In this thesis, I will study the qualitative properties of solutions of stochastic differential equations arising in applications by using the numerical methods. It contains two parts. In the first part, I will first review some of the basic theory of the stochastic calculus and the Ito-Taylor expansion for stochastic differential equations (SDEs). Then I will discuss some numerical schemes that come from the Ito-Taylor expansion including their order of convergence. In the second part, I will use some schemes to solve the stochastic Duffing equation, the stochastic Lorenz equation, the stochastic pendulum equation, and the stochastic equations which model the spread options.

The Burgers equation, and related partial differential equations (PDEs), can be numerically challenging for small values of the viscosity parameter. For example, these equations can develop discontinuous solutions (or solutions with large gradients) from smooth initial data. Aside from numerical stability issues, standard numerical methods can also give rise to spurious oscillations near these discontinuities. In this study, we consider an equivalent form of the Burgers equation given by Constantin and Iyer, whose solution can be written as the expected value of a stochastic differential equation. This equivalence is used to develop a numerical method for approximating solutions to Burgers equation. Our preliminary analysis of the algorithm reveals that it is a natural generalization of the method of characteristics and that it produces approximate solutions that actually improve as the viscosity parameter vanishes. We present three examples that compare our algorithm to a recently published reference method as well as the vanishing viscosity/entropy solution for decreasing values of the viscosity. / Master of Science / Burgers equation is a Partial Differential Equation (PDE) used to model how fluids evolve in time based on some initial condition and viscosity parameter. This viscosity parameter helps describe how the energy in a fluid dissipates. When studying partial differential equations, it is often hard to find a closed form solution to the problem, so we often approximate the solution with numerical methods. As our viscosity parameter approaches 0, many numerical methods develop problems and may no longer accurately compute the solution. Using random variables, we develop an approximation algorithm and test our numerical method on various types of initial conditions with small viscosity coefficients.

We present results that span three interconnected domains. Initially, our analysis is centred on Backward Stochastic Differential Equations (BSDEs) featuring time-delayed generators. Subsequently, we direct our interest towards Mean Field Games (MFGs) incorporating absorption aspects, with a focus on the corresponding Master Equation within a confined domain under the imposition of Dirichlet boundary conditions. The investigation culminates in exploring pertinent Machine Learning methodologies applied to financial and economic decision-making processes.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: We consider the utility portfolio optimization problem of an investor whose activities are influenced by an exogenous financial risk (like bad weather or energy shortage) in an incomplete financial market. We work with a fairly general non-Markovian model, allowing stochastic correlations between the underlying assets. This important problem in finance and insurance is tackled by means of backward stochastic differential equations (BSDEs), which have been shown to be powerful tools in stochastic control. To lay stress on the importance and the omnipresence of BSDEs in stochastic control, we present three methods to transform the control problem into a BSDEs. Namely, the martingale optimality principle introduced by Davis, the martingale representation and a method based on Itô-Ventzell’s formula. These approaches enable us to work with portfolio constraints described by closed, not necessarily convex sets and to get around the classical