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- ItemOptimal dual martingales and their stability; fast evaluation of Bermudan products via dual backward regression(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Schoenmakers, John G.M.; Huang, JunboLiteraturverz. In this paper we introduce and study the concept of optimal and surely optimal dual martingales in the context of dual valuation of Bermudan options. We provide a theorem which give conditions for a martingale to be surely optimal, and a stability theorem concerning martingales which are near to be surely optimal in a sense. Guided by these theorems we develop a regression based backward construction of such a martingale in a Wiener environment. In turn this martingale may be utilized for computing upper bounds by non-nested Monte Carlo. As a by-product, the algorithm also provides approximations to continuation values of the product, which in turn determine a stopping policy. Hence, we obtain lower bounds at the same time. The proposed algorithm is pure dual in the sense that it doesn't require an (input) approximation to the Snell envelope, is quite easy to implement, and in a numerical study we show that, regarding the computed upper bounds, it is comparable with the method of Belomestny, et. al. (2009).
- ItemTime discretization and Markovian iteration for coupled FBSDEs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Bender, Christian; Zhang, JianfengIn this paper we lay the foundation for a numerical algorithm to simulate high-dimensional coupled FBSDEs under weak coupling or monotonicity conditions. In particular we prove convergence of a time discretization and a Markovian iteration. The iteration differs from standard Picard iterations for FBSDEs in that the dimension of the underlying Markovian process does not increase with the number of iterations. This feature seems to be indispensable for an efficient iterative scheme from a numerical point of view. We finally suggest a fully explicit numerical algorithm and present some numerical examples with up to 10-dimensional state space.
- ItemForward-reverse EM algorithm for Markov chains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bayer, Christian; Mai, Hilmar; Schoenmakers, John G.M.We develop an EM algorithm for estimating parameters that determine the dynamics of a discrete time Markov chain evolving through a certain measurable state space. As a key tool for the construction of the EM method we develop forward-reverse representations for Markov chains conditioned on a certain terminal state. These representations may be considered as an extension of the earlier work [1] on conditional diffusions. We present several experiments and consider the convergence of the new EM algorithm.
- ItemForward and reverse representations for Markov chains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Milstein, Grigori N.; Schoenmakers, John G.M.; Spokoiny, VladimirIn this paper we carry over the concept of reverse probabilistic representations developed in Milstein, Schoenmakers, Spokoiny (2004) for diffusion processes, to discrete time Markov chains. We outline the construction of reverse chains in several situations and apply this to processes which are connected with jump-diffusion models and finite state Markov chains. By combining forward and reverse representations we then construct transition density estimators for chains which have root-N accuracy in any dimension and consider some applications.
- ItemPrimal-dual linear Monte Carlo algorithm for multiple stopping : an application to flexible caps(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Balder, Sven; Mahayni, Antje; Schoenmakers, John G.M.In this paper we consider the valuation of Bermudan callable derivatives with multiple exercise rights. We present in this context a new primal-dual linear Monte Carlo algorithm that allows for efficient simulation of lower and upper price bounds without using nested simulations (hence the terminology). The algorithm is essentially an extension of a primal-dual Monte Carlo algorithm for standard Bermudan options proposed in Schoenmakers et al (2011), to the case of multiple exercise rights. In particular, the algorithm constructs upwardly a system of dual martingales to be plugged into the dual representation of Schoenmakers (2010). At each level the respective martingale is constructed via a backward regression procedure starting at the last exercise date. The thus constructed martingales are finally used to compute an upper price bound. At the same time, the algorithm also provides approximate continuation functions which may be used to construct a price lower bound. The algorithm is applied to the pricing of flexible caps in a Hull White (1990) model setup. The simple model choice allows for comparison of the computed price bounds with the exact price which is obtained by means of a trinomial tree implementation. As a result, we obtain tight price bounds for the considered application. Moreover, the algorithm is generically designed for multi-dimensional problems and is tractable to implement.
- ItemSimulation of conditional diffusions via forward-reverse stochastic representations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bayer, Christian; Schoenmakers, John G.M.In this paper we derive stochastic representations for the finite dimensional distributions of a multidimensional diffusion on a fixed time interval, conditioned on the terminal state. The conditioning can be with respect to a fixed point or more generally with respect to some subset. The representations rely on a reverse process connected with the given (forward) diffusion as introduced in Milstein et al. [Bernoulli 10(2):281312, 2004] in the context of a forward-reverse transition density estimator. The corresponding Monte Carlo estimators have essentially root-N accuracy, hence they do not suffer from the curse of dimensionality. We provide a detailed convergence analysis and give a numerical example involving the realized variance in a stochastic volatility asset model conditioned on a fixed terminal value of the asset.
- ItemA variance-reduced electrothermal Monte Carlo method for semiconductor device simulation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Muscato, Orazio; Di Stefano, Vincenza; Wagner, WolfgangThis paper is concerned with electron transport and heat generation in semiconductor devices. An improved version of the electrothermal Monte Carlo method is presented. This modification has better approximation properties due to reduced statistical fluctuations. The corresponding transport equations are provided and results of numerical experiments are presented.
- ItemRobust optimal stopping(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Krätschmer, Volker; Ladkau, Marcel; Laeven, Roger J.A.; Schoenmakers, John G.M.; Stadje, MitjaThis paper studies the optimal stopping problem in the presence of model uncertainty (ambiguity). We develop a method to practically solve this problem in a general setting, allowing for general time-consistent ambiguity averse preferences and general payoff processes driven by jump-diffusions. Our method consists of three steps. First, we construct a suitable Doob martingale associated with the solution to the optimal stopping problem using backward stochastic calculus. Second, we employ this martingale to construct an approximated upper bound to the solution using duality. Third, we introduce backward-forward simulation to obtain a genuine upper bound to the solution, which converges to the true solution asymptotically. We analyze the asymptotic behavior and convergence properties of our method. We illustrate the generality and applicability of our method and the potentially significant impact of ambiguity to optimal stopping in a few examples.