Stochastic processes are mathematical objects that offer a probabilistic representation of how some quantities evolve in time. In this thesis we focus on estimating the trajectory and parameters of dynamical systems in cases where only indirect observations of the driving stochastic process are available. We have first explored means to use weekly recorded numbers of cases of Influenza to capture how the frequency and nature of contacts made with infected individuals evolved in time. The latter was modelled with diffusions and can be used to quantify the impact of varying drivers of epidemics as holidays, climate, or prevention interventions. Following this idea, we have estimated how the frequency of condom use has evolved during the intervention of the Gates Foundation against HIV in India. In this setting, the available estimates of the proportion of individuals infected with HIV were not only indirect but also very scarce observations, leading to specific difficulties. At last, we developed a methodology for fractional Brownian motions (fBM), here a fractional stochastic volatility model, indirectly observed through market prices. The intractability of the likelihood function, requiring augmentation of the parameter space with the diffusion path, is ubiquitous in this thesis. We aimed for inference methods robust to refinements in time discretisations, made necessary to enforce accuracy of Euler schemes. The particle Marginal Metropolis Hastings (PMMH) algorithm exhibits this mesh free property. We propose the use of fast approximate filters as a pre-exploration tool to estimate the shape of the target density, for a quicker and more robust adaptation phase of the asymptotically exact algorithm. The fBM problem could not be treated with the PMMH, which required an alternative methodology based on reparameterisation and advanced Hamiltonian Monte Carlo techniques on the diffusion pathspace, that would also be applicable in the Markovian setting.

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Generated by stochastic recursions, perpetuities encompass a vast range of discretetime financial behaviours. When focusing on the dramatic changes occurring in such processes, the analysis of threshold exceedances provides an extensive description of their underlying mechanisms. Asymptotically, an exceedance point process tends to a compound Poisson measure, highlighting a tendency to cluster. Now, the parameters of this limit law are known, but complex. Here, an empirical approach is adopted, and a class of explicit compound Poisson models developed, with a bound on the error, for the exceedance point process of a finite, multidimensional perpetuity. In a financial regulatory context, this provides a new way of examining the Value-at-Risk criterion for securities.

In the last 30 years, whilst there has been an explosion in our ability to make quantative predictions, less progress has been made in terms of building useful forecasts to aid decision support. In most real world systems, single point forecasts are fundamentally limited because they only simulate a single scenario and thus do not account for observational uncertainty. Ensemble forecasts aim to account for this uncertainty but are of limited use since it is unclear how they should be interpreted. Building probabilistic forecast densities is a theoretically sound approach with an end result that is easy to interpret for decision makers; it is not clear how to implement this approach given finite ensemble sizes and structurally imperfect models. This thesis explores methods that aid the interpretation of model simulations into predictions of the real world. This includes evaluation of forecasts, evaluation of the models used to make forecasts and the evaluation of the techniques used to interpret ensembles of simulations as forecasts. Bayes theorem is a fundamental relationship used to update a prior probability of the occurence of some event given new information. Under the assumption that each of the probabilities in Bayes theorem are perfect, it can be shown to make optimal use of the information available. Bayes theorem can also be applied to probability density functions and thus updating some previously constructed forecast density with a new one can be expected to improve forecast skill, as long as each forecast density gives a good representation of the uncertainty at that point in time. The relevance of the probability calculus, however, is in doubt when the forecasting system is imperfect, as is always the case in real world systems. Taking the view that we wish to maximise the logarithm of the probability density placed on the outcome, two new approaches to the combination of forecast densities formed at different lead times are introduced and shown to be informative even in the imperfect model scenario, that is a case where the Bayesian approach is shown to fail.

This thesis has two related aims: establishing tractable conditions for posterior consistency of statistical inference from non-IID data with an intractable likelihood, and developing Monte Carlo methodology for conducting such inference. Two prominent classes of models, jump diffusions and generalised coalescent processes, are considered throughout. Both are motivated by population genetics applications. Posterior consistency of nonparametric inference is established for joint inference of drift and compound Poisson jump components of unit volatility jump diffusions in arbitrary dimension under an identifiability assumption. This assumption is straightforward to verify in the diffusion case, but difficult to check in general for jump diffusions. A similar consistency result is established under somewhat weaker conditions for Λ-coalescent processes whenever time series data is available. I also show that Λ-coalescent inference cannot be consistent if observations are contemporaneous, in stark contrast to the more classical case of the Kingman coalescent. I also introduce the notion of reverse time sequential Monte Carlo (SMC), which has previously been applied to Kingman and Λ-coalescents. Here, reverse time SMC is presented as a generic algorithm, and general conditions under which it is effective are developed. In brief, it is well suited to integration over paths which begin at a mode of the target distribution, and terminate in the tails. These innovations are used to design new SMC algorithms for generalised coalescent processes, as well as non-coalescent examples including evaluating a containment probability of the hyperbolic diffusion, an overflow probability in a queueing model and finding an initial infection in an epidemic network model.

The zero-crossing problem is the determination of the probability density function of the intervals between the successive axis crossings of a stochastic process. This thesis studies the properties of the zero-crossings of stationary processes belonging to the symmetric-stable class of Gaussian and non-Gaussian type, corresponding to the stability index nu=2 and 0 < nu < 2 respectively.