duality theory of convex analysis. The solution of the optimization problem can then be simply read from the solution of the BSDE. An interesting feature of each of the different approaches is that the generator of the BSDE characterizing the control problem has a quadratic growth and depends on the form of the set of constraints. We review some recent advances on the theory of quadratic BSDEs and its applications. There is no general existence result for multidimensional quadratic BSDEs. In the one-dimensional case, existence and uniqueness strongly depend on the form of the terminal condition. Other topics of investigation are measure solutions of BSDEs, notably measure solutions of BSDE with jumps and numerical approximations. We extend the equivalence result of Ankirchner et al. (2009) between existence of classical solutions and existence of measure solutions to the case of BSDEs driven by a Poisson process with a bounded terminal condition. We obtain a numerical scheme to approximate measure solutions. In fact, the existing self-contained construction of measure solutions gives rise to a numerical scheme for some classes of Lipschitz BSDEs. Two numerical schemes for quadratic BSDEs introduced in Imkeller et al. (2010) and based, respectively, on the Cole-Hopf transformation and the truncation procedure are implemented and the results are compared. Keywords: BSDE, quadratic growth, measure solutions, martingale theory, numerical scheme, indifference pricing and hedging, non-tradable underlying, defaultable claim, utility maximization. / AFRIKAANSE OPSOMMING: Ons beskou die nuts portefeulje optimalisering probleem van ’n belegger wat se aktiwiteite beïnvloed word deur ’n eksterne finansiele risiko (soos onweer of ’n energie tekort) in ’n onvolledige finansiële mark. Ons werk met ’n redelik algemene nie-Markoviaanse model, wat stogastiese korrelasies tussen die onderliggende bates toelaat. Hierdie belangrike probleem in finansies en versekering is aangepak deur middel van terugwaartse stogastiese differensiaalvergelykings (TSDEs), wat blyk om ’n onderskeidende metode in stogastiese beheer te wees. Om klem te lê op die belangrikheid en alomteenwoordigheid van TSDEs in stogastiese beheer, bespreek ons drie metodes om die beheer probleem te transformeer na ’n TSDE. Naamlik, die martingale optimaliteits beginsel van Davis, die martingale voorstelling en ’n metode wat gebaseer is op ’n formule van Itô-Ventzell. Hierdie benaderings stel ons in staat om te werk met portefeulje beperkinge wat beskryf word deur geslote, nie noodwendig konvekse versamelings, en die klassieke dualiteit teorie van konvekse analise te oorkom. Die oplossing van die optimaliserings probleem kan dan bloot afgelees word van die oplossing van die TSDE. ’n Interessante kenmerk van elkeen van die verskillende benaderings is dat die voortbringer van die TSDE wat die beheer probleem beshryf, kwadratiese groei en afhanglik is van die vorm van die versameling beperkings. Ons herlei ’n paar onlangse vooruitgange in die teorie van kwadratiese TSDEs en gepaartgaande toepassings. Daar is geen algemene bestaanstelling vir multidimensionele kwadratiese TSDEs nie. In die een-dimensionele geval is bestaan ââen uniekheid sterk afhanklik van die vorm van die terminale voorwaardes. Ander ondersoek onderwerpe is maatoplossings van TSDEs, veral maatoplossings van TSDEs met spronge en numeriese benaderings. Ons brei uit op die ekwivalensie resultate van Ankirchner et al. (2009) tussen die bestaan van klassieke oplossings en die bestaan van maatoplossings vir die geval van TSDEs wat gedryf word deur ’n Poisson proses met begrensde terminale voorwaardes. Ons verkry ’n numeriese skema om oplossings te benader. Trouens, die bestaande self-vervatte konstruksie van maatoplossings gee aanleiding tot ’n numeriese skema vir sekere klasse van Lipschitz TSDEs. Twee numeriese skemas vir kwadratiese TSDEs, bekendgestel in Imkeller et al. (2010), en gebaseer is, onderskeidelik, op die Cole-Hopf transformasie en die afknot proses is geïmplementeer en die resultate word vergelyk.

