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The goal of this paper is to extend the results of Bayraktar and Young (2006) on minimizing an individual's probability of lifetime ruin; i.e. the probability that the individual goes bankrupt before dying. We consider a scenario in which the individual is allowed to invest in both a domestic bank account and a foreign bank account, with the exchange rate between the two currencies being modeled by geometric Brownian motion. Additionally, we impose the restriction that the individual is not allowed to borrow money, and assume that the individual's wealth is consumed at a constant rate. We derive formulas for the minimum probability of ruin as well as the individual's optimal investment strategy. We also give a few numerical examples to illustrate these results.

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.

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.

This dissertation provides a brief theoretical introduction to the Malliavin calculus leading to a particular application in finance. The Malliavin calculus concepts are used to aid in the simulation of the Greeks for financial contingent claims. Particular focus is placed on creating efficiency in the more exotic type option simulations, where no closed solution pricing formulae exist. Copyright / Dissertation (MSc)--University of Pretoria, 2009. / Mathematics and Applied Mathematics / unrestricted

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A stochastic differential equation with vanishing martingale term is studied. Specifically, given a domain D, the asymptotic scaling properties of both the exit time from the domain and the exit distribution are considered under the additional (non-standard) hypothesis that the initial condition also has a scaling limit. Methods from dynamical systems are applied to get more complete estimates than the ones obtained by the probabilistic large deviation theory. Two situations are completely analyzed. When there is a unique critical saddle point of the deterministic system (the system without random effects), and when the unperturbed system escapes the domain D in finite time. Applications to these results are in order. In particular, the study of 2-dimensional heteroclinic networks is closed with these results and shows the existence of possible asymmetries. Also, 1-dimensional diffusions conditioned to rare events are further studied using these results as building blocks. The approach tries to mimic the well known linear situation. The original equation is smoothly transformed into a very specific non-linear equation that is treated as a singular perturbation of the original equation. The transformation provides a classification to all 2-dimensional systems with initial conditions close to a saddle point of the flow generated by the drift vector field. The proof then proceeds by estimates that propagate the small noise nature of the system through the non-linearity. Some proofs are based on geometrical arguments and stochastic pathwise expansions in noise intensity series.

This thesis presents the prediction algorithm for the price of the share of risky asset. The price of the share is presented by dynamic model and observation is presented by the measurement model. Dynamic model is derived by using Stochastic Calculus. The algorithm is simulated by using Matlab.

Définies il y a quelques décennies, les martingales dans les variétés sont maintenant des objets bien connus. Des questions très simples restent en suspens cependant. Par exemple,étant donnée une variable aléatoire à valeurs dans une variété complète, et une filtration continue(dont toute martingale réelle possède une version continue), existe-t-il une martingale continue dans cette variété qui a pour valeur terminale donnée cette variable aléatoire ? Que dire des semimartingales de valeur terminale et dérives données ? Le but principal de cette thèse est d’apporter des réponses à ces questions. Sous des hypothèses de géométrie convexe, des réponses sont données dans les articles de Kendall (1990), Picard (1991), Picard (1994), Darling (1995) ou encore Arnaudon (1997). Le cas des semimartingales a plus largement été traité par Blache (2004). Les martingales dans les variétés permettent de définir les barycentres associés à une filtration, qui sont parfois plus simples à calculer que les barycentres usuels ou les moyennes, et qui possèdent une propriété d’associativité. Ils sont fortement reliés à la théorie du contrôle, à l’optimisation stochastique, ainsi qu’aux équations différentielles stochastiques rétrogrades (EDSRs). La résolution du problème avec des arguments géométriques donne par ailleurs des outils pour résoudre des EDSRs quadratiques multidimensionnelles.Au cours de cette thèse deux principales méthodes ont été employées pour étudier le problème de l’existence de martingale de valeur terminale donnée. La première est basée sur un algorithme stochastique. La variable aléatoire que l’on cherchera à atteindre sera d’abord déformée en une famille x (a) de classe C 1, et on se posera la question suivante : existe-t-il une martingale X(a) de valeur terminale x (a) ? Une méthode de tir, selon un principe similaire au tir géodésique déterministe, sera employée relativement au paramètre a en direction de la variable aléatoire x (a). La seconde est basée sur la résolution générale d’une EDSR multidimensionnelle à croissance quadratique. La principale problématique de cette partie sera d’adapter au cadre multidimensionnel une stratégie récente développée par Briand et Elie (2013) permettant de traiter les EDSRs quadratiques multidimensionnelles. Cette approche nouvelle permet de retrouver la plupart des résultats partiels obtenus par des méthodes différentes. Au-delà de l’intérêt unifiant,cette nouvelle approche ouvre la voie à de potentiels futurs travaux. / Defined several decades ago, martingales in manifolds are very canonical objects. About these objects very simple questions are still unresolved. For instance, given a random variable with values in a complete manifold and a continuous filtration (one with respect to which all real-valued martingales admit a continuous version), does there exist a continuous martingale in the manifold with terminal value given by this random variable ? What about semimartingales with prescribed drift and terminal value ? The main aim of this thesis is to provide answers to these questions. Under convex geometry assumption, answers are given in the articles of Kendall (1990), Picard (1991), Picard (1994), Darling (1995) or Arnaudon (1997). The case of semimartingales was widely treated by Blache (2004). The martingales in the manifolds make it possible to define the barycenters associated to a filtration, which are sometimes simpler to compute than the usual barycenters or averages, and which have an associative property. They are strongly related to control theory, stochastic optimization, and backward stochastic differential equations (BSDEs). Solving the problem with geometric arguments also gives tools for solving multidimensional quadratic EDSRs.During the thesis, two methods have been used for studying the problem of existence of a martingale with prescribed terminal value. The first one is based on a stochastic algorithm. The random variable that we try to reach will be deformed into a $mcC^1$-family $xi(a)$, and we deal with the following newer problem: does there exist a martingale $X(a)$ with terminal value $xi(a)$ ? A shooting method, using the same kind of principle as the deterministic geodesic shooting, will be used with respect to a parameter $a$ towards $xi(a)$.The second one is the resolution of a multidimensional quadratic BSDE. The aim of this part will be to adapt to the multidimensional framework a recent strategy developed by Briand and Elie (2013) to treat multidimensional quadratic BSDEs. This new approach makes it possible to rediscover the results obtained by different methods. Beyond the unification, this new approach paves the way for potential future works.