Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of Hamilton--Jacobi--Bellman equations

We analyse a class of nonoverlapping domain decomposition preconditioners for nonsymmetric linear systems arising from discontinuous Galerkin finite element approximation of fully nonlinear Hamilton--Jacobi--Bellman (HJB) partial differential equations. These nonsymmetric linear systems are uniformly bounded and coercive with respect to a related symmetric bilinear form, that is associated to a matrix $\mathbf{A}$. In this work, we construct a nonoverlapping domain decomposition preconditioner $\mathbf{P}$, that is based on $\mathbf{A}$, and we then show that the effectiveness of the preconditioner for solving the} nonsymmetric problems can be studied in terms of the condition number $\kappa(\mathbf{P}^{-1}\mathbf{A})$. In particular, we establish the bound $\kappa(\mathbf{P}^{-1}\mathbf{A}) \lesssim 1+ p^6 H^3 /q^3 h^3$, where $H$ and $h$ are respectively the coarse and fine mesh sizes, and $q$ and $p$ are respectively the coarse and fine mesh polynomial degrees. This represents the first such result for this class of methods that explicitly accounts for the dependence of the condition number on $q$; our analysis is founded upon an original optimal order approximation result between fine and coarse discontinuous finite element spaces. Numerical experiments demonstrate the sharpness of this bound. Although the preconditioners are not robust with respect to the polynomial degree, our bounds quantify the effect of the coarse and fine space polynomial degrees. Furthermore, we show computationally that these methods are effective in practical applications to nonsymmetric, fully nonlinear HJB equations under $h$-refinement for moderate polynomial degrees

B-Spline Based Methods: From Monotone Multigrid Schemes for American Options to Uncertain Volatility Models

In the first part of this thesis, we consider B-spline based methods for pricing American options in the Black-Scholes and Heston model. The difference between these two models is the assumption on the volatility of the underlying asset. While in the Black-Scholes model the volatility is assumed to be constant, the Heston model includes a stochastic volatility variable. The underlying problems are formulated as parabolic variational inequalities. Recall that, in finance, to determine optimal risk strategies, one is not only interested in the solution of the variational inequality, i.e., the option price, but also in its partial derivatives up to order two, the so-called Greeks. A special feature for these option price problems is that initial conditions are typically given as piecewise linear continuous functions. Consequently, we have derived a spatial discretization based on cubic B-splines with coinciding knots at the points where the initial condition is not differentiable. Together with an implicit time stepping scheme, this enables us to achieve an accurate pointwise approximation of the partial derivatives up to order two. For the efficient numerical solution of the discrete variational inequality, we propose a monotone multigrid method for (tensor product) B-splines with possible internal coinciding knots. Corresponding numerical results show that the monotone multigrid method is robust with respect to the refinement level and mesh size. In the second part of this thesis, we consider the pricing of a European option in the uncertain volatility model. In this model the volatility of the underlying asset is a priori unknown and is assumed to lie within a range of extreme values. Mathematically, this problem can be formulated as a one dimensional parabolic Hamilton-Jacobi-Bellman equation and is also called Black-Scholes-Barenblatt equation. In the resulting non-linear equation, the diffusion coefficient is given by a volatility function which depends pointwise on the second derivative. This kind of non-linear partial differential equation does not admit a weak H^1-formulation. This is due to the fact that the non-linearity depends pointwise on the second derivative of the solution and, thus, no integration by parts is possible to pass the partial derivative onto a test function. But in the discrete setting this pointwise second derivative can be approximated in H^1 by L^1-normalized B-splines. It turns out that the approximation of the volatility function leads to discontinuities in the partial derivatives. In order to improve the approximation of the solution and its partial derivatives for cubic B-splines, we develop a Newton like algorithm within a knot insertion step. Corresponding numerical results show that the convergence of the solution and its partial derivatives are nearly optimal in the L^2-norm, when the location of volatility change is approximated with desired accuracy

A penalty scheme for monotone systems with interconnected obstacles: convergence and error estimates

We present a novel penalty approach for a class of quasi-variational inequalities (QVIs) involving monotone systems and interconnected obstacles. We show that for any given positive switching cost, the solutions of the penalized equations converge monotonically to those of the QVIs. We estimate the penalization errors and are able to deduce that the optimal switching regions are constructed exactly. We further demonstrate that as the switching cost tends to zero, the QVI degenerates into an equation of HJB type, which is approximated by the penalized equation at the same order (up to a log factor) as that for positive switching cost. Numerical experiments on optimal switching problems are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method.Comment: Accepted for publication (in this revised form) in SIAM Journal on Numerical Analysi

Penalty Methods for the Solution of Discrete HJB Equations -- Continuous Control and Obstacle Problems

In this paper, we present a novel penalty approach for the numerical solution of continuously controlled HJB equations and HJB obstacle problems. Our results include estimates of the penalisation error for a class of penalty terms, and we show that variations of Newton's method can be used to obtain globally convergent iterative solvers for the penalised equations. Furthermore, we discuss under what conditions local quadratic convergence of the iterative solvers can be expected. We include numerical results demonstrating the competitiveness of our methods.Comment: 31 Pages, 7 Figure

