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Policy Gradient in Continuous Time

by Remi Munos, Michael Littman - Journal of Machine Learning Research , 2006
"... Policy search is a method for approximately solving an optimal control problem by performing a parametric optimization search in a given class of parameterized policies. In order to process a local optimization technique, such as a gradient method, we wish to evaluate the sensitivity of the perfo ..."
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Policy search is a method for approximately solving an optimal control problem by performing a parametric optimization search in a given class of parameterized policies. In order to process a local optimization technique, such as a gradient method, we wish to evaluate the sensitivity of the performance measure with respect to the policy parameters, the so-called policy gradient. This paper is concerned with the estimation of the policy gradient for continuous-time, deterministic state dynamics, in a reinforcement learning framework, that is, when the decision maker does not have a model of the state dynamics.

A duality approach for the weak approximation of stochastic differential equations, in "Annals of Applied Probability", vol. 16, n o 3

by Emmanuelle Clément, Arturo Kohatsu-higa, Damien Lamberton , 2006
"... In this article we develop a new methodology to prove weak approximation results for general stochastic differential equations. Instead of using a partial differential equation approach as is usually done for diffusions, the approach considered here uses the properties of the linear equation satisfi ..."
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In this article we develop a new methodology to prove weak approximation results for general stochastic differential equations. Instead of using a partial differential equation approach as is usually done for diffusions, the approach considered here uses the properties of the linear equation satisfied by the error process. This methodology seems to apply to a large class of processes and we present as an example the weak approximation of stochastic delay equations. 1. Introduction. The Euler
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ABSOLUTELY CONTINUOUS LAWS OF JUMP-DIFFUSIONS IN FINITE AND INFINITE DIMENSIONS WITH APPLICATIONS TO MATHEMATICAL FINANCE

by Barbara Forster, Eva Lütkebohmert, Josef Teichmann , 2007
"... Abstract. In mathematical Finance calculating the Greeks by Malliavin weights has proved to be a numerically satisfactory procedure for finite-dimensional Itô-diffusions. The existence of Malliavin weights relies on absolute continuity of laws of the projected diffusion process and a sufficiently re ..."
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Abstract. In mathematical Finance calculating the Greeks by Malliavin weights has proved to be a numerically satisfactory procedure for finite-dimensional Itô-diffusions. The existence of Malliavin weights relies on absolute continuity of laws of the projected diffusion process and a sufficiently regular density. In this article we first prove results on absolute continuity for laws of projected jump-diffusion processes in finite and infinite dimensions, and a general result on the existence of Malliavin weights in finite dimension. In both cases we assume Hörmander conditions and hypotheses on the invertibility of the so-called linkage operators. The message is that for the construction of numerical procedures for the calculation of the Greeks in fairly general jumpdiffusion cases one can proceed as in a pure diffusion case. We also show how the given results apply to infinite dimensional questions in mathematical Finance. There we start with a Vasiček model, and add – by pertaining no arbitrage – a jump diffusion component. We prove that we can obtain in this case an interest rate model, where the law of any projection is absolutely continuous with respect to Lebesgue measure on R M. 1.
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Discretization and simulation of Zakai equation

by Emmanuel Gobet, Jacques Printems - SIAM Journal on Numerical Analysis , 2006
"... this revised version: June 2006 This paper is concerned with numerical approximations for stochastic partial dif-ferential Zakai equation of nonlinear filtering problem. The approximation scheme is based on the representation of the solutions as weighted conditional distributions. We first accuratel ..."
Abstract - Cited by 8 (1 self) - Add to MetaCart
this revised version: June 2006 This paper is concerned with numerical approximations for stochastic partial dif-ferential Zakai equation of nonlinear filtering problem. The approximation scheme is based on the representation of the solutions as weighted conditional distributions. We first accurately analyse the error caused by an Euler type scheme of time discretization. Sharp error bounds are calculated: we show that the rate of convergence is in general of order δ (δ is the time step), but in the case when there is no correlation between the signal and the observation for the Zakai equation, the order of convergence becomes δ. This result is obtained by carefully employing techniques of Malliavin calculus. In a second step, we propose a simulation of the time discretization Euler scheme by a quantization approach. This formally consists in an approximation of the weighted conditional distribution by a conditional discrete distribution on finite supports. We provide error bounds and rate of convergence in terms of the number N of the grids of this support. These errors are minimal at some optimal grids which are computed by a recursive method based on Monte Carlo simulations. Finally, we illustrate our results with some numerical experiments arising from correlated Kalman-Bucy filter. Key words: Stochastic partial differential equations, nonlinear filtering, Zakai equation,
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Multi-step Richardson-Romberg Extrapolation: Remarks on Variance Control and Complexity

