The Mean Field Forward Backward Stochastic Differential Equations and Stochastic Partial Differential Equations

Jie Zhu; Hong Zhang; Li Zhou; Yuhang Feng

doi:doi:10.11648/j.pamj.20150403.20

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The Mean Field Forward Backward Stochastic Differential Equations and Stochastic Partial Differential Equations

Jie Zhu, Hong Zhang, Li Zhou, Yuhang Feng

Published in Pure and Applied Mathematics Journal (Volume 4, Issue 3)

Received: 14 May 2015 Accepted: 26 May 2015 Published: 6 June 2015

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Abstract

Since 1990 Pardoux and Peng, proposed the theory of backward stochastic differential equation Backward stochastic differential equation and is backward stochastic differential equations (short for FBSDE) theory has been widely research (see El Karoui, Peng and Cauenez, Ma and Yong, etc.) Generally, a backward stochastic differential equation is a type Ito stochastic differential equation and a coupling Pardoux - Peng and backward stochastic differential equation. Antonelli, Ma, Protter and Yong is backward stochastic differential equation for a series of research, and apply to the financial. One of the research direction is put forward by Hu and Peng first. Peng and Wu Peng and Shi made a further research, and Yong to a more detailed discussion of this method, by introducing the concept of the bridge, systematically studied the FBSDE continuity method. Because such a system can be applied to random Feynman - Kac of partial differential equations of research, And a double optimal control problem of stochastic control systems, we will be working in Peng and Shi further in-depth study on the basis of this category are backward stochastic differential equation. In this paper, we are considering various constraint conditions with backward stochastic differential equation.

Published in	Pure and Applied Mathematics Journal (Volume 4, Issue 3)
DOI	10.11648/j.pamj.20150403.20
Page(s)	120-127
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

FBSDE, Mean-Field Forward Backward, Stochastic Differential Equations, Stochastic Partial Differential Equations

References

[1]	Deepmala and H. K. Pathak, A study on some problems on existence of solutions for nonlinear functional-integral equations, Acta Mathematica Scientia, 33 B(5) (2013), 1305–1313.
[2]	Deepmala, A Study on Fixed Point Theorems for Nonlinear Contractions and its Applications, Ph.D. Thesis (2014), Pt. Ravishankar Shukla University, Raipur (Chhatisgarh) India – 492 010.
[3]	V.N. Mishra, Some Problems on Approximations of Functions in Banach Spaces, Ph.D. Thesis (2007), Indian Institute of Technology, Roorkee - 247 667, Uttarakhand, India.
[4]	V.N. Mishra, L.N. Mishra, Trigonometric Approximation of Signals (Functions) in Lp (p _ 1)− norm, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, no. 19, pp. 909 – 918.
[5]	L.N. Mishra, S.K. Tiwari, V.N. Mishra, I.A. Khan; Unique Fixed Point Theorems for Generalized Contractive Mappings in Partial Metric Spaces, Journal of Function Spaces, Volume 2015 (2015), Article ID 960827, 8 pages.
[6]	V.N. Mishra, M.L. Mittal, U. Singh, On best approximation in locally convex space, Varahmihir Journal of Mathematical Sciences India, Vol. 6, No.1, (2006), 43-48.
[7]	Deepmala, Existence Theorems for Solvability of a Functional Equation Arising in Dynamic Programming, Int. J. Math. Math. Sci., Vol. 2014, Article ID: 706585, 9 pages.
[8]	Deepmala, H.K. Pathak, On Solutions of Some Functional-Integral Equations in Banach Algebra, Research Journal of Science and Technology 5 (3), 358-362.
[9]	Z. Wu, Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems, J. Systems Sci. Math. Sci., 11 (1998), 249-259.
[10]	Z. Wu, Forward-backward stochastic differential equations with Brownian Motion and Process Poisson, Acta Math. Appl. Sonica, English Series, 15 (1999), 433-443.
[11]	Z. Wu, A maximum principle for partially observed optimal control of forward-backward stochastic control systems, Sci China Ser F, 53 (2010), 2205-2214.
[12]	Z. Wu and Z. Yu, Fully coupled forward-backward stochastic differential equations and related partial differential equations system, Chinese Ann Math Ser A, 25 (2004), 457-468
[13]	H. Xiao and G. Wang, A necessary condition for optimal control of initial coupled forward-backward stochastic differential equations with partial information, J. Appl. Math. Compute, 37 (2011), 347-359.
[14]	J. Xiong, An Introduction to Stochastic Filtering Theory, London, U.K.: Oxford University Press, 2008.

