This thesis considers three topics in stochastic control theory. Each of these topics is motivated by an application in finance. In each of the stochastic control problems formulated, the optimal strategy is characterised using dynamic programming. Closed form solutions are derived in a number of special cases. The first topic is about the market making problem in which a market maker manages his risk from inventory holdings of a certain asset. The magnitude of this inventory is stochastic with changes occurring due to client trading activity, and can be controlled by making small adjustments to the so-called skew, namely, the quoted price offered to the clients. After formulating the stochastic control problem, closed form solutions are derived for the special cases that arise if the asset price is modelled by a Brownian motion with drift or a geometric Brownian motion. In both cases the impact of skew is additive. The optimal controls are time dependent affine functions of the inventory size and the inventory process under the optimal skew is an Ornstein-Uhlenbeck process. As a result, the asset price is mean reverting around a reference rate. In the second topic the same framework is expanded to include a hedging control that can be used by the market maker to manage the inventory. In particular, the market impact is assumed to be of the Almgren and Chriss type. Explicit solutions are derived in the special case where the asset price follows a Brownian motion with drift. The third topic is about Merton’s portfolio optimisation problem with the additional feature that the risky asset price is modelled in a way that exhibits support and resistance levels. In particular, the risky asset price is modelled using a skew Brownian motion. After formulating the stochastic control problem, closed form solutions are derived.

Approximate Monte Carlo algorithms are not uncommon these days, their applicability is related to the possibility of controlling the computational cost by introducing some noise or approximation in the method. We focus on the stability properties of a particular approximate MCMC algorithm, which we term noisy Metropolis-Hastings. Such properties have been studied before in tandem with the pseudo-marginal algorithm, but under fairly strong assumptions. Here, we examine the noisy Metropolis-Hastings algorithm in more detail and explore possible corrective actions for reducing the introduced bias. In this respect, a novel approximate method is presented, motivated by the class of exact algorithms with randomised acceptance. We also discuss some applications and theoretical guarantees of this new approach.

The objective of this thesis is to further the understanding of the time inversion property for self-similar Markov processes. In particular, we focus upon seeking a full characterisation of the class of processes that enjoy the time inversion property. The first chapter in this thesis is a review of current literature in the areas that we use in the sequel. Chapter 2 provides a full characterisation of processes enjoying the time inversion property on R up to certain restrictions. Namely, we show that on R+, the only processes that enjoy the time inversion property are Bessel processes in the wide sense. Extending this characterisation to R, we show that we are necessarily restricted to variations of Bessel and Dunkl processes. We then give an expression of the semigroup density that all processes with the time inversion property must satisfy. In Chapter 3, we extend some of these results to Rn. We provide a restriction on the jump measure of processes with the time inversion property and show that ^ρ(Rt) is necessarily a Bessel process for a process Rt with the time inversion property and a defined function ^ρ. Finally, Chapter 4 extends the work of Vuolle-Apiala [2012] on the skew product representation and presents a methodology by which one can construct examples of processes with the time inversion property. This leads to several examples of particular interest.

Recently, there has been a great interest in methods that can decompose brain networks into clusters with similar connection patterns. However, most of the currently used clustering methods in neuroimaging are based on the stringent assumption that the cluster structure is modular, that is, the nodes are densely connected within clusters, but sparsely connected between clusters. Furthermore, multi-subject network data is typically fit by several subject-by-subject analyses, which are limited by the fact that there is no obvious way to combine the results for group comparisons, or on a group-averaged network analysis, which does not reflect the variability between subjects. In the first part of this thesis, we consider the analysis of a single binary-valued brain network using the Stochastic Blockmodel (Daudin et al., 2008) and compare it to the widely used clustering methods, Louvain and Spectral algorithms. For this, we use the Caenorhabditis elegans (C. elegans) worm nervous system as a model organism whose wealth of prior biological knowledge can be used to validate the functional relevance of network decompositions. We show that the ‘cores-in-modules’ decomposition of the worm brain network estimated by the Stochastic Blockmodel is more compatible with prior biological knowledge about the C. elegans than the purely modular decompositions found by the Louvain and Spectral algorithms. In the second part of this thesis, we propose three multi-subject extensions of Daudin et al.’s Stochastic Blockmodel that can estimate a common cluster structure across subjects. Two of these (non-trivial) models use subject specific covariates to model variation in connection rates in the data. The first and trivial model assumes no variability between subjects, the second model accounts for a global variability in connections between subjects, and the third model accounts for local variability in connections between subjects that can differ across individual within/between-cluster connectivity elements. In the third part of this thesis, we propose a mixed-effect multi-subject model which can account for the repeated-measures aspects of multi-subject network data by including a random intercept. For the second and third part of the thesis, we use intensive Monte Carlo simulations to investigate the accuracy of the estimated parameters as well as the validity of inference procedures. Furthermore, we illustrate the proposed models on a resting state fMRI dataset with two groups of subjects: healthy volunteers and individuals diagnosed with schizophrenia.