Fokker-Planck equations, along with stochastic differential equations, play vital roles in physics, population modeling, game theory and optimization (finite or infinite dimensional). In this thesis, we study three topics, both theoretically and computationally, centered around them. In part one, we consider the optimal transport for finite discrete states, which are on a finite but arbitrary graph. By defining a discrete 2-Wasserstein metric, we derive Fokker-Planck equations on finite graphs as gradient flows of free energies. By using dynamical viewpoint, we obtain an exponential convergence result to equilibrium. This derivation provides tools for many applications, including numerics for nonlinear partial differential equations and evolutionary game theory. In part two, we introduce a new stochastic differential equation based framework for optimal control with constraints. The framework can efficiently solve several real world problems in differential games and Robotics, including the path-planning problem. In part three, we introduce a new noise model for stochastic oscillators. With this model, we prove global boundedness of trajectories. In addition, we derive a pair of associated Fokker-Planck equations.

This thesis is about double product integrals with pseudo rotational generator, and aims to exhibit them as unitary implementors of Bogolubov transformations. We further introduce these concepts in this abstract and describe their roles in the thesis's chapters. The notion of product integral, (simple product integral, not double) is not a new one, but is unfamiliar to many a mathematician. Product integrals were first investigated by Volterra in the nineteenth century. Though often regarded as merely a notation for solutions of differential equations, they provide a priori a multiplicative analogue of the additive integration theories of Riemann, Stieltjes and Lebesgue. See Slavik [2007] for a historical overview of the subject. Extensions of the theory of product integrals to multiplicative versions of Ito and especially quantum Ito calculus were first studied by Hudson, Ion and Parthasarathy in the 1980's, Hudson et al. [1982]. The first developments of double product integrals was a theory of an algebraic kind developed by Hudson and Pulmannova motivated by the study of the solution of the quantum Yang-Baxter equation by the construction of quantum groups, see Hudson and Pulmaanova [2005]. This was a purely algebraic theory based on formal power series in a formal parameter. However, there also exists a developing analytic theory of double product integral. This thesis contributes to this analytic theory. The first papers in that direction are Hudson [2005b] and Hudson and Jones [2012]. Other motivations include quantum extension of Girsanov's theorem and hence a quantum version of the Black-Scholes model in finance. They may also provide a general model for causal interactions in noisy environments in quantum physics. From a different direction "causal" double products, (see Hudson [2005b]), have become of interest in connection with quantum versions of the Levy area, and in particular quantum Levy area formula (Hudson [2011] and Chen and Hudson [2013]) for its characteristic function. There is a close association of causal double products with the double products of rectangular type (Hudson and Jones [2012] pp 3). For this reason it is of interest to study "forwardforward" rectangular double products. In the first chapter we give our notation which will be used in the following chapters and we introduce some simple double products and show heuristically that they are the solution of two different quantum stochastic differential equations. For each example the order in which the products are taken is shown to be unimportant; either calculation gives the same answer. This is in fact a consequence of the so called multiplicative Fubini Theorem Hudson and Pulmaanova [2005]. In Chapter two we formally introduce the notion of product integral as a solution of two particular quantum stochastic differential equations. In Chapter three we introduce the Fock representation of the canonical commutation relations, and discuss the Stone-von Neumann uniqueness theorem. We define the notion of Bogolubov transformation (often called a symplectic automorphism, see Parthasarathy [1992] for example), implementation of these transformations by an implementor (a unitary operator) and introduce Shale's theorem which will be relevant to the following chapters. For an alternative coverage of Shale's Theorem, symplectic automorphism and their implementors see Derezinski [2003]. In Chapter four we study double product integrals of the pseudo rotational type. This is in contrast to double product integrals of the rotational type that have been studied in (Hudson and Jones [2012] and Hudson [2005b]). The notation of the product integral is suggestive of a natural discretisation scheme where the infinitesimals are replaced by discrete increments i.e. discretised creation and annihilation operators of quantum mechanics. Because of a weak commutativity condition, between the discretised creation and annihilation operators corresponding on different subintervals of R, the order of the factors of the product are unimportant (Hudson [2005a]), and hence the discrete product is well defined; we call this result the discrete multiplicative Fubini Theorem. It is also the case that the order in which the products are taken in the continuous (non-discretised case) does not matter (Hudson [2005a], Hudson and Jones [2012]). The resulting discrete double product is shown to be the implementor (a unitary operator) of a Bogolubov transformation acting on discretised creation and annihilation operators (Bogolubov transformations are invertible real linear operators on a Hilbert space that preserve the imaginary part of the inner product, but here we may regard them equivalently as liner transformations acting directly on creation and annihilations operators but preserving adjointness and commutation relations). Unitary operators on the same Hilbert space are a subgroup of the group of Bogolubov transformations. Essentially Bogolubov transformations are used to construct new canonical pairs from old ones (In the literature Bogolubov transformations are often called symplectic automorphisms). The aforementioned Bogolubov transformation (acting on the discretised creation and annihilation operators) can be embedded into the space L2(R+) L2(R+) and limits can be taken resulting in a limiting Bogolubov transformation in the space L2(R+) L2(R+). It has also been shown that the resulting family of Bogolubov transformation has three important properties, namely bi-evolution, shift covariance and time-reversal covariance, see (Hudson [2007]) for a detailed description of these properties. Subsequently we show rigorously that this transformation really is a Bogolubov transformation. We remark that these transformations are Hilbert-Schmidt perturbations of the identity map and satisfy a criterion specified by Shale's theorem. By Shale's theorem we then know that each Bogolubov transformation is implemented in the Fock representation of the CCR. We also compute the constituent kernels of the integral operators making up the Hilbert-Schmidt operators involved in the Bogolubov transformations, and show that the order in which the approximating discrete products are taken has no bearing on the final Bogolubov transformation got by the limiting procedure, as would be expected from the multiplicative Fubini Theorem. In Chapter five we generalise the canonical form of the double product studied in Chapter four by the use of gauge transformations. We show that all the theory of Chapter four carries over to these generalised double product integrals. This is because there is unitary equivalence between the Bogolubov transformation got from the generalised canonical form of the double product and the corresponding original one. In Chapter six we make progress towards showing that a system of implementors of this family of Bogolubov transformations can be found which inherits properties of the original family such as being a bi-evolution and being covariant under shifts. We make use of Shales theorem (Parthasarathy [1992] and Derezinski [2003]). More specifically, Shale's theorem ensures that each Bogolubov transformation of our system is implemented by a unitary operator which is unique to with multiplicaiton by a scalar of modulus 1. We expect that there is a unique system of implementors, which is a bi-evolution, shift covariant, and time reversal covariant (i.e. which inherits the properties of the corresponding system of Bogolubov transformation). This is partly on-going research. We also expect the implementor of the Bogolubov transformation to be the original double product. In Evans [1988], Evan's showed that the the implementor of a Bogolubov transformation in the simple product case is indeed the simple product. If given more time it might be possible to adapt Evan's result to the double product case.

