In this thesis, our research focus on a weak second order stochastic Runge–Kutta method applied to a system of stochastic differential equations known as the Gatheral Model. We approximate numerical solutions to this system and investigate the rate of convergence of our method. Both call and put options are priced using Monte-Carlo simulation to investigate the order of convergence. The numerical results show that our method is consistent with the theoretical order of convergence of the Monte-Carlo simulation. However, in terms of the Runge-Kutta method, we cannot accept the consistency of our method with the theoretical order of convergence without further research.

There exist many numerical methods for numerical solutions of the systems of stochastic differential equations. We choose the method of deterministic quadrature formulae proposed by Müller–Gronbach, and Yaroslavtseva in 2016. The idea is to apply a simplified version of the cubature in Wiener space. We explain the method and check how good it works in the simplest case of the classical Black–Scholes model.

In this paper, we develop a model for predation movements of a subsocial spider species, Anelosimus studiosus. We expand on a previous model to include multiple spider interaction on the web as well as a latency period during predation. We then use the model to test different spatial configurations to determine the optimal spacing of spiders within a colony for successful capture during predation. The model simulations indicate that spiders uniformly spacing out along the edge of the web results in the most successful predation strategy. This is similar to the behavior observed by Ross (2013) in which it was determined to be statistically significant that during certain times of the day, spiders were positioned along the edge more than expected under complete spatial randomness.

We consider two distinct techniques for estimating random parameters in random differential equation (RDE) models. In one approach, the solution to a RDE is represented by a collection of solution trajectories in the form of sample deterministic equations. In a second approach we employ pointwise equivalent stochastic differential equation (SDE) representations for certain RDEs. Each of the approaches is tested using deterministic model comparison techniques for a logistic growth model which is viewed as a special case of a more general Bernoulli growth model. We demonstrate efficacy of the preferred method with experimental data using algae growth model comparisons.

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: This work employs perturbation techniques to price and hedge financial derivatives in a stochastic volatility framework. Fouque et al. [44] model volatility as a function of two processes operating on different time-scales. One process is responsible for the fast-fluctuating feature of volatility and corresponds to the slow time-scale and the second is for slowfluctuations or fast time-scale. The former is an Ergodic Markov process and the latter is a strong solution to a Lipschitz stochastic differential equation. This work mainly involves modelling, analysis and estimation techniques, exploiting the concept of mean reversion of volatility. The approach used is robust in the sense that it does not assume a specific volatility model. Using singular and regular perturbation techniques on the resulting PDE a first-order price correction to Black-Scholes option pricing model is derived. Vital groupings of market parameters are identified and their estimation from market data is extremely efficient and stable. The implied volatility is expressed as a linear (affine) function of log-moneyness-tomaturity ratio, and can be easily calibrated by estimating the grouped market parameters from the observed implied volatility surface. Importantly, the same grouped parameters can be used to price other complex derivatives beyond the European and American options, which include Barrier, Asian, Basket and Forward options. However, this semi-analytic perturbative approach is effective for longer maturities and unstable when pricing is done close to maturity. As a result a more accurate technique, the decomposition pricing approach that gives explicit analytic first- and second-order pricing and implied volatility formulae is discussed as one of the current alternatives. Here, the method is only employed for European options but an extension to other options could be an idea for further research. The only requirements for this method are integrability and regularity of the stochastic volatility process. Corrections to [3] remarkable work are discussed here. / AFRIKAANSE OPSOMMING: Hierdie werk gebruik steuringstegnieke om finansiële afgeleide instrumente in ’n stogastiese wisselvalligheid raamwerk te prys en te verskans. Fouque et al. [44] gemodelleer wisselvalligheid as ’n funksie van twee prosesse wat op verskillende tyd-skale werk. Een proses is verantwoordelik vir die vinnig-wisselende eienskap van die wisselvalligheid en stem ooreen met die stadiger tyd-skaal en die tweede is vir stadig-wisselende fluktuasies of ’n vinniger tyd-skaal. Die voormalige is ’n Ergodiese-Markov-proses en die laasgenoemde is ’n sterk oplossing vir ’n Lipschitz stogastiese differensiaalvergelyking. Hierdie werk behels hoofsaaklik modellering, analise en skattingstegnieke, wat die konsep van terugkeer to die gemiddelde van die wisseling gebruik. Die benadering wat gebruik word is rubuust in die sin dat dit nie ’n aanname van ’n spesifieke wisselvalligheid model maak nie. Deur singulêre en reëlmatige steuringstegnieke te gebruik op die PDV kan ’n eerste-orde pryskorreksie aan die Black-Scholes opsie-waardasiemodel afgelei word. Belangrike groeperings van mark parameters is geïdentifiseer en hul geskatte waardes van mark data is uiters doeltreffend en stabiel. Die geïmpliseerde onbestendigheid word uitgedruk as ’n lineêre (affiene) funksie van die log-geldkarakter-tot-verval verhouding, en kan maklik gekalibreer word deur gegroepeerde mark parameters te beraam van die waargenome geïmpliseerde wisselvalligheids vlak. Wat belangrik is, is dat dieselfde gegroepeerde parameters gebruik kan word om ander komplekse afgeleide instrumente buite die Europese en Amerikaanse opsies te prys, dié sluit in Barrier, Asiatiese, Basket en Stuur opsies. Hierdie semi-analitiese steurings benadering is effektief vir langer termyne en onstabiel wanneer pryse naby aan die vervaldatum beraam word. As gevolg hiervan is ’n meer akkurate tegniek, die ontbinding prys benadering wat eksplisiete analitiese eerste- en tweede-orde pryse en geïmpliseerde wisselvalligheid formules gee as een van die huidige alternatiewe bespreek. Hier word slegs die metode vir Europese opsies gebruik, maar ’n uitbreiding na ander opsies kan’n idee vir verdere navorsing wees. Die enigste vereistes vir hierdie metode is integreerbaarheid en reëlmatigheid van die stogastiese wisselvalligheid proses. Korreksies tot [3] se noemenswaardige werk word ook hier bespreek.