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APA Style

Jie Zhu, Hong Zhang, Li Zhou, Yuhang Feng. (2015). The Mean Field Forward Backward Stochastic Differential Equations and Stochastic Partial Differential Equations. Pure and Applied Mathematics Journal, 4(3), 120-127. https://doi.org/10.11648/j.pamj.20150403.20

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ACS Style

Jie Zhu; Hong Zhang; Li Zhou; Yuhang Feng. The Mean Field Forward Backward Stochastic Differential Equations and Stochastic Partial Differential Equations. Pure Appl. Math. J. 2015, 4(3), 120-127. doi: 10.11648/j.pamj.20150403.20

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AMA Style

Jie Zhu, Hong Zhang, Li Zhou, Yuhang Feng. The Mean Field Forward Backward Stochastic Differential Equations and Stochastic Partial Differential Equations. Pure Appl Math J. 2015;4(3):120-127. doi: 10.11648/j.pamj.20150403.20

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@article{10.11648/j.pamj.20150403.20,
  author = {Jie Zhu and Hong Zhang and Li Zhou and Yuhang Feng},
  title = {The Mean Field Forward Backward Stochastic Differential Equations and Stochastic Partial Differential Equations},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {3},
  pages = {120-127},
  doi = {10.11648/j.pamj.20150403.20},
  url = {https://doi.org/10.11648/j.pamj.20150403.20},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150403.20},
  abstract = {Since 1990 Pardoux and Peng, proposed the theory of backward stochastic differential equation Backward stochastic differential equation and is backward stochastic differential equations (short for FBSDE) theory has been widely research (see El Karoui, Peng and Cauenez, Ma and Yong, etc.) Generally, a backward stochastic differential equation is a type Ito stochastic differential equation and a coupling Pardoux - Peng and backward stochastic differential equation. Antonelli, Ma, Protter and Yong is backward stochastic differential equation for a series of research, and apply to the financial. One of the research direction is put forward by Hu and Peng first. Peng and Wu Peng and Shi made a further research, and Yong to a more detailed discussion of this method, by introducing the concept of the bridge, systematically studied the FBSDE continuity method. Because such a system can be applied to random Feynman - Kac of partial differential equations of research, And a double optimal control problem of stochastic control systems, we will be working in Peng and Shi further in-depth study on the basis of this category are backward stochastic differential equation. In this paper, we are considering various constraint conditions with backward stochastic differential equation.},
 year = {2015}
}

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TY  - JOUR
T1  - The Mean Field Forward Backward Stochastic Differential Equations and Stochastic Partial Differential Equations
AU  - Jie Zhu
AU  - Hong Zhang
AU  - Li Zhou
AU  - Yuhang Feng
Y1  - 2015/06/06
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150403.20
DO  - 10.11648/j.pamj.20150403.20
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 120
EP  - 127
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20150403.20
AB  - Since 1990 Pardoux and Peng, proposed the theory of backward stochastic differential equation Backward stochastic differential equation and is backward stochastic differential equations (short for FBSDE) theory has been widely research (see El Karoui, Peng and Cauenez, Ma and Yong, etc.) Generally, a backward stochastic differential equation is a type Ito stochastic differential equation and a coupling Pardoux - Peng and backward stochastic differential equation. Antonelli, Ma, Protter and Yong is backward stochastic differential equation for a series of research, and apply to the financial. One of the research direction is put forward by Hu and Peng first. Peng and Wu Peng and Shi made a further research, and Yong to a more detailed discussion of this method, by introducing the concept of the bridge, systematically studied the FBSDE continuity method. Because such a system can be applied to random Feynman - Kac of partial differential equations of research, And a double optimal control problem of stochastic control systems, we will be working in Peng and Shi further in-depth study on the basis of this category are backward stochastic differential equation. In this paper, we are considering various constraint conditions with backward stochastic differential equation.
VL  - 4
IS  - 3
ER  -