Data assimilation schemes are methods to estimate true underlying state of the physical systems of interest by combining the theoretical knowledge about the underlying system with available observations of the state. However, in most of the physical systems the observations often are noisy and only partially available. In the first part of this thesis we study the case of sequential data assimilation scheme, when the underlying system is nonlinear chaotic and the observations are partial and noisy. We produce a rigorous and quantitative analysis of data assimilation process for fixed observation modes. We also introduce a novel method of dynamically rearranging observation modes, leading to the requirement of fewer observation modes while maintaining the accuracy of the data assimilation process. In the second part of the thesis we focus on 4DVAR data assimilation scheme which is a variational method. 4DVAR data assimilation is a method that solves a variational problem; given a set of observations and a numerical model for the underlying physical system together with a priori information on the initial condition to estimate the initial condition for the underlying model. We propose a hybrid data assimilation scheme where, we consider the 3DVAR scheme for the model as the constraint on the variational form, rather than constraining the variational form with the original model. We observe that this method reduces the computational cost of the minimization of the 4DVAR variational form, however, it introduces a bias in the estimate of the initial condition. We then explore how the results can be extended to weak constraint 4DVAR.

Periodic behaviours in continuous media can be described with great power and economy using conceptual machinery such as the notion of a field. However periodic effects can also be `observed' in collections of discrete objects, be they individuals sending emails, fire-flies signalling to attract mates, synapses firing in the brain or photons emerging from a cavity. The origin of periodic behaviours becomes more difficult to identify and interpret in these instances; particularly for systems whose individual components are fundamentally stochastic and memoryless. This thesis describes how periodic behaviour can emerge from intrinsic fluctuations in a fully discrete system that is completely isolated from any external coherent forcing. This thesis identifies the essential elements required to produce naturally emerging periodic behaviours in a collection of interacting `particles' which are constrained to a finite set of `states', represented by the nodes of a network. The network can be identified with a type of a spatial structure throughout which particles can move by spontaneously jumping between nodes. The particles interact by affecting the rate at which other particles jump. In such systems it is the collective ensemble of particles, rather than the individual particles themselves, that exhibit periodic behaviours. The existence or non-existence of such collective periodic behaviours is attributed to the structure of the network and the form of interaction between particles that together describe the microscopic dynamics of the system. This thesis develops a methodology for deriving the macroscopic description of the ensemble of particles from the microscopic dynamics that govern the behaviour of individual particles and uses this to find key ingredients for collective periodic behaviour. In order for periodic behaviours to emerge and persist it is necessary that the microscopic dynamics be irreversible and hence violate the principle of detailed balance. However such a condition is not sufficient and irreversibility must also manifest on the macroscopic level. Simple systems that admit collective periodic behaviours are presented, analysed and used to hypothesise on the essential elements needed for such behaviour. Important general results are then proven. It is necessary that the network have more than two nodes and directed edges such that particles jump between states at different rates in both directions. Perhaps most significantly, it is demonstrated that collective periodic behaviours are possible without invoking `action at a distance' - there need not be a field providing a mechanism for the interactions between particles.

Condensation is an emergent phenomenon in complex systems that is observed in both physical and social sciences, from granular polydisperse spheres to macroeconomic studies. The critical behaviour of condensation in such systems is of continual interest in research. In this thesis we study this in the context of interacting particle systems, in particular the recently introduced explosive condensation process. We firstly provide a review of the mathematical foundations of interacting particle systems from the aspects of Markov processes. This includes the formulation of factorised hop rates, stationary product measures, the equivalence of ensembles and how these properties are related to condensation. Subsequently, we give a review of key interacting particle systems of interest, namely the zero-range process, inclusion process and the explosive condensation process. We then introduce two models that have similar stationary weights scaling as the explosive condensation process and include them in our study in the thermodynamic limit. The density and the maximum site occupation are derived under the stationary distribution, and from this we are able to identify the choice of parameters that could lead to a phase transition. Exact results for these models using the generator are di�cult to obtain. For the main results of this study, we therefore analyse the formation of condensate using a heuristic approach. The microscopic interactions leading to the formation of an explosive condensate are structurally studied, and this leads to a comprehensive model with a timescale analysis. The time to condensation is shown to vanish as the thermodynamic limit is reached, depending on the choice of parameter values. Throughout the thesis, theoretical results are supported by Monte Carlo simulation and numerical calculations where appropriate. A modification of the conventional Gillespie algorithm is proposed. The new algorithm improves e�ciencies but is also able to preserve key stochastic properties, and is used throughout the simulation of the main findings.

This thesis studies the risk-sensitive control problem for a class of non-linear stochastic systems and its financial applications. The nonlinearity is of the square- root type, and is inspired by applications. The problems of optimal investment and consumption are also considered under several different assumptions on the stochastic interest rate and stochastic volatility. At the beginning, we systematically investigate the nonlinearity of risk-sensitive control problem. It consists of quadratic and square-root terms in the state. Such an optimal control problem can be solved in an explicit closed form by the completion of squares method. As an application of the risk-sensitive control in financial mathematics, the optimal investment problem will be described in the Chapter 4. A new interest rate, which follows the stochastic process with mixed Cox- Ingersoll-Ross (CIR) model and quadratic affine term structure model (QATSM) is introduced. Such an interest rate model admits an explicit price for the zero- coupon bond. In Chapter 5, we consider a portfolio optimization problem on an infinite time horizon. The stochastic interest rate consists not only of the quadratic terms, but also of the square-root terms. On the other hand, the double square root process is also introduced to establish the interest rate model. Under some sufficient conditions, the unique solution of the optimal investment problem is found in an explicit closed form. Furthermore, the optimal consumption problem is considered in Chapter 6 and 7. It can be solved in an explicit closed form via the methods of completion of squares and the change of measure. We provide a detailed discussion on the existence of the optimal trading strategies. Such trading strategies can be deduced for both finite and infinite time horizon cases.