In this thesis, we consider some popular stochastic differential equation models used in finance, such as the Vasicek Interest Rate model, the Heston model and a new fractional Heston model. We discuss how to perform inference about unknown quantities associated with these models in the Bayesian framework. We describe sequential importance sampling, the particle filter and the auxiliary particle filter. We apply these inference methods to the Vasicek Interest Rate model and the standard stochastic volatility model, both to sample from the posterior distribution of the underlying processes and to update the posterior distribution of the parameters sequentially, as data arrive over time. We discuss the sensitivity of our results to prior assumptions. We then consider the use of Markov chain Monte Carlo (MCMC) methodology to sample from the posterior distribution of the underlying volatility process and of the unknown model parameters in the Heston model. The particle filter and the auxiliary particle filter are also employed to perform sequential inference. Next we extend the Heston model to the fractional Heston model, by replacing the Brownian motions that drive the underlying stochastic differential equations by fractional Brownian motions, so allowing a richer dependence structure across time. Again, we use a variety of methods to perform inference. We apply our methodology to simulated and real financial data with success. We then discuss how to make forecasts using both the Heston and the fractional Heston model. We make comparisons between the models and show that using our new fractional Heston model can lead to improve forecasts for real financial data.

This paper studies the endogenous evolution of investment behaviour under the various macroeconomic circumstances, which might be relatively constructed by free-float, fixed and target zone regimes as the economic stability policy. It applies the issues of stock price target zone policy to a simultaneous stochastic differential equation system. We construct the stochastic macro model which utilized the basic conception of Dornbusch [1976] with the different price adjustment mechanism. In addition, we intend to apply the topological method which used by Miller and Weller [1991] to analyze the general economic property from the non-recursive model. The main purpose of this paper is to discuss how the public’s expectation affects the dynamic loci of commodity and stock price when the public agents have the perfect or imperfect credibility. We utilize this model to investigate whether stock price target zone regime will have honeymoon effect or not, when the government announce to execute the stock price target zone policy in the various situations. Moreover, we discuss whether stock price target zone can simultaneously stabilize other variables in the different situations.

