Stratified Regression Monte-Carlo Scheme for Semilinear PDEs and BSDEs with Large Scale Parallelization on GPUs

Gobet, Emmanuel; López-Salas, José G.; Turkedjiev, Plamen; Vázquez, Carlos

doi:10.1137/16m106371x

Cited by 53 publications

(45 citation statements)

References 19 publications

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“…, sigma , beta , p , a , t0 , x0 , T , psi )46 BP = BranchingProcess ( mu , sigma , beta , p , t0 , x0 , T ); * ( a ( k )/ p ( k ) )^( BP {2}( k ) );55 elseif a ( k )~= 0 56 error ( 'a ( k ) zero but p ( k ) non -zero ' ); , sigma , beta , p , t0 , x0 , T ) 64 BP = cell (2 ,1); 65 BP {2} = p *0; 66 tau = exprnd (1/ beta ); 67 new_t0 = min ( tau + t0 , T ); 68 delta_t = new_t0 -t0 ; 69 m = size ( sigma ,2); 70 new_x0 = x0 + mu * delta_t + sigma * sqrt ( delta_t )* randn (m ,1); tmp , wh ic h_ no nl in ea ri ty ] = max ( mnrnd (1 , p )); 75 BP {2}( w hi ch _n on li ne ar it y ) = BP {2}( whi ch _n on li ne ar it y ) + 1; 76 for k =1: which_nonlinearity -1 77 tmp = BranchingProcess (... 78 mu , sigma , beta , p , new_t0 , new_x0 , T ); 79 BP {1} = [ BP {1} tmp {1} ]; 80 BP {2} = BP {2} + tmp {2}; 81 end Matlab code 2: A Matlab code for the Branching diffusion method used in Subsection 4.5. Python code for the deep 2BSDE method used in Subsection 4The following Python code is based on the Python code in E, Han, & Jentzen [33, Subsection 6.1].…”

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confidence: 99%

Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations

2019

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High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (i) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (ii) a merged formulation of the PDE and the 2BSDE problem, (iii) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (iv) a stochastic gradient descent-type optimization procedure. Numerical results obtained using TensorFlow in Python illustrate the efficiency and the accuracy of the method in the cases of a 100-dimensional Black-Scholes-Barenblatt equation, 1 arXiv:1709.05963v1 [math.NA] 18 Sep 2017 a 100-dimensional Hamilton-Jacobi-Bellman equation, and a nonlinear expectation of a 100-dimensional G-Brownian motion.

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confidence: 99%

Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations

2019

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“…As pointed out in the introduction, many (time) discretization schemes and several (spacial) numerical approximation method of the solution of such as BSDE are proposed in the literature (we refer for example to [1,3,7,11,14,10,2,20]). Our aim in this section is to test the performances of our method to the numerical scheme proposed in [20].…”

Section: Approximation Of Bsdementioning

confidence: 99%

Product Markovian Quantization of a Diffusion Process with Applications to Finance

Fiorin

Pagès²,

Sagna

2018

Methodol Comput Appl Probab

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We introduce a new approach to quantize the Euler scheme of an R d -valued diffusion process. This method is based on a Markovian and componentwise product quantization and allows us, from a numerical point of view, to speak of fast online quantization in dimension greater than one since the product quantization of the Euler scheme of the diffusion process and its companion weights and transition probabilities may be computed quite instantaneously. We show that the resulting quantization process is a Markov chain, then, we compute the associated companion weights and transition probabilities from (semi-) closed formulas. From the analytical point of view, we show that the induced quantization errors at the k-th discretization step t k is a cumulative of the marginal quantization error up to time t k . Numerical experiments are performed for the pricing of a Basket call option, for the pricing of a European call option in a Heston model and for the approximation of the solution of backward stochastic differential equations to show the performances of the method.

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“…This is a considerable reduction of computational time, especially as the dimension q grows and correspondingly the dimension of the basis for approximating z i grows. Highlighting the computational time may be not the most important issues when solving BSDEs: in some situations (memory constraint in large dimension [19] or in parallel computing [17], calibration on small data [16]), the number of paths to be used is imposed and the objectives become to obtain the most efficient algorithm with a given Monte Carlo simulation effort. In that case, the use of ISMWDP type scheme for reducing statistical error is certainly advantageous.…”

Section: Concluding Remarks On the Numerical Experimentsmentioning

confidence: 99%

Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations

Gobet

Turkedjiev²

2017

Stochastic Processes and their Applications

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Please cite this article as: E. Gobet, P. Turkedjiev, Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations, Stochastic Processes and their Applications (2016), http://dx.doi.org/10. 1016/j.spa.2016.07.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractWe design an importance sampling scheme for backward stochastic differential equations (BSDEs) that minimizes the conditional variance occurring in least-squares MonteCarlo (LSMC) algorithms. The Radon-Nikodym derivative depends on the solution of BSDE, and therefore it is computed adaptively within the LSMC procedure. To allow robust error estimates w.r.t. the unknown change of measure, we properly randomize the initial value of the forward process. We introduce novel methods to analyze the error: firstly, we establish norm stability results due to the random initialization; secondly, we develop refined concentration-of-measure techniques to capture the variance reduction. Our theoretical results are supported by numerical experiments.

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Stratified Regression Monte-Carlo Scheme for Semilinear PDEs and BSDEs with Large Scale Parallelization on GPUs

Cited by 53 publications

References 19 publications

Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations

Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations

Product Markovian Quantization of a Diffusion Process with Applications to Finance

Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations

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