This thesis contains a discussion of four problems arising from the application of stochastic differential equations and real option theory to investment decision problems in a continuous-time framework. It is based on four papers written jointly with the author’s supervisor. In the first problem, we study an evolutionary stock market model in a continuous-time framework where uncertainty in dividends is produced by a single Wiener process. The model is an adaptation to a continuous-time framework of a discrete evolutionary stock market model developed by Evstigneev, Hens and Schenk-Hoppé (2006). We consider the case of fix-mix strategies and derive the stochastic differential equations which determine the evolution of the wealth processes of the various market players. The wealth dynamics for various initial set-ups of the market are simulated. In the second problem, we apply an entry-exit model in real option theory to study concessionary agreements between a private company and a state government to run a privatised business or project. The private company can choose the time to enter into the agreement and can also choose the time to exit the agreement if the project becomes unprofitable. An early termination of the agreement by the company might mean that it has to pay a penalty fee to the government. Optimal times for the company to enter and exit the agreement are calculated. The dynamics of the project are assumed to follow either a geometric mean reversion process or geometric Brownian motion. A comparative analysis is provided. Particular emphasis is given to the role of uncertainty and how uncertainty affects the average time that the concessionary agreement is active. The effect of uncertainty is studied by using Monte Carlo simulation. In the third problem, we study numerical methods for solving stochastic optimal control problems which are linear in the control. In particular, we investigate methods based on spline functions for solving the two-point boundary value problems that arise from the method of dynamic programming. In the general case, where only the value function and its first derivative are guaranteed to be continuous, piecewise quadratic polynomials are used in the solution. However, under certain conditions, the continuity of the second derivative is also guaranteed. In this case, piecewise cubic polynomials are used in the solution. We show how the computational time and memory requirements of the solution algorithm can be improved by effectively reducing the dimension of the problem. Numerical examples which demonstrate the effectiveness of our method are provided. Lastly, we study the situation where, by partial privatisation, a government gives a private company the opportunity to invest in a government-owned business. After payment of an initial instalment cost, the private company’s investments are assumed to be flexible within a range [0, k] while the investment in the business continues. We model the problem in a real option framework and use a geometric mean reversion process to describe the dynamics of the business. We use the method of dynamic programming to determine the optimal time for the private company to enter and pay the initial instalment cost as well as the optimal dynamic investment strategy that it follows afterwards. Since an analytic solution cannot be obtained for the dynamic programming equations, we use quadratic splines to obtain a numerical solution. Finally we determine the optimal degree of privatisation in our model from the perspective of the government.

No description available.

We study dynamical systems in the complex plane under the effect of constant noise. We show for a wide class of polynomial equations that the ergodic property is valid in the associated stochastic perturbation if and only if the noise added is in the direction transversal to all unstable trajectories of the deterministic system. This has the interpretation that noise in the "right" direction prevents the process from being unstable: a fundamental, but not well-understood, geometric principle which seems to underlie many other similar equations. The result is proven by using Lyapunov functions and geometric control theory.

This thesis studies abstract stochastic differential equations on Banach spaces. The well-posedness of abstract stochastic differential equations on such spaces is a recent result of van Neerven, Veraar and Weis, based on the theory of stochastic integration of Banach space valued processes constructed by the same authors. We study existence and uniqueness for solutions of stochastic differential equations with (possibly infinite) delay in their inputs on UMD Banach spaces. Such problems are also known as functional differential equations or delay differential equations. We show that the methods of van Neerven et al. extend to such problems if the initial history of the system lies in a space of a type introduced by Hale and Kato. The results are essentially of a fixed point type, both autonomous and non-autonomous cases are discussed and an example is given. We also study some long time properties of solutions to these stochastic differential equations on general Banach spaces. We show the existence of solutions to stochastic problems with almost periodicity in a weak or distributional sense. Results are again given for both autonomous and non-autonomous cases and depend heavily on estimates for R-bounds of operator families developed by Veraar. An example is given for a second order differential operator on a domain in ℝ^d. Finally we consider the existence of invariant measures for such problems. This extends recent work of van Gaans in Hilbert spaces to Banach spaces of type 2.