The following thesis is divided in two main topics. The first part studies variations of optimal prediction problems introduced in Shiryaev, Zhou and Xu (2008) and Du Toit and Peskir (2009) to a randomized terminal-time set up and different families of utility measures. The work presents optimal stopping rules that apply under different criteria, introduces a numerical technique to build approximations of stopping boundaries for fixed terminal time problems and suggest previously reported stopping rules extend to certain generalizations of measures. The second part of the thesis is concerned with analysing optimal wealth allocation techniques within a defaultable financial market similar to Bielecki and Jang (2007). It studies a portfolio optimization problem combining a continuous time jump market and a defaultable security; and presents numerical solutions through the conversion into a Markov Decision Process and characterization of its value function as a unique fixed point to a contracting operator. This work analyses allocation strategies under several families of utilities functions, and highlights significant portfolio selection differences with previously reported results.

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Markov chains offer a rigorous mathematical framework to describe systems that exhibit stochastic behaviour, as they are supported by a plethora of methodologies to analyse their properties. Stochastic process algebras are high-level formalisms, where systems are represented as collections of interacting components. This compositional approach to modelling allows us to describe complex Markov chains using a compact high-level specification. There is an increasing need to investigate the properties of complex systems, not only in the field of computer science, but also in computational biology. To explore the stochastic properties of large Markov chains is a demanding task in terms of computational resources. Approximating the stochastic properties can be an effective way to deal with the complexity of large models. In this thesis, we investigate methodologies to approximate the stochastic behaviour of Markovian process algebra models. The discussion revolves around two main topics: approximate state-space aggregation and stochastic simulation. Although these topics are different in nature, they are both motivated by the need to efficiently handle complex systems. Approximate Markov chain aggregation constitutes the formulation of a smaller Markov chain that approximates the behaviour of the original model. The principal hypothesis is that states that can be characterised as equivalent can be adequately represented as a single state. We discuss different notions of approximate state equivalence, and how each of these can be used as a criterion to partition the state-space accordingly. Nevertheless, approximate aggregation methods typically require an explicit representation of the transition matrix, a fact that renders them impractical for large models. We propose a compositional approach to aggregation, as a means to efficiently approximate complex Markov models that are defined in a process algebra specification, PEPA in particular. Regarding our contributions to Markov chain simulation, we propose an accelerated method that can be characterised as almost exact, in the sense that it can be arbitrarily precise. We discuss how it is possible to sample from the trajectory space rather than the transition space. This approach requires fewer random samples than a typical simulation algorithm. Most importantly, our approach does not rely on particular assumptions with respect to the model properties, in contrast to otherwise more efficient approaches.

This thesis studies three models: Multi-type TASEP in discrete time, long-range last- passage percolation on the line and convoy formation in a travelling servers model. All three models are relatively easy to state but they show a very rich and interesting behaviour. The TASEP is a basic model for a one-dimensional interacting particle system with non-reversible dynamics. We study some aspects of the TASEP in discrete time and compare the results to recently obtained results for the TASEP in continuous time. In particular we focus on stationary distributions for multi-type models, speeds of second- class particles, collision probabilities and the speed process. We consider various natural update rules.

In this work we apply the Adomian decomposition method to an orientation aggregation problem modelling the time distribution of filaments. We find analytical solutions under certain specific criteria and programmatically implement the Adomian method to two variants of the orientation aggregation model. We extend the utility of the Adomian decomposition method beyond its original capability to enable it to converge to more than one solution of a nonlinear problem and further to be used as a corrector in path following bifurcation problems.

