with respect to self-adjointness. This theory is used to re-formulate the quantum stochastic calculus of Hudson and Parthasarathy. Integrals of chaos-matrix processes are defined using the Hitsuda-Skorokhod integral and Malliavin gradient,following Lindsay and Belavkin. A new way of defining adaptedness is developed and the consequent quantum product Ito formula is used to provide a genuine functional Ito formula for polynomials in a large class of unbounded processes, which include the Poisson process and Brownian motion. A new type of adaptedness, known as $\Omega$-adaptedness, is defined. We show that quantum stochastic integrals of $\Omega$-adapted processes are well-behaved; for instance, bounded processes have bounded integrals. We solve the appropriate modification of the evolution equation of Hudson and Parthasarathy: $U(t)=I+\int_{0}^{t}E(s)\mathrm{d}\Lambda(s)+F(s)\mathrm{d} A(s)+ G(s)U(s)\mathrm{d} A^{\dagger}(s)+H(s)U(s)\mathrm{d} s, $ where the coefficients are time-dependent, bounded, $\Omega$-adapted processes acting on the whole Fock space. We show that the usual conditions on the coefficients, viz. $(E,F,G,H)=(W-I,L,-WL^{*},iK+\mbox{$\frac{1}{2}$}LL^{*})$ where $W$ is unitary and $K$ self-adjoint, are necessary and sufficient conditions for the solution to be unitary. This is a very striking result when compared to the adapted case.

The heat content of a domain D of &Ropf;^d is defined as</sp> < p >E(s) = &int;_D u(s,x)dx, where u is the solution to the heat equation with zero initial condition and unit Dirichlet boundary condition. This thesis studies the behaviour of E(s) for small s with a particular emphasis on the case where $D$ is a planar domain whose boundary is a random Koch curve. When &part;D is spatially homogeneous, we show that we can recover the upper and lower Minkowski dimensions of &part;D from E(s). Furthermore, in some cases where the Minkowski dimension does exist, finer fluctuations can be recovered and the heat content is controlled by s^&alpha; exp{f (log(1/s)} for small s, for some positive &alpha; and some regularly varying function f. When &part;D is statistically self-similar, the heat content asymptotics are studied using a law of large numbers for the general branching process, and we show that the Minkowski dimension and content of &part;D exist and can be recovered from E(s). More precisely the heat content has an almost sure expansion E(s) = c₁ s^&alpha; N_&infin; + o(s^&alpha;), a.s. for small s, for some positive c₁ and &alpha; and a positive random variable N_&infin; with unit expectation. To study the fluctuations around these asymptotics, we prove a central limit theorem for the general branching process. The proof follows a standard Taylor expansion argument and relies on the independence built into the general branching process. The limiting distribution established here is reminiscent of those arising in central limit theorems for martingales. When &part;D is a statistically self-similar Cantor subset of &Ropf;, we discuss examples where we have and fail to have a central limit theorem for the heat content. We conclude with an open question about the fluctuations of the heat content when &part;D is a statistically self-similar Koch curve.

This thesis consists of two parts. The first part (Chapters 2-4) considers multilevel Monte Carlo for option pricing in finite activity jump-diffusion models. We use a jump-adapted Milstein discretisation for constant rate cases and with the thinning method for bounded state-dependent rate cases. Multilevel Monte Carlo estimators are constructed for Asian, lookback, barrier and digital options. The computational efficiency is numerically demonstrated and analytically justified. The second part (Chapter 5) deals with option pricing problems in exponential Lévy models where the increments of the underlying process can be directly simulated. We discuss several examples: Variance Gamma, Normal Inverse Gaussian and alpha-stable processes and present numerical experiments of multilevel Monte Carlo for Asian, lookback, barrier options, where the running maximum of the Lévy process involved in lookback and barrier payoffs is approximated using discretely monitored maximum. To analytically verify the computational complexity of multilevel method, we also prove some upper bounds on L^p convergence rate of discretely monitored error for a broad class of Lévy processes.