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Author Information

Jie Zhu

School of Information, Beijing Wuzi University, Beijing, China
Hong Zhang

School of Information, Beijing Wuzi University, Beijing, China
Li Zhou

School of Information, Beijing Wuzi University, Beijing, China
Yuhang Feng

Insurance Department, Central University of Finance and Economics, Beijing, China

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Plain Text BibTeX RIS

APA Style

Jie Zhu, Hong Zhang, Li Zhou, Yuhang Feng. (2015). The Mean Field Forward Backward Stochastic Differential Equations and Stochastic Partial Differential Equations. Pure and Applied Mathematics Journal, 4(3), 120-127. https://doi.org/10.11648/j.pamj.20150403.20

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ACS Style

Jie Zhu; Hong Zhang; Li Zhou; Yuhang Feng. The Mean Field Forward Backward Stochastic Differential Equations and Stochastic Partial Differential Equations. Pure Appl. Math. J. 2015, 4(3), 120-127. doi: 10.11648/j.pamj.20150403.20

Copy | Download

AMA Style

Jie Zhu, Hong Zhang, Li Zhou, Yuhang Feng. The Mean Field Forward Backward Stochastic Differential Equations and Stochastic Partial Differential Equations. Pure Appl Math J. 2015;4(3):120-127. doi: 10.11648/j.pamj.20150403.20

Copy | Download

@article{10.11648/j.pamj.20150403.20,
  author = {Jie Zhu and Hong Zhang and Li Zhou and Yuhang Feng},
  title = {The Mean Field Forward Backward Stochastic Differential Equations and Stochastic Partial Differential Equations},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {3},
  pages = {120-127},
  doi = {10.11648/j.pamj.20150403.20},
  url = {https://doi.org/10.11648/j.pamj.20150403.20},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150403.20},
  abstract = {Since 1990 Pardoux and Peng, proposed the theory of backward stochastic differential equation Backward stochastic differential equation and is backward stochastic differential equations (short for FBSDE) theory has been widely research (see El Karoui, Peng and Cauenez, Ma and Yong, etc.) Generally, a backward stochastic differential equation is a type Ito stochastic differential equation and a coupling Pardoux - Peng and backward stochastic differential equation. Antonelli, Ma, Protter and Yong is backward stochastic differential equation for a series of research, and apply to the financial. One of the research direction is put forward by Hu and Peng first. Peng and Wu Peng and Shi made a further research, and Yong to a more detailed discussion of this method, by introducing the concept of the bridge, systematically studied the FBSDE continuity method. Because such a system can be applied to random Feynman - Kac of partial differential equations of research, And a double optimal control problem of stochastic control systems, we will be working in Peng and Shi further in-depth study on the basis of this category are backward stochastic differential equation. In this paper, we are considering various constraint conditions with backward stochastic differential equation.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - The Mean Field Forward Backward Stochastic Differential Equations and Stochastic Partial Differential Equations
AU  - Jie Zhu
AU  - Hong Zhang
AU  - Li Zhou
AU  - Yuhang Feng
Y1  - 2015/06/06
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150403.20
DO  - 10.11648/j.pamj.20150403.20
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 120
EP  - 127
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20150403.20
AB  - Since 1990 Pardoux and Peng, proposed the theory of backward stochastic differential equation Backward stochastic differential equation and is backward stochastic differential equations (short for FBSDE) theory has been widely research (see El Karoui, Peng and Cauenez, Ma and Yong, etc.) Generally, a backward stochastic differential equation is a type Ito stochastic differential equation and a coupling Pardoux - Peng and backward stochastic differential equation. Antonelli, Ma, Protter and Yong is backward stochastic differential equation for a series of research, and apply to the financial. One of the research direction is put forward by Hu and Peng first. Peng and Wu Peng and Shi made a further research, and Yong to a more detailed discussion of this method, by introducing the concept of the bridge, systematically studied the FBSDE continuity method. Because such a system can be applied to random Feynman - Kac of partial differential equations of research, And a double optimal control problem of stochastic control systems, we will be working in Peng and Shi further in-depth study on the basis of this category are backward stochastic differential equation. In this paper, we are considering various constraint conditions with backward stochastic differential equation.
VL  - 4
IS  - 3
ER  -

Copy | Download