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- ItemSemitractability of optimal stopping problems via a weighted stochastic mesh algorithm(Oxford [u.a.] : Wiley-Blackwell, 2020) Belomestny, Denis; Kaledin, Maxim; Schoenmakers, JohnIn this paper, we propose a Weighted Stochastic Mesh (WSM) algorithm for approximating the value of discrete- and continuous-time optimal stopping problems. In this context, we consider tractability of such problems via a useful notion of semitractability and the introduction of a tractability index for a particular numerical solution algorithm. It is shown that in the discrete-time case the WSM algorithm leads to semitractability of the corresponding optimal stopping problem in the sense that its complexity is bounded in order by (Formula presented.) with (Formula presented.) being the dimension of the underlying Markov chain. Furthermore, we study the WSM approach in the context of continuous-time optimal stopping problems and derive the corresponding complexity bounds. Although we cannot prove semitractability in this case, our bounds turn out to be the tightest ones among the complexity bounds known in the literature. We illustrate our theoretical findings by a numerical example. © 2020 Wiley Periodicals LLC
- ItemGeneralized Post-Widder inversion formula with application to statistics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Belomestny, Denis; Mai, Hilmar; Schoenmakers, JohnIn this work we derive an inversion formula for the Laplace transform of a density observed on a curve in the complex domain, which generalizes the well known Post-Widder formula. We establish convergence of our inversion method and derive the corresponding convergence rates for the case of a Laplace transform of a smooth density. As an application we consider the problem of statistical inference for variance-mean mixture models.We construct a nonparametric estimator for the mixing density based on the generalized Post-Widder formula, derive bounds for its root mean square error and give a brief numerical example.
- ItemTwo-particle models for the estimation of the mean and standard deviation of concentrations in coastal waters(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Spivakovskaya, Daria; Heemink, Arnold; Schoenmakers, JohnIn this paper we study the mean and standard deviation of concentrations using random walk models. Two-particle models that takes into account the space correlation of the turbulence are introduced and some properties of the distribution of the particle concentration are studied. In order to reduce the CPU time of the calculation a new estimator based on reverse time diffusion is applied. This estimator has been introduced recently by Milstein, Schoenmakers, and Spokoiny (2004). Some numerical aspects of the implementation are discussed for relative simple test problems and finally a realistic application to predict the spreading of the pollutant in the Dutch coastal zone is described.
- ItemSensitivities for Bermudan options by regression methods(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Belomestny, Denis; Milstein, Grigori N.; Schoenmakers, JohnIn this article we propose several pathwise and finite difference based methods for calculating sensitivities of Bermudan options using regression methods and Monte Carlo simulation. These methods rely on conditional probabilistic representations which allows, in combination with a regression approach, an efficient simultaneous computation of sensitivities at all initial positions. Assuming that the price of a Bermudan option can be evaluated sufficiently accurate, we develop a method for constructing deltas based on least squares. We finally propose a testing procedure for assessing the performance of the developed methods.
- ItemEffiziente Methoden zur Bestimmung von Risikomaßen : Schlussbericht ; Projekt des BMBF-Förderprogramm "Neue Mathematische Verfahren in Industrie und Dienstleistungen"(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2004) Schoenmakers, John; Spokoiny, Vladimir; Reiß, Oliver; Zacherias-Langhans, Johan-Hinrich[no abstract available]
- ItemSolving linear parabolic rough partial differential equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Bayer, Christian; Belomestny, Denis; Redmann, Martin; Riedel, Sebastian; Schoenmakers, JohnWe study linear rough partial differential equations in the setting of [Friz and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path W of Hölder regularity with 1=3 < 1=2. Based on a stochastic representation of the solution of the rough partial differential equation, we propose a regression Monte Carlo algorithm for spatio-temporal approximation of the solution. We provide a full convergence analysis of the proposed approximation method which essentially relies on the new bounds for the higher order derivatives of the solution in space. Finally, a comprehensive simulation study showing the applicability of the proposed algorithm is presented.
- ItemRepresentations for optimal stopping under dynamic monetary utility functionals(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Krätschmer, Volker; Schoenmakers, JohnIn this paper we consider the optimal stopping problem for general dynamic monetary utility functionals. Sufficient conditions for the Bellman principle and the existence of optimal stopping times are provided. Particular attention is payed to representations which allow for a numerical treatment in real situations. To this aim, generalizations of standard evaluation methods like policy iteration, dual and consumption based approaches are developed in the context of general dynamic monetary utility functionals. As a result, it turns out that the possibility of a particular generalization depends on specific properties of the utility functional under consideration.
- ItemA fully adaptive interpolated stochastic sampling method for random PDEs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Anker, Felix; Bayer, Christian; Eigel, Martin; Neumann, Johannes; Schoenmakers, JohnA numerical method for the fully adaptive sampling and interpolation of PDE with random data is presented. It is based on the idea that the solution of the PDE with stochastic data can be represented as conditional expectation of a functional of a corresponding stochastic differential equation (SDE). The physical domain is decomposed subject to a non-uniform grid and a classical Euler scheme is employed to approximately solve the SDE at grid vertices. Interpolation with a conforming finite element basis is employed to reconstruct a global solution of the problem. An a posteriori error estimator is introduced which provides a measure of the different error contributions. This facilitates the formulation of an adaptive algorithm to control the overall error by either reducing the stochastic error by locally evaluating more samples, or the approximation error by locally refining the underlying mesh. Numerical examples illustrate the performance of the presented novel method.
- ItemPricing CMS spreads in the Libor market model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Belomestny, Denis; Kolodko, Anastasia; Schoenmakers, JohnWe present two approximation methods for pricing of CMS spread options in Libor market models. Both approaches are based on approximating the underlying swap rates with lognormal processes under suitable measures. The first method is derived straightforwardly from the Libor market model. The second one uses a convexity adjustment technique under a linear swap model assumption. A numerical study demonstrates that both methods provide satisfactory approximations of spread option prices and can be used for calibration of a Libor market model to the CMS spread option market.
- ItemRegression based duality approach to optimal control with application to hydro electricity storage(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Hildebrand, Roland; Schoenmakers, John; Zhang, Jianing; Dickmann, FabianIn this paper we consider the problem of optimal control of stochastic processes. We employ the dual martingale method brought forward in [Brown, Smith, and Sun, 2010]. The martingale constituting the solution of the dual problem is determined by linear regression within a Monte-Carlo approach. We apply the solution algorithm to a model of a hydro electricity storage and production system coupled with a model of the electricity wholesale market.