Polyhedral Newton-min algorithms for complementarity problems

Abstract : The semismooth Newton method is a very efficient approach for computing a zero of a large class of nonsmooth equations. When the initial iterate is sufficiently close to a regular zero and the function is strongly semismooth, the generated sequence converges quadratically to that zero, while the iteration only requires to solve a linear system. If the first iterate is far away from a zero, however, it is difficult to force its convergence using linesearch or trust regions because a semismooth Newton direction may not be a descent direction of the associated least-square merit function, unlike when the function is differentiable. We explore this question in the particular case of a nonsmooth equation reformulation of the nonlinear complementarity problem, using the minimum function. We propose a globally convergent algorithm using a modification of a semismooth Newton direction that makes it a descent direction of the least-square function. Instead of requiring that the direction satisfies a linear system, it must be a feasible point of a convex polyhedron; hence, it can be computed in polynomial time. This polyhedron is defined by the often very few inequalities, obtained by linearizing pairs of functions that have close negative values at the current iterate; hence, somehow, the algorithm feels the proximity of a “negative kink” of the minimum function and acts accordingly. In order to avoid as often as possible the extra cost of having to find a feasible point of a polyhedron, a hybrid algorithm is also proposed, in which the Newton-min direction is accepted if a sufficient-descent-like criterion is satisfied, which is often the case in practice. Global convergence to regular points is proved

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Numerical Methods for Nonlinear Equations in Option Pricing

This thesis explores numerical methods for solving nonlinear partial differential equations (PDEs) that arise in option pricing problems. The goal is to develop or identify robust and efficient techniques that converge to the financially relevant solution for both one and two factor problems. To illustrate the underlying concepts, two nonlinear models are examined in detail: uncertain volatility and passport options. For any nonlinear model, implicit timestepping techniques lead to a set of discrete nonlinear equations which must be solved at each timestep. Several iterative methods for solving these equations are tested. In the cases of uncertain volatility and passport options, it is shown that the frozen coefficient method outperforms two different Newton-type methods. Further, it is proven that the frozen coefficient method is guaranteed to converge for a wide class of one factor problems. A major issue when solving nonlinear PDEs is the possibility of multiple solutions. In a financial context, convergence to the viscosity solution is desired. Conditions under which the one factor uncertain volatility equations are guaranteed to converge to the viscosity solution are derived. Unfortunately, the techniques used do not apply to passport options, primarily because a positive coefficient discretization is shown to not always be achievable. For both uncertain volatility and passport options, much work has already been done for one factor problems. In this thesis, extensions are made for two factor problems. The importance of treating derivative estimates consistently between the discretization and an optimization procedure is discussed. For option pricing problems in general, non-smooth data can cause convergence difficulties for classical timestepping techniques. In particular, quadratic convergence may not be achieved. Techniques for restoring quadratic convergence for linear problems are examined. Via numerical examples, these techniques are also shown to improve the stability of the nonlinear uncertain volatility and passport option problems. Finally, two applications are briefly explored. The first application involves static hedging to reduce the bid-ask spread implied by uncertain volatility pricing. While static hedging has been carried out previously for one factor models, examples for two factor models are provided. The second application uses passport option theory to examine trader compensation strategies. By changing the payoff, it is shown how the expected distribution of trading account balances can be modified to reflect trader or bank preferences

Unified analysis of discontinuous Galerkin and C0-interior penalty finite element methods for Hamilton-Jacobi-Bellman and Isaacs equations

We provide a unified analysis of a posteriori and a priori error bounds for a broad class of discontinuous Galerkin and C0-IP finite element approximations of fully nonlinear second-order elliptic Hamilton–Jacobi–Bellman and Isaacs equations with Cordes coefficients. We prove the existence and uniqueness of strong solutions in H2 of Isaacs equations with Cordes coefficients posed on bounded convex domains. We then show the reliability and efficiency of computable residual-based error estimators for piecewise polynomial approximations on simplicial meshes in two and three space dimensions. We introduce an abstract framework for the a priori error analysis of a broad family of numerical methods and prove the quasi-optimality of discrete approximations under three key conditions of Lipschitz continuity, discrete consistency and strong monotonicity of the numerical method. Under these conditions, we also prove convergence of the numerical approximations in the small-mesh limit for minimal regularity solutions. We then show that the framework applies to a range of existing numerical methods from the literature, as well as some original variants. A key ingredient of our results is an original analysis of the stabilization terms. As a corollary, we also obtain a generalization of the discrete Miranda–Talenti inequality to piecewise polynomial vector fields

A generalized complementarity approach to solving real option problems

This article provides a unified framework for analyzing a wide variety of real option problems. These problems include the frequently studied, simple real option problems, as described in More specifically, all of the present real option problems can be directly solved by reformulating their optimality condition as a dynamical system of generalized linear complementarity problems (GLCPs). This enables us to develop an efficient and robust algorithm for solving a broad range of real option problems in a unified manner, exploiting recent advances in the theory of complementarity problems. JEL classification: C61; C63; D92; E22; G3