<p>Stochastic information, to be understood as &quot;information gained by the application of stochastic methods&quot;, is proposed as a tool in the assessment of changes in climate.</p> <p>This thesis aims at demonstrating that stochastic information can improve the consideration and reduction of uncertainty in the assessment of changes in climate. The thesis consists of three parts. In part one, an indicator is developed that allows the determination of the proximity to a critical threshold. In part two, the tolerable windows approach (TWA) is extended to a probabilistic TWA. In part three, an integrated assessment of changes in flooding probability due to climate change is conducted within the TWA.</p> <p>The thermohaline circulation (THC) is a circulation system in the North Atlantic, where the circulation may break down in a saddle-node bifurcation under the influence of climate change. Due to uncertainty in ocean models, it is currently very difficult to determine the distance of the THC to the bifurcation point. We propose a new indicator to determine the system's proximity to the bifurcation point by considering the THC as a stochastic system and using the information contained in the fluctuations of the circulation around the mean state. As the system is moved closer to the bifurcation point, the power spectrum of the overturning becomes &quot;redder&quot;, i.e. more energy is contained in the low frequencies. Since the spectral changes are a generic property of the saddle-node bifurcation, the method is not limited to the THC, but it could also be applicable to other systems, e.g. transitions in ecosystems. </p> <p>In part two, a probabilistic extension to the tolerable windows approach (TWA) is developed. In the TWA, the aim is to determine the complete set of emission strategies that are compatible with so-called guardrails. Guardrails are limits to impacts of climate change or to climate change itself. Therefore, the TWA determines the &quot;maneuvering space&quot; humanity has, if certain impacts of climate change are to be avoided. Due to uncertainty it is not possible to definitely exclude the impacts of climate change considered, but there will always be a certain probability of violating a guardrail. Therefore the TWA is extended to a probabilistic TWA that is able to consider &quot;probabilistic uncertainty&quot;, i.e. uncertainty that can be expressed as a probability distribution or uncertainty that arises through natural variability.</p> <p>As a first application, temperature guardrails are imposed, and the dependence of emission reduction strategies on probability distributions for climate sensitivities is investigated. The analysis suggests that it will be difficult to observe a temperature guardrail of 2&deg;C with high probabilities of actually meeting the target.</p> <p>In part three, an integrated assessment of changes in flooding probability due to climate change is conducted. A simple hydrological model is presented, as well as a downscaling scheme that allows the reconstruction of the spatio-temporal natural variability of temperature and precipitation. These are used to determine a probabilistic climate impact response function (CIRF), a function that allows the assessment of changes in probability of certain flood events under conditions of a changed climate. </p> <p>The assessment of changes in flooding probability is conducted in 83 major river basins. Not all floods can be considered: Events that either happen very fast, or affect only a very small area can not be considered, but large-scale flooding due to strong longer-lasting precipitation events can be considered. Finally, the probabilistic CIRFs obtained are used to determine emission corridors, where the guardrail is a limit to the fraction of world population that is affected by a predefined shift in probability of the 50-year flood event. This latter analysis has two main results. The uncertainty about regional changes in climate is still very high, and even small amounts of further climate change may lead to large changes in flooding probability in some river systems.</p> / <p>Stochastische Information, zu verstehen als &quot;Information, die durch die Anwendung stochastischer Methoden gewonnen wird&quot;, wird als Hilfsmittel in der Bewertung von Klima&auml;nderungen vorgeschlagen.