There are two sources of information available in empirical research in finance: one corresponding to historical data and the other to prices currently observed in the markets. When proposing a model, it is desirable to use information from both sources. However in modern finance, where stochastic differential equations have been one of the main modeling tools, the common models are typically different for historical data and for current market data. The former are usually assumed to be time homogeneous, while the latter are typically time in-homogeneous. This practice can be explained by the fact that a time-homogeneous model is stationary and easier to estimate, while time-inhomogeneous model are required in order to replicate market data sufficiently well without creating arbitrage opportunities. In this thesis, we study methods of statistical inference, both parametric and non-parametric, for stochastic differential equations with time-dependent parameters. In the first part, we propose a new class of stochastic differential equation with time-dependent drift and diffusion terms, where some of the parameters change according to a hidden Markov process. We show that under some technical conditions this innovative way of modeling switching times renders the resulting model stationary. We also explore different approaches to estimate parameters in our proposed model. Our simulation studies demonstrate that the parameters of the model can be efficiently estimated by using a version of the filtering method proposed in the literature. We illustrate our model and the proposed estimation method by applying them to interest rate data, and we detect significant time variations in early 1980s, when targets of the monetary policy in the United States were changed. One of the known drawbacks of parametric models is the risk of model misspecification. In the second part of the thesis, we allow the drift to be time-dependent and nonparametric, and our objective is to estimate it using a single trajectory of the process. The main idea underlying this method is to approximate the time-dependent function with a sequence of polynomials. Since we can estimate efficiently only a finite number of parameters for any finite length of data, in our method we propose to relate the number of parameters to the length of the observed trajectory. This idea is similar to the method of sieves proposed by Grenander (Abstract Inference, 1981). The asymptotic analysis that we present is based on the assumption that the length of available data $T$ increases to infinity. We investigate two cases, one is a Brownian motion with time-dependent drift and the other corresponds to a class of mean-reverting stochastic differential equations with time-dependent mean-reversion level. In both cases we prove asymptotic consistency and normality of a modified maximum likelihood estimator of the projected time-dependent component. The main challenge in proving our results in the second case stems from two features of the problem: one is due to the fact that coefficients of projections change with $T$ and the other is related to the confounding effect between the mean-reversion speed and the level function. By applying our method to the same interest rate data we use in the first part, we find another evidence of time-variation in the drift term.

Although modern society is critically reliant on power grids, even modern power grids are subject to unavoidable outages due to storms, lightning strikes, and equipment failures. The situation in developing countries is even worse, with frequent load shedding lasting several hours a day due to unreliable generation. We study the use of battery storage to allow a set of homes in a single residential neighbour- hood to avoid power outages. Due to the high cost of storage, our goal is to choose the smallest battery size such that, with high target probability, there is no loss of power despite a grid out- age. Recognizing that the most common approach today for mitigating outages is to use a diesel generator (genset), we study the related problem of minimizing the carbon footprint of genset operation. Drawing on recent results, we model both problems as buffer sizing problems that can be ad- dressed using stochastic network calculus. We show that this approach greatly improves battery sizing in contrast to prior approaches. Specifically, a numerical study shows that, for a neigh- bourhood of 100 homes, our approach computes a battery size, which is less than 10% more than the minimum possible size necessary to satisfy a one day in ten years loss probability (2.7 ∗ 10^4 ). Moreover, we are able to estimate the carbon footprint reduction, compared to an exact numerical analysis, within a factor of 1.7. We also study the genset scheduling problem when the rate of genset fuel consumption is given by an affine function instead of a linear function of the current power. We give alternate scheduling, an online scheduling strategy that has a competitive ratio of (k1 G/C +k2)/(k1+k2) , where G is the genset capacity, C is the battery charging rate, and k1, k2 are the affine function constants. Numerically, we show that for a real industrial load alternate scheduling is very close to the offline optimal strategy.

Performance and stability of many iterative algorithms such as stochastic gradient descent largely depend on a fixed and scalar step-size parameter. Use of a fixed and scalar step-size value may lead to limited performance in many problems. We study several existing step-size adaptation algorithms in nonstationary, supervised learning problems using simulated and real-world data. We discover that effectiveness of the existing step-size adaptation algorithms requires tuning of a meta parameter across problems. We introduce a new algorithm - Autostep - by combining several new techniques with an existing algorithm, and demonstrate that it can effectively adapt a vector step-size parameter on all of our training and test problems without tuning its meta parameter across them. Autostep is the first step-size adaptation algorithm that can be used in widely different problems with the same setting of all of its parameters.