This thesis consists of two parts, in both of which we consider approximations of rough stochastic PDEs and investigate convergence properties of the approximate solutions. In the first part we use the theory of (controlled) rough paths to define a solution for one-dimensional stochastic PDEs of Burgers type driven by an additive space-time white noise. We prove that natural numerical approximations of these equations converge to the solution of a corrected continuous equation and that their optimal convergence rate in the uniform topology (in probability) is arbitrarily close to 1/2 . In the second part of the thesis we develop a general framework for spatial discretisations of parabolic stochastic PDEs whose solutions are provided in the framework of the theory of regularity structures and which are functions in time. As an application, we show that the dynamical �43 model on the dyadic grid converges after renormalisation to its continuous counterpart. This result in particular implies that, as expected, the �43 measure is invariant for this equation and that the lifetime of its solutions is almost surely infinite for almost every initial condition.

We consider the (1+1) dimensional Laplacian model with pinning interaction. This is a probabilistic model for a polymer or an interface that is attracted to the zero line. Without the pinning interaction, the Laplacian model is a Gaussian field (Φi)iEΛN, where ΛN = {1, 2, ..., N - 1}. The covariance matrix of this field is given by the inverse of Φ -> 1/2 ENi=0(ΔΦi)2, where Δ is the discrete Laplacian. Furthermore the values at {-1, 0, N, N+1} are fixed boundary values. The pinning interaction is introduced by giving the field a reward each time it touches the zero line. Depending on the reward the model with pinning and the one without pinning show different behaviour. Caravenna and Deuschel [10] study the localisation behaviour of the polymer. The model is delocalised if the number of times a typical field touches the zero line is of order o(N). The authors of [10] show that for zero boundary conditions there is a critical reward such that for smaller rewards the model is delocalised whilst for larger rewards the model is localised. In this thesis we study the behaviour of the empirical profile of the field. We show that for non zero boundary conditions there is a critical reward such that for smaller rewards the empirical profile for the model with pinning and the one for the model without pinning behave in the same way whilst for larger rewards the empirical profile of the model with pinning interaction is attracted to the zero line.

This thesis develops computationally efficient methodology in two areas. Firstly, we consider a particularly challenging class of discretely observed continuous-time point-process models. For these, we analyse and improve an existing filtering algorithm based on sequential Monte Carlo (smc) methods. To estimate the static parameters in such models, we devise novel particle Gibbs samplers. One of these exploits a sophisticated non-entred parametrisation whose benefits in a Markov chain Monte Carlo (mcmc) context have previously been limited by the lack of blockwise updates for the latent point process. We apply this algorithm to a Lévy-driven stochastic volatility model. Secondly, we devise novel Monte Carlo methods – based around pseudo-marginal and conditional smc approaches – for performing optimisation in latent-variable models and more generally. To ease the explanation of the wide range of techniques employed in this work, we describe a generic importance-sampling framework which admits virtually all Monte Carlo methods, including smc and mcmc methods, as special cases. Indeed, hierarchical combinations of different Monte Carlo schemes such as smc within mcmc or smc within smc can be justified as repeated applications of this framework.