</p> <p>Das Ziel dieser Doktorarbeit ist es, zu zeigen, dass stochastische Information die Ber&uuml;cksichtigung und Reduktion von Unsicherheit in der Bewertung des Klimawandels verbessern kann. Die Arbeit besteht aus drei Teilen. Im ersten Teil wird ein Indikator entwickelt, der die Bestimmung des Abstandes zu einem kritischen Grenzwert ermöglicht. Im zweiten Teil wird der &quot;tolerable windows approach&quot; (TWA) zu einem probabilistischen TWA erweitert. Im dritten Teil wird eine integrierte Absch&auml;tzung der Ver&auml;nderung von &Uuml;berflutungswahrscheinlichkeiten im Rahmen des TWA durchgeführt.</p> <p>Die thermohaline Zirkulation (THC) ist ein Zirkulationssystem im Nordatlantik, in dem die Zirkulation unter Einfluss des Klimawandels in einer Sattel-Knoten Bifurkation abreißen kann. Durch Unsicherheit in Ozeanmodellen ist es gegenw&auml;rtig kaum m&ouml;glich, den Abstand des Systems zum Bifurkationspunkt zu bestimmen. Wir schlagen einen neuen Indikator vor, der es erm&ouml;glicht, die N&auml;he des Systems zum Bifurkationspunkt zu bestimmen. Dabei wird die THC als stochastisches System angenommen, und die Informationen, die in den Fluktuationen der Zirkulation um den mittleren Zustand enthalten sind, ausgenutzt. Wenn das System auf den Bifurkationspunkt zubewegt wird, wird das Leistungsspektrum &quot;roter&quot;, d.h. die tiefen Frequenzen enthalten mehr Energie. Da diese spektralen Ver&auml;nderungen eine allgemeine Eigenschaft der Sattel-Knoten Bifurkation sind, ist die Methode nicht auf die THC beschränkt, sondern weitere Anwendungen k&ouml;nnten m&ouml;glich sein, beispielsweise zur Erkennung von &Uuml;berg&auml;ngen in &Ouml;kosystemen.</p> <p>Im zweiten Teil wird eine probabilistische Erweiterung des &quot;tolerable windows approach&quot; (TWA) entwickelt. Das Ziel des TWA ist die Bestimmung der Menge der Emissionsreduktionsstrategien, die mit sogenannten Leitplanken kompatibel sind. Diese Leitplanken sind Begrenzungen der Auswirkungen des Klimawandels, oder des Klimawandels selber. Der TWA bestimmt daher den Spielraum, den die Menschheit hat, wenn bestimmte Auswirkungen des Klimawandels vermieden werden sollen. Durch den Einfluss von Unsicherheit ist es aber nicht m&ouml;glich, die betrachteten Auswirkungen des Klimawandels mit Sicherheit auszuschlie&szlig;en, sondern es existiert eine gewisse Wahrscheinlichkeit, dass die Leitplanke verletzt wird. Der TWA wird daher zu einem probabilistischen TWA weiterentwickelt, der es erm&ouml;glicht, &quot;probabilistische Unsicherheit&quot;, also Unsicherheit, die durch eine Wahrscheinlichkeitsverteilung ausgedr&uuml;ckt werden kann, oder die durch den Einfluß von nat&uuml;rlicher Variabilit&auml;t entsteht, zu ber&uuml;cksichtigen.</p> <p>Als erste Anwendung werden Temperaturleitplanken betrachtet, und die Abh&auml;ngigkeit der Emissionsreduktionsstrategien von Wahrscheinlichkeitsverteilungen über die Klimasensitivit&auml;t wird bestimmt. Die Analyse ergibt, dass die Einhaltung einer Temperaturleitplanke von 2&deg;C sehr schwierig wird, wenn man hohe Wahrscheinlichkeiten des Einhaltens der Leitplanke fordert.</p> <p>Im dritten Teil wird eine integrierte Absch&auml;tzung der &Auml;nderungen von &Uuml;berflutungswahrscheinlichkeiten unter Einfluss des Klimawandels durchgef&uuml;hrt. Ein einfaches hydrologisches Modell wird vorgestellt, sowie ein Skalierungsansatz, der es erm&ouml;glicht, die raum-zeitliche nat&uuml;rliche Variabilit&auml;t von Temperatur und Niederschlag zu rekonstruieren. Diese werden zur Bestimmung einer probabilistischen Klimawirkungsfunktion genutzt, einer Funktion, die es erlaubt, die Ver&auml;nderungen der Wahrscheinlichkeit bestimmter &Uuml;berflutungsereignisse unter Einfluss von Klima&auml;nderungen abzusch&auml;tzen.</p> <p>Diese Untersuchung der Ver&auml;nderung von &Uuml;berflutungswahrscheinlichkeiten wird in 83 großen Flusseinzugsgebieten durchgef&uuml;hrt. Nicht alle Klassen von &Uuml;berflutungen k&ouml;nnen dabei ber&uuml;cksichtigt werden: Ereignisse, die entweder sehr schnell vonstatten gehen, oder die nur ein kleines Gebiet betreffen, k&ouml;nnen nicht ber&uuml;cksichtigt werden, aber großfl&auml;chige &Uuml;berflutungen, die durch starke, langanhaltende Regenf&auml;lle hervorgerufen werden, k&ouml;nnen ber&uuml;cksichtigt werden. Zuguterletzt werden die bestimmten Klimawirkungsfunktion dazu genutzt, Emissionskorridore zu bestimmen, bei denen die Leitplanken Begrenzungen des Bev&ouml;lkerungsanteils, der von einer bestimmten Ver&auml;nderung der Wahrscheinlichkeit eines 50-Jahres-Flutereignisses betroffen ist, sind. Letztere Untersuchung hat zwei Hauptergebnisse. Die Unsicherheit von regionalen Klima&auml;nderungen ist immer noch sehr hoch, und au&szlig;erdem k&ouml;nnen in einigen Flusssystemen schon kleine Klima&auml;nderungen zu gro&szlig;en &Auml;nderungen der &Uuml;berflutungswahrscheinlichkeit f&uuml;hren.</p>