Mawson Lakes is a new suburban housing development, situated 12 kms from the city of Adelaide in South Australia. The developers, the Mawson Lakes Joint Venture (MLJV), and the local council, the City of Salisbury, intend to capture all stormwater entering the site and recondition all wastewater. The water will then be supplied to residents and businesses for non-potable usage. Modelling the behaviour of the Mawson Lakes catchment under extreme conditions such as drought and prolonged periods of high rainfall will allow the project team to determine optimal water management strategies for the catchment. -- abstract / Thesis (PhDMathematics)--University of South Australia, 2004.

Plasma polymerisation (PP) is an emerging processing technology with immense potential for future industrial applications, which is increasingly being used for the fabrication of functional coatings on polymeric substrates. In this technique, the solid polymeric film is directly deposited onto the substrate surface to create a new surface of very interesting and unique properties. PP utilizes gas phase chemistries in low pressure environment to produce well-defined high quality films in controllable and tunable fashion. A major advantage of this process is that it is an environmental safety technique and strategically superior compared to other thin film deposition techniques such as spin coating and spray coating. In eneral, the quality of the plasma polymer film can be controlled, precisely and reproducible. However, mechanism of the coating under plasma polymerisation is complex and has not yet been completely understood. / Thesis (PhD)--University of South Australia, 1999

This thesis is concerned with stochastic heat equation with memory and nonlinear energy supply. The main motivation to study such systems comes from Thermodynamics, see [85]. The main objective of this work is to study the existence and uniqueness of solutions to such equations and to investigate some fundamental properties of solutions like continuous dependence on initial conditions. In our approach we follow the seminal papers by Da Prato and Clement [10], where the stochastic heat equation with memory is tranformed into an integral equation in a function space and the so-called mild solutions are studied. In the aforementioned papers only linear equations with additive noise were investigated. The main contribution of this work is the extension of this approach to nonlinear equations. Our main tools are the theory of stochastic convolutions as developed in [33] and the theory of resolvent kernels for deterministic linear heat equations with memory, see[10]. Since the solution at time t depends on the whole history of the process up to time t, the resolvent kernel does not define a semigroup of operators in the state space of the process and therefore a ???standard??? theory of stochastic evolution equations as presented in the monograph [33] does not apply. A more delicate analysis of the resolvent kernles and the associated stochastic convolutions is needed. We will describe now content of this thesis in more detail. Introductory Chapters 1 and 2 collect some basic and essentially well known facts about the Wiener process, stochastic integrals, stochastic convolutions and integral kernels. However, some results in Chapter 2 dealing with stochastic convolution with respect to non-homogenous Wiener process are extensions of the existing theory. The main results of this thesis are presented in Chapters 3 and 4. In Chapter 3 we prove the existence and uniqueness of solutions to heat equations with additive noise and either Lipschitz or dissipative nonlinearities. In both cases we prove the continuous dependence of solutions on initial conditions. In Chapter 4 we prove the existence and uniqueness of solutions and continuous dependence on initial conditions for equations with multiplicative noise. The diffusion coefficients defined by unbounded operators are allowed.

Includes summary / Bibliography: leaves 108-112 / vii, 112 leaves : ill ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 1989

Under the democratic systems of government instilled in many sovereign states, the party in government maintains a constitutional right to call an early election. While the constitution states that there is a maximum period between elections, early elections are frequently called. This right to call an early election gives the government a control to maximize its remaining life in power. The optimal control for the government is found by locating an exercise boundary that indicates whether or not a premature election should be called. This problem draws upon the body of literature on optimal stopping problems and stochastic control. Morgan Poll’s two-party-preferred data are used to model the behaviour of the poll process and a mean reverting Stochastic Differential Equation (SDE) is fitted to these data. Parameters of this SDE are estimated using the Maximum Likelihood Estimation (MLE) Method. Analytic analysis of the SDE for the poll process is given and it will be proven that there is a unique solution to the SDE subject to some conditions. In the first layer, a discrete time model is developed by considering a binary control for the government, viz. calling an early election or not. A comparison between a three-year and a four-year maximum term is also given. A condition when the early exercise option is removed, which leads to a fixed term government such as in the USA is also considered. In the next layer, the possibility for the government to use some control tools that are termed as ‘boosts’ to induce shocks to the opinion polls by making timely policy announcements or economic actions is also considered. These actions will improve the government’s popularity and will have some impacts upon the early-election exercise boundary. An extension is also given by allowing the government to choose the size of its ‘boosts’ to maximize its expected remaining life in power. In the next layer, a continuous time model for this election timing is developed by using a martingale approach and Ito’s Lemma which leads to a problem of solving a partial differential equation (PDE) along with some boundary conditions. Another condition considered is when the government can only call an election and the opposition can apply ‘boosts’ to raise its popularity or just to pull government’s popularity down. The ultimate case analysed is when both the government and the opposition can use ‘boosts’ and the government still has option to call an early election. In these two cases a game theory approach is employed and results are given in terms of the expected remaining life in power and the probability of calling and using ‘boosts’ at every time step and at certain level of popularity.

