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53
Optimal switching with application to energy tolling agreements
, 2005
"... We consider the problem of optimal switching with finite horizon. This special case of stochastic impulse control naturally arises during analysis of operational flexibility of exotic energy derivatives. The current practice for such problems relies on Markov decision processes that have poor dimens ..."
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Cited by 24 (4 self)
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We consider the problem of optimal switching with finite horizon. This special case of stochastic impulse control naturally arises during analysis of operational flexibility of exotic energy derivatives. The current practice for such problems relies on Markov decision processes that have poor dimensionscaling properties, or on strips of spark spread options that ignore the operational constraints of the asset. To overcome both of these limitations, we propose a new framework based on recursive optimal stopping. Our model demonstrates that the optimal dispatch policies can be described with the aid of ‘switching boundaries’, similar to standard American options. In turn, this provides new insight regarding the qualitative properties of the value function. Our main contribution is a new method of numerical solution based on Monte Carlo regressions. The scheme uses dynamic programming to simultaneously approximate the optimal switching times along all the simulated paths. Convergence analysis is carried out and numerical results are illustrated with a variety of concrete
Pricing asset scheduling flexibility using optimal switching
 Applied Mathematical Finance
, 2008
"... www.umich.edu / mludkov We study the financial engineering aspects of operational flexibility of energy assets. The current practice relies on a representation that uses strips of European sparkspread options, ignoring the operational constraints. Instead, we propose a new approach based on a stoch ..."
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Cited by 22 (5 self)
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www.umich.edu / mludkov We study the financial engineering aspects of operational flexibility of energy assets. The current practice relies on a representation that uses strips of European sparkspread options, ignoring the operational constraints. Instead, we propose a new approach based on a stochastic impulse control framework. The model reduces to a cascade of optimal stopping problems and directly demonstrates that the optimal dispatch policies can be described with the aid of ‘switching boundaries’, similar to the free boundaries of standard American options. Our main contribution is a new method of numerical solution relying on Monte Carlo regressions. The scheme uses dynamic programming to efficiently approximate the optimal dispatch policy along the simulated paths. Convergence analysis is carried out and results are illustrated with a variety of concrete examples. We benchmark and compare our scheme to alternative numerical methods.
Decomposition of multistage stochastic programs with recombining scenario trees
, 2007
"... This paper presents a decomposition approach for linear multistage stochastic programs, that is based on the concept of recombining scenario trees. The latter, widely applied in Mathematical Finance, may prevent the node number of the scenario tree to grow exponentially with the number of time stage ..."
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Cited by 14 (4 self)
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This paper presents a decomposition approach for linear multistage stochastic programs, that is based on the concept of recombining scenario trees. The latter, widely applied in Mathematical Finance, may prevent the node number of the scenario tree to grow exponentially with the number of time stages. It is shown how this property may be exploited within a nonMarkovian framework and under timecoupling constraints. Being close to the wellestablished Nested Benders Decomposition, our approach uses the special structure of recombining trees for simultaneous cutting plane approximations. Convergence is proved and stopping criteria are deduced. Techniques for the generation of suitable scenario trees and some numerical examples are presented.
A forwardbackward stochastic algorithm for quasilinear PDEs
 Ann. Appl. Probab
, 2006
"... Abstract. In this paper, we improve the forwardbackward algorithm for quasilinear PDEs introduced in Delarue and Menozzi (2006). The new discretization scheme takes advantage of the standing regularity properties of the true solution through an interpolation procedure. For the convergence analysis ..."
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Cited by 10 (0 self)
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Abstract. In this paper, we improve the forwardbackward algorithm for quasilinear PDEs introduced in Delarue and Menozzi (2006). The new discretization scheme takes advantage of the standing regularity properties of the true solution through an interpolation procedure. For the convergence analysis, we also exploit the optimality of the square Gaussian quantization used to approximate the conditional expectations involved. The resulting bound for the error is closely related to the Hölder exponent of the second order spatial derivatives of the true solution and turns out to be more satisfactory than the one previously established. 1.
Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions
 Available on http://hal.archivesouvertes.fr/hal00642685, 2013. EJP
, 2015
"... Abstract. We design a numerical scheme for solving the Multi stepforward Dynamic Programming (MDP) equation arising from the timediscretization of backward stochastic differential equations. The generator is assumed to be locally Lipschitz, which includes some cases of quadratic drivers. When the ..."
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Cited by 9 (5 self)
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Abstract. We design a numerical scheme for solving the Multi stepforward Dynamic Programming (MDP) equation arising from the timediscretization of backward stochastic differential equations. The generator is assumed to be locally Lipschitz, which includes some cases of quadratic drivers. When the large sequence of conditional expectations is computed using empirical leastsquares regressions, under general conditions we establish an upper bound error as the average, rather than the sum, of local regression errors only, suggesting that our error estimation is tight. Despite the nested regression problems, the interdependency errors are justified to be at most of the order of the statistical regression errors (up to logarithmic factor). Finally, we optimize the algorithm parameters, depending on the dimension and on the smoothness of value functions, in the limit as the time mesh size goes to zero and compute the complexity needed to achieve a given accuracy.
ON NUMERICAL APPROXIMATIONS OF FORWARDBACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS
, 2008
"... A numerical method for a class of forwardbackward stochastic differential equations (FBSDEs) is proposed and analyzed. The method is designed around the four step scheme [J. Douglas, Jr., J. Ma, and P. Protter, Ann. Appl. Probab., 6 (1996), pp. 940–968] but with a Hermitespectral method to approxi ..."
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Cited by 8 (1 self)
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A numerical method for a class of forwardbackward stochastic differential equations (FBSDEs) is proposed and analyzed. The method is designed around the four step scheme [J. Douglas, Jr., J. Ma, and P. Protter, Ann. Appl. Probab., 6 (1996), pp. 940–968] but with a Hermitespectral method to approximate the solution to the decoupling quasilinear PDE on the whole space. A rigorous synthetic error analysis is carried out for a fully discretized scheme, namely a firstorder scheme in time and a Hermitespectral scheme in space, of the FBSDEs. Equally important, a systematical numerical comparison is made between several schemes for the resulting decoupled forward SDE, including a stochastic version of the Adams–Bashforth scheme. It is shown that the stochastic version of the Adams–Bashforth scheme coupled with the Hermitespectral method leads to a convergence rate of 3 (in time) which is better than those in previously published 2 work for the FBSDEs.
A variance reduction technique using a quantized Brownian motion as a control variate
 J. Comput. Finance
"... Abstract. This article presents a new variance reduction technique for diffusion processes where a control variate is constructed using a quantization of the coefficients of the KarhunenLoève decomposition of the underlying Brownian motion. This method may be indeed used for other Gaussian proces ..."
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Cited by 7 (0 self)
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Abstract. This article presents a new variance reduction technique for diffusion processes where a control variate is constructed using a quantization of the coefficients of the KarhunenLoève decomposition of the underlying Brownian motion. This method may be indeed used for other Gaussian processes. 1.
Optimal stopping for partially observed piecewisedeterministic Markov processes. Stochastic Process
 Appl
"... Benôıte de Saporta François Dufour This paper deals with the optimal stopping problem under partial observation for piecewisedeterministic Markov processes. We first obtain a recursive formulation of the optimal filter process and derive the dynamic programming equation of the partially observed ..."
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Cited by 6 (1 self)
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Benôıte de Saporta François Dufour This paper deals with the optimal stopping problem under partial observation for piecewisedeterministic Markov processes. We first obtain a recursive formulation of the optimal filter process and derive the dynamic programming equation of the partially observed optimal stopping problem. Then, we propose a numerical method, based on the quantization of the discretetime filter process and the interjump times, to approximate the value function and to compute an actual optimal stopping time. We prove the convergence of the algorithms and bound the rates of convergence.
The steepest descent method for forwardbackward SDEs
 Electron. J. Probab
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