We investigate an explicit tamed Euler scheme of stochastic differential equation with random coefficients driven by Lévy noise, which has super-linear drift coefficient. The strong convergence property of the tamed Euler scheme is proved when drift coefficient satisfies one-sided local Lipschitz condition whereas diffusion and jump coefficients satisfy local Lipschitz conditions. A rate of convergence for the tamed Euler scheme is recovered when local Lipschitz conditions are replaced by global Lipschitz conditions and drift satisfies polynomial Lipschitz condition. These findings are consistent with those of the classical Euler scheme. New methodologies are developed to overcome challenges arising due to the jumps and the randomness of the coefficients. Moreover, as an application of these findings, a tamed Euler scheme is proposed for the stochastic delay differential equation driven by Lévy noise with drift coefficient that grows super-linearly in both delay and non-delay variables. The strong convergence property of the tamed Euler scheme for such SDDE driven by Lévy noise is studied and rate of convergence is shown to be consistent with that of the classical Euler scheme. Finally, an explicit tamed Milstein scheme with rate of convergence arbitrarily close to one is developed to approximate the stochastic differential equation driven by Lévy noise (without random coefficients) that has super-linearly growing drift coefficient.

Orientador: Pedro Jose Catuogno / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatística e Computação Cientifica / Made available in DSpace on 2018-08-14T14:39:55Z (GMT). No. of bitstreams: 1 Castrequini_RafaelAndretto_M.pdf: 934326 bytes, checksum: e4c45bc1efde09bbe52710c44eab8bbf (MD5) Previous issue date: 2009 / Resumo: Introduzimos a teoria dos p-rough paths seguindo a abordagem de M. Gubinelli, conhecida por integração algébrica. Durante toda a dissertação nos restringimos ao caso 1 </= p < 3, o que e suficiente para lidar com trajetórias do movimento Browniano e aplicações ao Cálculo Estocástico. Em seguida, estudamos as equações diferenciais associadas aos rough paths, onde nós conectamos a abordagem de A. M. Davie (as equações) e a abordagem de M. Gubinelli (as integrais). No final da dissertação, aplicamos a teoria de rough path ao cálculo estocástico, mais precisamente relacionando as integrais de Itô e Stratonovich com a integral ao longo de caminhos. / Abstract: We introduce p-Rough Path Theory following M. Gubinelli_s approach, as known as algebraic integration. Throughout this masters thesis, we are concerned only in the case where 1 </= p < 3, witch is enough to deal with trajectories of a Brownnian motion and some applications to Stochastic Calculus. Afterwards, we study differential equations related to rough paths, where we connect the approach of A. M. Davie to equations with the approach of M. Gubinelli to integrals. At the end of this work, we apply the theory of rough paths to stochastic calculus, more precisely, we related the integrals of Itô and Stratonovich to integral along paths. / Mestrado / Sistemas estocasticos / Mestre em Matemática