In this thesis, the transmission of spike trains in a neuron model is studied in order to obtain a better understanding of the role played by stochastic activity, i.e. uncorrelated spikes, in the auditory pathway. Fluctuations of the neuron membrane potential are given by a first-order stochastic differential equation, using a leaky integrate-and-fire model. In contrast to most previous studies the model has a finite number of synapses, and the usual diffusion approximation does not hold. / The input signal is modeled by spike trains with spiking times described by inhomogenous Poisson processes. The membrane potential is a shot noise process for which statistical properties are derived with a Gaussian approximation. The statistics of the output spike train are obtained by using the property that a pool of a large number of output spike trains can be modeled by an inhomogeneous Poisson process. It is shown that, under certain conditions, the addition of uncorrelated input spikes, i.e. noise, can enhance the transmission of periodic temporal information. This phenomenon, called stochastic resonance, is demonstrated analytically and supported by computer simulations. / Results are compared with those obtained from the traditional leaky integrate-and- fire neuron receiving a continuous waveform input. The shot-noise property of the membrane potential, which implies that its variance is de facto modulated by the input stimulus, is shown to enhance the phenomenon of stochastic resonance. Indeed, for a given average noise level, a modulated noise gives a higher output signal-to-noise ratio than an unmodulated noise with the same average amplitude. / The derivation is then extended to certain polyperiodic stimuli mimicking vowel sounds. The fact that the addition of uncorrelated input spikes can enhance the transmission of information is discussed in the context of cochlear implants. The results provide supportive evidence to the postulate that a cochlear implant speech coding strategy that elicits stochastic firing neural activity might benefit the user.

Under the democratic systems of government instilled in many sovereign states, the party in government maintains a constitutional right to call an early election. While the constitution states that there is a maximum period between elections, early elections are frequently called. This right to call an early election gives the government a control to maximize its remaining life in power. The optimal control for the government is found by locating an exercise boundary that indicates whether or not a premature election should be called. This problem draws upon the body of literature on optimal stopping problems and stochastic control. Morgan Poll’s two-party-preferred data are used to model the behaviour of the poll process and a mean reverting Stochastic Differential Equation (SDE) is fitted to these data. Parameters of this SDE are estimated using the Maximum Likelihood Estimation (MLE) Method. Analytic analysis of the SDE for the poll process is given and it will be proven that there is a unique solution to the SDE subject to some conditions. In the first layer, a discrete time model is developed by considering a binary control for the government, viz. calling an early election or not. A comparison between a three-year and a four-year maximum term is also given. A condition when the early exercise option is removed, which leads to a fixed term government such as in the USA is also considered. In the next layer, the possibility for the government to use some control tools that are termed as ‘boosts’ to induce shocks to the opinion polls by making timely policy announcements or economic actions is also considered. These actions will improve the government’s popularity and will have some impacts upon the early-election exercise boundary. An extension is also given by allowing the government to choose the size of its ‘boosts’ to maximize its expected remaining life in power. In the next layer, a continuous time model for this election timing is developed by using a martingale approach and Ito’s Lemma which leads to a problem of solving a partial differential equation (PDE) along with some boundary conditions. Another condition considered is when the government can only call an election and the opposition can apply ‘boosts’ to raise its popularity or just to pull government’s popularity down. The ultimate case analysed is when both the government and the opposition can use ‘boosts’ and the government still has option to call an early election. In these two cases a game theory approach is employed and results are given in terms of the expected remaining life in power and the probability of calling and using ‘boosts’ at every time step and at certain level of popularity.