by Gilles Pagès , 2007
"... We propose a multi-step Richardson-Romberg extrapolation method for the computation of expectations Ef(X T) of a diffusion (Xt) t∈[0,T] when the weak time discretization error induced by the Euler scheme admits an expansion at an order R ≥ 2. The complexity of the estimator grows as R 2 (instead of ..."
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We propose a multi-step Richardson-Romberg extrapolation method for the computation of expectations Ef(X T) of a diffusion (Xt) t∈[0,T] when the weak time discretization error induced by the Euler scheme admits an expansion at an order R ≥ 2. The complexity of the estimator grows as R 2 (instead of 2 R in the classical method) and its variance is asymptotically controlled by considering some consistent Brownian increments in the underlying Euler schemes. Some Monte Carlo simulations were carried with path-dependent options (lookback, barrier) which support the conjecture that their weak time discretization error also admits an expansion (in a different scale). Then an appropriate Richardson-Romberg extrapolation seems to outperform the Euler scheme with Brownian bridge.
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Discretization and simulation for a class of SPDE’s with applications to Zakai and McKean–Vlasov equations

by Emmanuel Gobet, Jacques Printems - Univ. Paris 6 (France , 2005
"... This paper is concerned with numerical approximations for a class of nonlinear stochastic partial differential equations: Zakai equation of nonlinear filtering problem and McKean-Vlasov type equations. The approximation scheme is based on the re-presentation of the solutions as weighted conditional ..."
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This paper is concerned with numerical approximations for a class of nonlinear stochastic partial differential equations: Zakai equation of nonlinear filtering problem and McKean-Vlasov type equations. The approximation scheme is based on the re-presentation of the solutions as weighted conditional distributions. We first accurately analyse the error caused by an Euler type scheme of time discretization. Sharp error bounds are calculated: we show that the rate of convergence is in general of order√ δ (δ is the time step), but in the case when there is no correlation between the signal and the observation for the Zakai equation, the order of convergence becomes δ. This result is obtained by carefully employing techniques of Malliavin calculus. In a second step, we propose a simulation of the time discretization Euler scheme by a quantization approach. This formally consists in an approximation of the weighted conditional distribution by a conditional discrete distribution on finite supports. We provide error bounds and rate of convergence in terms of the number N of the grids of this support. These errors are minimal at some optimal grids which are computed by a recursive method based on Monte Carlo simulations. Finally, we illustrate our results with some numerical experiments arising from correlated Kalman-Bucy filter and Burgers equation.

Malliavin Calculus in Finance

by Arturo Kohatsu-higa, Miquel Montero , 2003
"... This article is an introduction to Malliavin Calculus for practitioners. ..."
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This article is an introduction to Malliavin Calculus for practitioners.

Fractional smoothness and applications in Finance

by Stefan Geiss, Emmanuel Gobet , 2010
"... This overview article concerns the notion of fractional smoothness of random variables of the form g(XT), where X = (Xt)t∈[0,T] is a certain diffusion process. Wereviewtheconnectiontothereal interpolation theory,giveexamples and applications of this concept. The applications in stochastic finance ma ..."
Abstract - Cited by 5 (3 self) - Add to MetaCart
This overview article concerns the notion of fractional smoothness of random variables of the form g(XT), where X = (Xt)t∈[0,T] is a certain diffusion process. Wereviewtheconnectiontothereal interpolation theory,giveexamples and applications of this concept. The applications in stochastic finance mainly concern the analysis of discrete time hedging errors. We close the review by indicating some further developments.

CALCULATING THE GREEKS BY CUBATURE FORMULAS

by Josef Teichmann , 2004
"... We provide cubature formulas for the calculation of derivatives of expected values in the spririt of Terry Lyons and Nicolas Victoir. In financial mathematics derivatives of option prices with respect to initial values, so called Greeks, are of particular importance as hedging parameters. Cubature ..."
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We provide cubature formulas for the calculation of derivatives of expected values in the spririt of Terry Lyons and Nicolas Victoir. In financial mathematics derivatives of option prices with respect to initial values, so called Greeks, are of particular importance as hedging parameters. Cubature formulas allow to calculate these quantities very quickly. Simple examples are added to the theoretical exposition.

SEQUENTIAL MONTE CARLO METHODS FOR DIFFUSION PROCESSES

by Ajay Jasra, Arnaud Doucet
"... Abstract. In this paper we show how to use sequential Monte Carlo (SMC) methods ([7, 13]) to compute expectations of functionals of diffusions at a given time and the gradients of these quantities w.r.t the initial condition of the process. In some cases, using the exact simulation of diffusions met ..."
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Abstract. In this paper we show how to use sequential Monte Carlo (SMC) methods ([7, 13]) to compute expectations of functionals of diffusions at a given time and the gradients of these quantities w.r.t the initial condition of the process. In some cases, using the exact simulation of diffusions methodology ([3]), there is no time discretization error, otherwise the methods use Euler discretization. We illustrate our approach on both high and low dimensional problems from optimal control, and establish that our approach substantially out-performs standard Monte Carlo methods typically adopted in the literature. The methods developed here are appropriate for solving a certain class of partial differential equations (PDEs) as well as for option pricing and hedging.
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