Reflected Backward Stochastic Difference Equations and Optimal Stopping Problems under g-expectation

In this paper, we study reflected backward stochastic difference equations (RBSDEs for short) with finitely many states in discrete time. The general existence and uniqueness result, as well as comparison theorems for the solutions, are established under mild assumptions. The connections between RBSDEs and optimal stopping problems are also given. Then we apply the obtained results to explore optimal stopping problems under $g$-expectation. Finally, we study the pricing of American contingent claims in our context.Comment: 29 page

Ergodic backward stochastic difference equations

We consider ergodic backward stochastic differential equations in a discrete time setting, where noise is generated by a finite state Markov chain. We show existence and uniqueness of solutions, along with a comparison theorem. To obtain this result, we use a Nummelin splitting argument to obtain ergodicity estimates for a discrete time Markov chain which hold uniformly under suitable perturbations of its transition matrix. We conclude with an application of this theory to a treatment of an ergodic control problem

Approximating solutions of backward doubly stochastic differential equations with measurable coefficients using a time discretization scheme

It has been shown that backward doubly stochastic differential equations (BDSDEs) provide a probabilistic representation for a certain class of nonlinear parabolic stochastic partial differential equations (SPDEs). It has also been shown that the solution of a BDSDE with Lipschitz coefficients can be approximated by first discretizing time and then calculating a sequence of conditional expectations. Given fixed points in time and space, this approximation has been shown to converge in mean square. In this thesis, we investigate the approximation of solutions of BDSDEs with coefficients that are measurable in time and space using a time discretization scheme with a view towards applications to SPDEs. To achieve this, we require the underlying forward diffusion to have smooth coefficients and we consider convergence in a norm which includes a weighted spatial integral. This combination of smoother forward coefficients and weaker norm allows the use of an equivalence of norms result which is key to our approach. We additionally take a brief look at the approximation of solutions of a class of infinite horizon BDSDEs with a view towards approximating stationary solutions of SPDEs. Whilst we remain agnostic with regards to the implementation of our discretization scheme, our scheme should be amenable to a Monte Carlo simulation based approach. If this is the case, we propose that in addition to being attractive from a performance perspective in higher dimensions, such an approach has a potential advantage when considering measurable coefficients. Specifically, since we only discretize time and effectively rely on simulations of the underlying forward diffusion to explore space, we are potentially less vulnerable to systematically overestimating or underestimating the effects of coefficients with spatial discontinuities than alternative approaches such as finite difference or finite element schemes that do discretize space. Another advantage of the BDSDE approach is that it is possible to derive an upper bound on the error of our method for a fairly broad class of conditions in a single analysis. Furthermore, our conditions seem more general in some respects than is typically considered in the SPDE literature

Impulse Control in Finance: Numerical Methods and Viscosity Solutions

The goal of this thesis is to provide efficient and provably convergent numerical methods for solving partial differential equations (PDEs) coming from impulse control problems motivated by finance. Impulses, which are controlled jumps in a stochastic process, are used to model realistic features in financial problems which cannot be captured by ordinary stochastic controls. The dynamic programming equations associated with impulse control problems are Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) Other than in certain special cases, the numerical schemes that come from the discretization of HJBQVIs take the form of complicated nonlinear matrix equations also known as Bellman problems. We prove that a policy iteration algorithm can be used to compute their solutions. In order to do so, we employ the theory of weakly chained diagonally dominant (w.c.d.d.) matrices. As a byproduct of our analysis, we obtain some new results regarding a particular family of Markov decision processes which can be thought of as impulse control problems on a discrete state space and the relationship between w.c.d.d. matrices and M-matrices. Since HJBQVIs are nonlocal PDEs, we are unable to directly use the seminal result of Barles and Souganidis (concerning the convergence of monotone, stable, and consistent numerical schemes to the viscosity solution) to prove the convergence of our schemes. We address this issue by extending the work of Barles and Souganidis to nonlocal PDEs in a manner general enough to apply to HJBQVIs. We apply our schemes to compute the solutions of various classical problems from finance concerning optimal control of the exchange rate, optimal consumption with fixed and proportional transaction costs, and guaranteed minimum withdrawal benefits in variable annuities

Impulse Control in Finance: Numerical Methods and Viscosity Solutions

The goal of this thesis is to provide efficient and provably convergent numerical methods for solving partial differential equations (PDEs) coming from impulse control problems motivated by finance. Impulses, which are controlled jumps in a stochastic process, are used to model realistic features in financial problems which cannot be captured by ordinary stochastic controls. The dynamic programming equations associated with impulse control problems are Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) Other than in certain special cases, the numerical schemes that come from the discretization of HJBQVIs take the form of complicated nonlinear matrix equations also known as Bellman problems. We prove that a policy iteration algorithm can be used to compute their solutions. In order to do so, we employ the theory of weakly chained diagonally dominant (w.c.d.d.) matrices. As a byproduct of our analysis, we obtain some new results regarding a particular family of Markov decision processes which can be thought of as impulse control problems on a discrete state space and the relationship between w.c.d.d. matrices and M-matrices. Since HJBQVIs are nonlocal PDEs, we are unable to directly use the seminal result of Barles and Souganidis (concerning the convergence of monotone, stable, and consistent numerical schemes to the viscosity solution) to prove the convergence of our schemes. We address this issue by extending the work of Barles and Souganidis to nonlocal PDEs in a manner general enough to apply to HJBQVIs. We apply our schemes to compute the solutions of various classical problems from finance concerning optimal control of the exchange rate, optimal consumption with fixed and proportional transaction costs, and guaranteed minimum withdrawal benefits in variable annuities