In the presented thesis, we build the theoretical framework for the development of temperature-based weather derivatives market in China. Our research is divided into two separate studies due to their different scopes. In the first study, we focus on the determination of the most precise model for temperature-based weather derivative modeling and pricing in China. To achieve this objective, a heuristic comparison of the new stochastic seasonal variation (SSV) model with three established empirical temperature and pricing models, i.e. the Alaton model [1], the CAR model [2] and the Spline model [3] is conducted. Comparison criteria include residual normality, residual auto-correlation function (ACF), Akaike information criterion (AIC), relative errors, and stability of price behaviors. The results show that the SSV model dominates the other three models by providing both a more precise fitting of the temperature process and more stable price behaviors. In the second study, novel forms of temperature indices are proposed and analyzed both on the city level and the climatic zone level, with the aim to provide a contract-selecting scheme that increases the risk management efficiency in the agricultural sector of China. Performances of the newly-introduced indices are investigated via an efficiency test which considers the root mean square loss (RMSL), the value at risk (VaR) and the certainty-equivalent revenues (CERs). According to the results, agricultural risk management on the city scale can be optimized by using the absolute-deviation growth degree-day (GDD) index. On the other hand, it is suggested that climatic zone-based contracts can be more efficient compared with city-based contracts. The recommended contract-selection scheme is to purchase climatic zone-based average GDD contracts in climatic zone II, and to purchase climatic zone-based optimal-weighted GDD contracts in climatic zone I or III.

Continuous random processes are used to model a huge variety of real world phenomena. In particular, the zero-crossings of such processes find application in modelling processes of diffusion, meteorology, genetics, finance and applied probability. Understanding the zero-crossings behaviour improves prediction of phenomena initiated by a threshold crossing, as well as extremal problems where the turning points of the process are of interest. To identify the Probability Density Function (PDF) for the times between successive zero-crossings of a stochastic process is a challenging problem with a rich history. This thesis considers the effect of an oscillatory auto-correlation function on the zero-crossings of a Gaussian process. Examining statistical properties of the number of zeros in a fixed time period, it is found that increasing the rate of oscillations in the auto-correlation function results in more ‘deterministic’ realisations of the process. The random interval times between successive zeros become more regular, and the variance is reduced. Accurate calculation of the variance is achieved through analysing the correlation between intervals,which numerical simulations show can be anti-correlated or correlated, depending on the rate of oscillations in the auto-correlation function. The persistence exponent describes the tail of the inter-event PDF, which is steeper where zero-crossings occur more regularly. It exhibits a complex phenomenology, strongly influenced by the oscillatory nature of the auto-correlation function. The interplay between random and deterministic components of a system governs its complexity. In an ever-more complex world, the potential applications for this scale of ‘regularity’ in a random process are far reaching and powerful.

In the analysis of movement patterns of animals, stochastic processes play an important role, providing us with a variety of tools to examine, model and simulate their behaviour. In this thesis we focus on the foraging of specific animals - bumblebees - and analyse experimental data to understand the influence of changes in the bumblebees’ environment on their search flights. Starting with a discussion of main classes of stochastic models useful for the description of foraging animals, we then look at a multitude of environmental factors influencing the dynamics of animals in their search for food. With this background we examine flight data of foraging bumblebees obtained from a laboratory experiment by stochastic analyses. The main point of interest of this analysis is the description, modelling and understanding of the data with respect to the influence of predatory threats on the bumblebee’s foraging search flights. After this detail-oriented view on interactions of bumblebees with food sources and predators in the experimental data, we develop a generalized reorientation model. By extracting the necessary information from the data, we arrive at a generalized correlated random walk foraging model for bumblebee flights, which we discuss and compare to the experimental data via simulations. We finish with a discussion of anomalous fluctuation relations and some results on spectral densities of autocorrelation functions. While this part is not directly related to the analysis of foraging, it concerns a closely related class of stochastic processes described by Langevin equations with non- trivial autocorrelation functions analyse experimental data to understand the influence of changes in the bumblebees’ environment on their search flights. Starting with a discussion of main classes of stochastic models useful for the description of foraging animals, we then look at a multitude of environmental factors influencing the dynamics of animals in their search for food. With this background we examine flight data of foraging bumblebees obtained from a laboratory experiment by stochastic analyses. The main point of interest of this analysis is the description, modelling and understanding of the data with respect to the influence of predatory threats on the bumblebee’s foraging search flights. After this detail-oriented view on interactions of bumblebees with food sources and predators in the experimental data, we develop a generalized reorientation model. By extracting the necessary information from the data, we arrive at a generalized correlated random walk foraging model for bumblebee flights, which we discuss and compare to the experimental data via simulations. We finish with a discussion of anomalous fluctuation relations and some results on spectral densities of autocorrelation functions. While this part is not directly related to the analysis of foraging, it concerns a closely related class of stochastic processes described by Langevin equations with nontrivial autocorrelation functions.