The main contribution of this thesis is the introduction of Bayesian quantile regression for hidden Markov models, especially when we have to deal with extreme quantile regression analysis, as there is a limited research to inference conditional quantiles for hidden Markov models, under a Bayesian approach. The first objective is to compare Bayesian extreme quantile regression and the classical extreme quantile regression, with the help of simulated data generated by three specific models, which only differ in the error term’s distribution. It is also investigated if and how the error term’s distribution affects Bayesian extreme quantile regression, in terms of parameter and confidence intervals estimation. Bayesian extreme quantile regression is performed by implementing a Metropolis-Hastings algorithm to update our parameters, while the classical extreme quantile regression is performed by using linear programming. Moreover, the same analysis and comparison is performed on a real data set. The results provide strong evidence that our method can be improved, by combining MCMC algorithms and linear programming, in order to obtain better parameter and confidence intervals estimation. After improving our method for Bayesian extreme quantile regression, we extend it by including hidden Markov models. First, we assume a discrete time finite state-space hidden Markov model, where the distribution associated with each hidden state is a) a Normal distribution and b) an asymmetric Laplace distribution. Our aim is to explore the number of hidden states that describe the extreme quantiles of our data sets and check whether a different distribution associated with each hidden state can affect our estimation. Additionally, we also explore whether there are structural changes (breakpoints), by using break-point hidden Markov models. In order to perform this analysis we implement two new MCMC algorithms. The first one updates the parameters and the hidden states by using a Forward-Backward algorithm and Gibbs sampling (when a Normal distribution is assumed), and the second one uses a Forward-Backward algorithm and a mixture of Gibbs and Metropolis-Hastings sampling (when an asymmetric Laplace distribution is assumed). Finally, we consider hidden Markov models, where the hidden state (latent variables) are continuous. For this case of the discrete-time continuous state-space hidden Markov model we implement a method that uses linear programming and the Kalman filter (and Kalman smoother). Our methods are used in order to analyze real interest rates by assuming hidden states, which represent different financial regimes. We show that our methods work very well in terms of parameter estimation and also in hidden state and break-point estimation, which is very useful for the real life applications of those methods.

The theory of large deviations can help to shed light on systems in non-equilibrium statistical mechanics and, more generically, on non-reversible stochastic processes. For this purpose, we target trajectories in space time rather than static configurations and study time-extensive observables. This suggests that the details of the evolution law such as the presence of time correlations take on a major role. In this thesis, we investigate selected models with stochastic dynamics that incorporate memory by means of different mechanisms, devise a numerical approach for such models, and quantify to what extent the memory affects the large deviation functionals. The results are relevant for real-world situations, where simplified memoryless (Markovian) models may not always be appropriate. After an original introduction to the mathematics of stochastic processes, we explore, analytically and numerically, an open-boundary zero-range process which incorporates memory by means of hidden variables that affect particle congestion. We derive the exact solution for the steady state of the one-site system, as well as a mean-field approximation for larger one-dimensional lattices. Then, we focus on the large deviation properties of the particle current in such a system. This reveals that the time correlations can be apparently absorbed in a memoryless description for the steady state and the small fluctuation regime. However, they can dramatically alter the probability of rare currents. Different regimes are separated by dynamical phase transitions. Subsequently, we address systems in which the memory cannot be encoded in hidden variables or the waiting-time distributions depend on the whole trajectory. Here, the difficulty in obtaining exact analytical results is exacerbated. To tackle these systems, we have proposed a version of the so-called 'cloning' algorithm for the evaluation of large deviations that can be applied consistently for both Markovian and non-Markovian dynamics. The efficacy of this approach is confirmed by numerical results for some of the rare non-Markovian models whose large deviation functions can be obtained exactly. We finally adapt this machinery to a technological problem, specifically the performance evaluation of communication systems, where temporal correlations and large deviations are important.

