A High-Order Numerical Method for BSPDEs with Applications to Mathematical Finance
Author: 
Yunzhang LiAuthors Info & Claims
Yunzhang LiAuthors Info & ClaimsPages 147 - 178
Abstract
In this paper, we propose a local discontinuous Galerkin (LDG) method for backward stochastic partial differential equations (BSPDEs), which is a high-order numerical scheme. We prove the $L^2$-stability of the numerical scheme. For the superparabolic BSPDEs, the optimal error estimates are obtained for Cartesian meshes with $Q^k$ elements, and the suboptimal error estimates are derived for triangular meshes with $P^k$ elements. We also prove the suboptimal error estimates for the degenerate BSPDEs. Numerical examples in one and two space dimensions are given to display the performance of the LDG method. As an application in mathematical finance, the numerical scheme is applied to approximate the hedging price of a contingent claim and the corresponding optimal hedging strategy.
References
[1]
V. Barbu, A. Răşcanu, and G. Tessitore, Carleman estimates and controllability of linear stochastic heat equations, Appl. Math. Optim., 47 (2003), pp. 97--120.
[2]
F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131 (1997), pp. 267--279.
[3]
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics Appl. Math. 40, SIAM, Philadelphia, 2002, https://doi.org/10.1137/1.9780898719208. [Unabridged republication of the work first published by North-Holland, Amsterdam, New York, Oxford, 1978.]
[4]
B. Cockburn, S. Hou, and C.-W. Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case, Math. Comp., 54 (1990), pp. 545--581.
[5]
B. Cockburn, G. Kanschat, I. Perugia, and D. Schötzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal., 39 (2001), pp. 264--285, https://doi.org/10.1137/S0036142900371544.
[6]
B. Cockburn, S. Y. Lin, and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems, J. Comput. Phys., 84 (1989), pp. 90--113.
[7]
B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework, Math. Comp., 52 (1989), pp. 411--435.
[8]
B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), pp. 2440--2463, https://doi.org/10.1137/S0036142997316712.
[9]
B. Cockburn and C.-W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems, J. Comput. Phys., 141 (1998), pp. 199--224.
[10]
K. Du and Q. Meng, A revisit to $W^n_2$-theory of super-parabolic backward stochastic partial differential equations in $\Bbb R^d$, Stochastic Process. Appl., 120 (2010), pp. 1996--2015.
[11]
K. Du, S. Tang, and Q. Zhang, $W^{m,p}$-solution $(p\ge 2)$ of linear degenerate backward stochastic partial differential equations in the whole space, J. Differential Equations, 254 (2013), pp. 2877--2904.
[12]
N. El Karoui, S. Peng, and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), pp. 1--71.
[13]
N. Englezos and I. Karatzas, Utility maximization with habit formation: Dynamic programming and stochastic PDEs, SIAM J. Control Optim., 48 (2009), pp. 481--520, https://doi.org/10.1137/070686998.
[14]
S. W. He, J. G. Wang, and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Kexue Chubanshe (Science Press), Beijing; CRC Press, Boca Raton, FL, 1992.
[15]
Y. Hu, J. Ma, and J. Yong, On semi-linear degenerate backward stochastic partial differential equations, Probab. Theory Related Fields, 123 (2002), pp. 381--411.
[16]
P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, in Mathematical Aspects of Finite Elements in Partial Differential Equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), Publication No. 33, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89--123.
[17]
Y. Li, C.-W. Shu, and S. Tang, A discontinuous Galerkin method for stochastic conservation laws, SIAM J. Sci. Comput., 42 (2020), pp. A54--A86, https://doi.org/10.1137/19M125710X.
[18]
Y. Li, C.-W. Shu, and S. Tang, An ultra-weak discontinuous Galerkin method with implicit-explicit time-marching for generalized stochastic KdV equations, J. Sci. Comput., 82 (2020), 61.
[19]
Y. Li, C.-W. Shu, and S. Tang, A local discontinuous Galerkin method for nonlinear parabolic SPDEs, ESAIM Math. Model. Numer. Anal., 55 (2021), pp. S187--S223.
[20]
Y. Li and S. Tang, Approximation of backward stochastic partial differential equations by a splitting-up method, J. Math. Anal. Appl., 493 (2021), 124518.
[21]
J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications, Stochastic Process. Appl., 70 (1997), pp. 59--84.
[22]
P. E. Protter, Stochastic Integration and Differential Equations, 2nd ed., Appl. Math. (N.Y.) 21 [continued as Stoch. Model. Appl. Probab.], Springer-Verlag, Berlin, 2004.
[23]
S. Tang, The maximum principle for partially observed optimal control of stochastic differential equations, SIAM J. Control Optim., 36 (1998), pp. 1596--1617, https://doi.org/10.1137/S0363012996313100.
[24]
S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations, SIAM J. Control Optim., 48 (2009), pp. 2191--2216, https://doi.org/10.1137/050641508.
[25]
J. Yan and C.-W. Shu, A local discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal., 40 (2002), pp. 769--791, https://doi.org/10.1137/S0036142901390378.
[26]
X. Yang and W. Zhao, Finite element methods for nonlinear backward stochastic partial differential equations and their error estimates, Adv. Appl. Math. Mech., 12 (2020), pp. 1457--1480.
[27]
X. Y. Zhou, A duality analysis on stochastic partial differential equations, J. Funct. Anal., 103 (1992), pp. 275--293.
[28]
X. Y. Zhou, On the necessary conditions of optimal controls for stochastic partial differential equations, SIAM J. Control Optim., 31 (1993), pp. 1462--1478, https://doi.org/10.1137/0331068.
Recommendations
A Discontinuous Galerkin Method for Stochastic Conservation Laws
In this paper we present a discontinuous Galerkin (DG) method to approximate stochastic conservation laws, which is an efficient high-order scheme. We study the stability for the semidiscrete DG methods for fully nonlinear stochastic equations. Error ...
An Ultra-Weak Discontinuous Galerkin Method with Implicit–Explicit Time-Marching for Generalized Stochastic KdV Equations
AbstractIn this paper, an ultra-weak discontinuous Galerkin (DG) method is developed to solve the generalized stochastic Korteweg–de Vries (KdV) equations driven by a multiplicative temporal noise. This method is an extension of the DG method for purely ...
A high-order discontinuous Galerkin method for Itô stochastic ordinary differential equations
In this paper, we develop a high-order discontinuous Galerkin (DG) method for strong solution of Itô stochastic ordinary differential equations (SDEs) driven by one-dimensional Wiener processes. Motivated by the DG method for deterministic ordinary ...
Comments
Information & Contributors
Information
Published In
© 2022, Society for Industrial and Applied Mathematics.
Publisher
Society for Industrial and Applied Mathematics
United States
Publication History
Published: 01 January 2022
Author Tags
Author Tags
Qualifiers
- Research-article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 13 Jan 2025