Barabási-Albert random graph models are a class of evolving random graphs that are frequently used to model social networks with scale-free degree distributions. It has been shown that Barabási-Albert random graph models have asymptotic scale-free degree distributions as the size of the graph tends to infinity. Real world networks, however, have finite size so it is important to know how close the degree distribution of a Barabási-Albert random graph of a given size is to its asymptotic distribution. Stein’s method is chosen as one main method for obtaining explicit bounds for the distance between distributions. We derive a new version of Stein’s method for a class of scale-free distributions and apply the method to a Barabási-Albert random graph. We compare the evolution of a sequence of Barabási-Albert random graphs with continuous time stochastic processes motivated by Yule’s model for evolution. Through a coupling of the models we bound the total variation distance between their degree distributions. Using these bounds, we extend degree distribution bounds that we find for specific models within the scheme to find bounds for every member of the scheme. We apply the Azuma-Hoeffding inequality and Chernoff bounds to find bounds between the degree sequences of the random graph models and the given scale-free distribution. These bounds prove that the degree sequences converge completely (and therefore also converge almost surely) to our scale-free distribution. We discuss the relationship between the random graph processes and the Chinese restaurant process. Aided by the construction of an inhomogeneous Markov chain, we apply our results for the degree distribution in a Barabási-Albert random graph to a particular statistic of the Chinese restaurant process. Finally, we explore how our methods can be adapted and extended to other evolving random graph processes. We study a Bernoulli evolving random graph process, for which we bound the distance between its degree distribution and a geometric distribution and we bound the distance between the number of triangles in the graph and a normal distribution.

In this thesis, we consider a class of stochastic dynamics running backwards, so called backward stochastic differential equations (BSDEs) in the literature. We demonstrate BSDEs can be reformulated as functional differential equations defined on path spaces, and therefore solving BSDEs is equivalent to solving the associated functional differential equations. With such observation we can solve BSDEs on general filtered probability space satisfying the usual conditions, and in particular without the requirement of the martingale representation. We further solve the above functional differential equations numerically, and propose a numerical scheme based on the time discretization and the Picard iteration. This in turn also helps us solve the associated BSDEs numerically. In the second part of the thesis, we consider a class of BSDEs with quadratic growth (QBSDEs). By using the functional differential equation approach introduced in this thesis and the idea of the Cole-Hopf transformation, we first solve the scalar case of such QBSDEs on general filtered probability space satisfying the usual conditions. For a special class of QBSDE systems (not necessarily scalar) in Brownian setting, we do not use such Cole-Hopf transformation at all, and instead introduce the weak solution method, which is to use the strong solutions of forward backward stochastic differential equations (FBSDEs) to construct the weak solutions of such QBSDE systems. Finally we apply the weak solution method to a specific financial problem in the credit risk setting, where we modify the Merton's structural model for credit risk by using the idea of indifference pricing. The valuation and the hedging strategy are characterized by a class of QBSDEs, which we solve by the weak solution method.

We present a unifying framework for the global optimization of functions which are expensive to evaluate. The framework is based on a Bayesian interpretation of radial basis function interpolation which incorporates existing methods such as Kriging, Gaussian process regression and neural networks. This viewpoint enables the application of Bayesian decision theory to derive a sequential global optimization algorithm which can be extended to include existing algorithms of this type in the literature. By posing the optimization problem as a sequence of sampling decisions, we optimize a general cost function at each stage of the algorithm. An extension to multi-stage decision processes is also discussed. The key idea of the framework is to replace the underlying expensive function by a cheap surrogate approximation. This enables the use of existing branch and bound techniques to globally optimize the cost function. We present a rigorous analysis of the canonical branch and bound algorithm in this setting as well as newly developed algorithms for other domains including convex sets. In particular, by making use of Lipschitz continuity of the surrogate approximation, we develop an entirely new algorithm based on overlapping balls. An application of the framework to the integration of expensive functions over rectangular domains and spherical surfaces in low dimensions is also considered. To assess performance of the framework, we apply it to canonical examples from the literature as well as an industrial model problem from oil reservoir simulation.