Recent experiments have shown the fundamental role that random fluctuations play in many chemical systems in living cells, such as gene regulatory networks. Mathematical models are thus indispensable to describe such systems and to extract relevant biological information from experimental data. Recent decades have seen a considerable amount of modelling effort devoted to this task. However, current methodologies still present outstanding mathematical and computational hurdles. In particular, models which retain the discrete nature of particle numbers incur necessarily severe computational overheads, greatly complicating the tasks of characterising statistically the noise in cells and inferring parameters from data. In this thesis we study analytical approximations and inference methods for stochastic reaction dynamics. The chemical master equation is the accepted description of stochastic chemical reaction networks whenever spatial effects can be ignored. Unfortunately, for most systems no analytic solutions are known and stochastic simulations are computationally expensive, making analytic approximations appealing alternatives. In the case where spatial effects cannot be ignored, such systems are typically modelled by means of stochastic reaction-diffusion processes. As in the non-spatial case an analytic treatment is rarely possible and simulations quickly become infeasible. In particular, the calibration of models to data constitutes a fundamental unsolved problem. In the first part of this thesis we study two approximation methods of the chemical master equation; the chemical Langevin equation and moment closure approximations. The chemical Langevin equation approximates the discrete-valued process described by the chemical master equation by a continuous diffusion process. Despite being frequently used in the literature, it remains unclear how the boundary conditions behave under this transition from discrete to continuous variables. We show that this boundary problem results in the chemical Langevin equation being mathematically ill-defined if defined in real space due to the occurrence of square roots of negative expressions. We show that this problem can be avoided by extending the state space from real to complex variables. We prove that this approach gives rise to real-valued moments and thus admits a probabilistic interpretation. Numerical examples demonstrate better accuracy of the developed complex chemical Langevin equation than various real-valued implementations proposed in the literature. Moment closure approximations aim at directly approximating the moments of a process, rather then its distribution. The chemical master equation gives rise to an infinite system of ordinary differential equations for the moments of a process. Moment closure approximations close this infinite hierarchy of equations by expressing moments above a certain order in terms of lower order moments. This is an ad hoc approximation without any systematic justification, and the question arises if the resulting equations always lead to physically meaningful results. We find that this is indeed not always the case. Rather, moment closure approximations may give rise to diverging time trajectories or otherwise unphysical behaviour, such as negative mean values or unphysical oscillations. They thus fail to admit a probabilistic interpretation in these cases, and care is needed when using them to not draw wrong conclusions. In the second part of this work we consider systems where spatial effects have to be taken into account. In general, such stochastic reaction-diffusion processes are only defined in an algorithmic sense without any analytic description, and it is hence not even conceptually clear how to define likelihoods for experimental data for such processes. Calibration of such models to experimental data thus constitutes a highly non-trivial task. We derive here a novel inference method by establishing a basic relationship between stochastic reaction-diffusion processes and spatio-temporal Cox processes, two classes of models that were considered to be distinct to each other to this date. This novel connection naturally allows to compute approximate likelihoods and thus to perform inference tasks for stochastic reaction-diffusion processes. The accuracy and efficiency of this approach is demonstrated by means of several examples. Overall, this thesis advances the state of the art of modelling methods for stochastic reaction systems. It advances the understanding of several existing methods by elucidating fundamental limitations of these methods, and several novel approximation and inference methods are developed.

In the work we estimate the rate of convergence of the Wong-Zakai type of approximations for SDEs and SPDEs. Two cases are studied: SDEs in finite dimensional settings and evolution stochastic systems (SDEs in the infinite dimensional case). The latter result is applied to the second order SPDEs of parabolic type and the filtering problem. Roughly, the result is the following. Let Wn be a sequence of continuous stochastic processes of finite variation on an interval [0, T]. Assume that for some a > 0 the processes Wn converge almost surely in the supremum norm in [0, T] to W with the rate n-k for each k < a. Then the solutions Un of the differential equations with Wn converge almost surely in the supremum norm in [0, T] to the solution u of the "Stratonovich" SDE with W with the same rate of convergence, n-k for each k < a, in the case of SDEs and with the rate of convergence n-k/2 for each k < a, in the case of evolution systems and SPDEs. In the final chapter we verify that the two most common approximations of the Wiener process, smoothing and polygonal approximation, satisfy the assumptions made in the previous chapters.

In both physical and social sciences, we usually use controlled differential equation to model various continuous evolving system; describing how a response y relates to another process x called control. For regular controls x, the unique existence of the response y is guaranteed while it would never be the case for non-smooth controls via the classical approach. Besides, uniform closeness of controls may not imply closeness of their corresponding responses. Theory of rough paths provides a solution to both concerns. Since the creation of rough path theory, it enjoys a fruitful development and finds wide applications in stochastic analysis. In particular, rough path theory provides an effective method to study irregularity of curves and its geometric consequences in relation to integration of differential forms. In the chapter 2, we demonstrate the power of rough path theory in classical complex analysis by showing the rough path nature of the boundaries of a class of Holder's domains; as an immediate application, we extend the classical Gauss-Green's theorem. Until recently, there has been only limited research on applications of theory of rough paths to high dimensional geometry. It is clear to us that many geometric objects, in some senses appearing as solids, are actually comprised of filaments. In the chapter 3, two basic results in the theory of rough paths which will motivate later development of my thesis has been included. In the chapters 4 and 5, we identify a sensible way to do geometric calculus via those filaments (more precisely, space-filling rough paths) in dimension 3. In a recent joint work of Hambly and Lyons, they have shown that every rectifiable path can be completely characterized, up to tree-like deformation, by an algebraic object called the signature, tensor of all iterated integrals, of the path. It is clear that all tree-like deformation of the path would not change its topological features. For instance, the number of times a planar loop of finite length winds around a point (not lying on the path) is unaltered if one deforms the path in tree-like ways. Therefore, it should be plausible to extract this topological information out from the signature of the loop since the signature is a complete algebraic invariant. In the chapter 6, we express the winding number of a nice loop (respectively linking number of a pair of nice loops) as a linear functional of the signature of the loop (respectively signatures of the pair of loops).