Multilevel Ansatz zur Bewertung von Bermuda Optionen

The Multilevel approach has been introduced into stochastics by Heinrich 2001 and Giles 2008. It is an idea about how to reduce the complexity of Monte Carlo simulations, if the precision and computational time of these simulations depend on a parameter. In this work, the Multilevel approach will be applied to approximate the fair price of a Bermudan option. The latter is a financial option that gives the holder the right to get an amount of money depending on a stochastic process at one of finitely many exercise dates. The strategy when to exercise the option should therefore by optimized. The task is now to find stochastic methods that calculate a lower and an upper bound for the fair price of such an option. If the performance of such methods is measured via its mean-squared error epsilon, the calculation of lower bounds has a complexity of epsilon^{-3} in the usual case. Here, the usual case is the combination of a good-natured problem and the use of a stochastic mesh method. By using the Multilevel approach, we can reduce the complexity down to epsilon^{-2.5}. In other cases, the reduction of complexity can be even of order epsilon^{-1} instead of epsilon^{-0.5}. In order to find upper bounds, we use the dual method from Rogers 2002 and Haugh and Kogan 2004. It expresses the fair price of the option as a minimization problem over a set of martingales. Andersen and Broadie 2004 exploit this idea by approximating martingales with nested simulations. These nested simulations lead to a high computational complexity. Depending on the problem, this complexity can be of order epsilon^{-3} or even epsilon^{-4}. The Multilevel approach reduces this order in each case to epsilon^{-2} up to a logarithmic factor.Der Multilevel Ansatz wurde durch die Arbeiten von Heinrich 2001 und Giles 2008 in der Stochastik populär. Es handelt sich dabei um eine Technik, die die Komplexität einer Monte-Carlo Simulation reduzieren kann, wenn deren Rechenzeit und Präzision von einem Parameter abhängt. Dieser Ansatz wird im Folgenden angewendet, um die Berechnung des fairen Preises einer Bermuda-Finanzoption zu berechnen. Letztere ist ein Derivat, das dem Besitzer das Recht gibt, einen von einem stochastischen Prozess abhängigen Geldbetrag an einem von endlich vielen gegebenen Zeitpunkten zu erhalten. Deshalb sollte der Zeitpunkt, an dem der Besitzer die Option einlöst, optimal gewählt werden. Die Aufgabe besteht nun darin, mit stochastischen Methoden eine obere und eine untere Schranke für den fairen Preis einer solchen Option zu berechnen. Bewertet man die Qualität einer solchen Methode mithilfe des mittleren quadratischen Fehlers epsilon, so ergibt sich für die Berechnung unterer Schranken eine Komplexität von epsilon^{-3} im gewöhnlichen Fall. Unter dem gewöhnlichen Fall ist ein gut gestelltes Problem und die Verwendung eines stochastischen Netzes zu verstehen. Mit Hilfe des Multilevel Ansatzes lässt sich in diesem Fall eine Komplexität von epsilon^{-2.5} erreichen. Die Verbesserung um den Faktor epsilon^{-0.5} kann in anderen Fällen sogar bis zu epsilon^{-1} betragen. Um obere Schranken für den Wert einer Bermuda Option zu berechnen, kann die duale Formulierung nach Rogers 2002 und Haugh und Kogan 2004 benutzt werden. Diese drückt den Wert der Option als Minimierungsproblem über einer Menge adaptierter Martingale aus. Andersen und Broadie 2004 nutzen in ihrer Arbeit diese Formulierung, indem sie das optimale Martingal mit Hilfe von geschachtelten Simulationen approximieren. Diese geschachtelten Simulationen führen zu hohem Rechenaufwand, welcher erneut durch den Multilevel Ansatz reduziert werden kann. Je nach Problemstellung kann das Problem eine Komplexität von epsilon^{-3} oder sogar epsilon^{-4} aufweisen. Der Multilevel Ansatz senkt die Komplexität in jedem Fall (bis auf einen logarithmischen Faktor) auf epsilon^{-2}

Techniques and approaches for pricing American option

This thesis is concerned with the theory of optimal stopping and martingale optimal transport, and its applications to the pricing and hedging of American-type contingent claims. In the first chapter we revisit the classical optimal stopping problem in continuous time and explore a delicate connection between semimartingale and Markovian formulations of the problem. More specifically, in the Markovian setting we are motivated by the question of whether the value function, corresponding to the optimal stopping problem, belongs to a certain class of functions (i.e. the domain of the extended or martingale generator) associated to the underlying Markov process. We show that the answer follows naturally from the fundamental property of the value process in a more general, semimartingale setting. We investigate applications of these results to the dual formulation of the optimal stopping problem and the classical smooth fit principle. The goal of the second chapter is to study the problem in martingale optimal transport, which is to move mass from a starting law (on R) to a terminal law (on R) in a way which respects the martingale property. One method is the `shadow embedding' of Beiglbock and Juillet [10]. Using the potential functions of the starting and terminal laws, we show how to explicitly construct the associated shadow measure. We also discuss the properties of the left-curtain martingale coupling, which is a coupling that arises (via shadow measure) from a certain parametrisation of the marginals. This coupling turns out to be optimal for the novel optimal martingale transport with stopping problem studied in the third chapter. The third chapter studies the problem of finding the highest robust or model independent price of the American put option given the prices of liquid European options, in a simple (but non-trivial) two time period setting. Combining ideas from the theory of optimal stopping and martingale optimal transport, we find, under some simplifying but still general conditions on the given data, the optimal model and the optimal stopping time. We also explicitly calculate the cheapest superhedging trading strategy. In the fourth chapter our goal is to find a specific geometric description of the left-curtain martingale coupling, which can be viewed as a martingale counterpart of the monotone Hoeffding-Frechet coupling in the classical optimal transport. While this is of independent interest, we also show that this generalised martingale coupling maximises the price of the American put option (studied in the third chapter under some simplifying assumptions)