Shortest path problems arise in a variety of applications ranging from transportation planning to network routing among others. One group of these problems involves finding shortest paths in graphs where the edge weights are defined by probability distributions. While some research has addressed the problem of finding a single shortest path, no research has been done on finding multiple paths in such graphs. This thesis addresses the problem of finding paths for multiple robots through a graph in which the edge weights represent the probability that each edge will fail. The objective is to find paths for n robots that maximize the probability that at least k of them will arrive at the destination. If we make certain restrictions on the edge weights and topology of the graph, this problem can be solved in O(n log n)time. If we restrict only the topology, we can find approximate solutions which are still guaranteed to be better than the single most reliable path.

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.

It is a dream of Systems-Biology to efficiently simulate an entire cell on a computer. The potential medical and biological applications of such a tool are immense, and so are the challenges to accomplish it. At the level of a cell, the number of reacting molecules is so low that stochastic effects can be crucial in deciding the system-level behaviour of the cell. Despite the recent development of many new and hybrid stochastic approaches, exact stochastic simulation algorithms are still needed, and are widely employed in most current biochemical simulation packages. Unfortunately, the performance of these algorithms scales badly with the number of reactions. It is shown that this is especially the case for hubs and scale-free networks. This is worrying because hubs are an important component of biochemical systems, and it is widely suspected that biochemical networks are scale-free. It is shown that the scalability issue in these algorithms is due to the high interdependency between reactions. A general method for arbitrarily reducing this interdependency is introduced, and it is shown how it can be used for many classes of simulation processes. This is applied to one of the fastest algorithms currently, the Next Reaction Method. The resulting algorithm, the Reactant-Margin Method, is tested on a wide range of hub sizes and shown to be asymptotically faster than the current best algorithms. Hybrid versions of the Reactant-Margin Method and the Next Reaction Method are also compared on a real biological model - the Lambda-Phage virus, and the new algorithm is again shown to perform better. The problems inherent in the hybridization are also shown to be more exactly and efficiently handled in the Reactant-Margin framework than in the Next-Reaction Method framework. Finally, a software tool called GeNIV is introduced. This GUI-based biochemical modelling and simulation tool is an embodiment of a mechanistic-representation philosophy. It is implements the Reactant Margin and Next Reaction hybrid algorithms, and has a simple representation system for gene-state occupancy and their subsequent biochemical reactions. It is also novel in that it translates the graphical model into Javacode which is compiled and executed for simulation.

It has been almost eighty years since quantum mechanics emerged as a complete theory, yet debates about how should quantum mechanics be interpreted still occur. Interpretations are many and varied, some taking us as fundamental in determining reality (orthodox interpretation), while others proposing that reality exists outside of us, but it is a lot more complicated than that implied by classical mechanics. In this thesis I am going to try to provide new light on this debate by investigating dynamics under both the orthodox and modal interpretation. In particular I will answer the question what is the interpretation of non-Markovian stochastic Schrodinger equations? I conclude that under the orthodox view these equations have only a numerical interpretation. They provide a rule for calculating the state of the system at time t if we made a measurement on the bath (a collection of oscillators {ak}) at that time, yielding results {zk}. However in the modal view they have a meaning: non-Markovian stochastic Schrodinger equations represent the evolution of the system part of the property state of the universe (bath + system).

"July 12, 2002" / Bibliography: p. 130-137 / viii, 160 leaves ; 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, 2002

Thesis (M. Phil.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

<p>In this paper we investigate pricing of variance swaps contracts. The</p><p>literature is mostly dedicated to the pricing using replication with</p><p>portfolio of vanilla options. In some papers the valuation with stochastic</p><p>volatility models is discussed as well. Stochastic volatility is becoming</p><p>more and more interesting to the investors. Therefore we decided to</p><p>perform valuation with the Heston stochastic volatility model, as well</p><p>as by using replication strategy.</p><p>The thesis was done at SunGard Front Arena, so for testing the replica-</p><p>tion strategy Front Arena software was used. For calibration and testing</p><p>of the Heston model we used MatLab.</p>

Electricity markets around the world have gone through, or are currently in a deregulation phase. As a result, power companies that formerly enjoyed a monopoly are now facing risks. In order to cover (hedge) these risks, futures markets have emerged, in parallel with the spot price markets. Then, markets of more complex derived products have appeared to better hedge the risk exposures of power suppliers and consumers. An Affine Jump Diffusion (AJD) framework is presented here to coherently model the dynamics of the spot price of electricity and all the futures contracts. The non-storability of electricity makes it indeed impossible to use it in hedging strategies. Futures contracts, however, are standard financial contracts that can be stored and used in hedging strategies. We thus propose to consider the set of futures contracts as the primary commodities to be modelled and jointly estimate the parameters of the spot and futures prices based on their historical time series. The estimation is done by Maximum Likelihood, using a Kalman Filter recursive algorithm that has been updated to account for non-Gaussian errors. This procedure has been applied to the German European Energy index (EEX) based in Frankfurt for electricity, to the Brent for Crude oil, and to the NBP for natural gas. The AJD framework is very powerful because the characteristic function of the underlying stochastic variables can be obtained just by solving a system of complex valued ODEs. We took advantage of this feature and developed a novel approach to estimate expectations of arbitrary functions of random variables that does not require the probability density function of the stochastic variables, but instead, their characteristic function. This approach, relying on the Parseval Identity, provided closed form solutions for options with payoff functions that have an analytical Fourier transform. In particular, European calls, puts and spread options could be computed as well as the value of multi-fuel power plants that can be viewed as an option to exchange the most economic fuel of the moment against electricity. A numerical procedure has also been developed for options with payoff functions that do not have an analytical Fourier transform. This numerical approach is indeed using a Fast Fourier Transform of the payoff function, and can be used in Dynamic Programming algorithms to price contracts with endogenous exercise strategies. Finally, it is showed that the (mathematical) partial derivatives of these contracts, often referred to as the Greeks, could also be computed at low cost. This allows to build hedging strategies to shape the risk profile of a given producer, or consumer.

In an e-retailing setting, the efficient utilization of inventory, storage space, and labor is paramount to achieving high levels of customer service and company profits. To optimize the storage space and labor, a retailer will split the warehouse into two storage regions with different densities. One region is for picking customer orders and the other to hold reserve stock. As a consequence, the inventory system for the warehouse is a multi-item two-stage, serial system. We investigate the problem when demand is stochastic and the objective is to minimize the total expected average cost under some space constraints. We generate an approximate formulation and solution procedure for a periodic review, nested ordering policy, and provide managerial insights on the trade-offs. In addition, we extend the formulation to account for shipping delays and advanced order information. / Singapore-MIT Alliance (SMA)

Albert H. Nuttall. / "May 26, 1958." Issued also as a thesis, M.I.T. Dept. of Electrical Engineering, May 19, 1958. / Bibliography: p. 57. / Army Signal Corps Contract DA36-039-sc-64637. Dept. of the Army Task 3-99-06-108 and Project 3-99-00-100.