Analysis of Backward Fuzzy Doubly Stochastic Differential Equations
Keywords:Backward Doubly Fuzzy Stochastic Differential Equations, Backward Fuzzy Stochastic Differential Equations, Approximation solution, differential equations.
In this paper, we propose a new formula for the backward doubly fuzzy stochastic differential equations (BFDSDEs), In the beginning, we present some basic concepts, definitions, and Hypotheses to obtain the numerical scheme for BFDSDEs, as our scheme depends on the partition of interval [0, T]. In our work, we prove that under Lipschitz conditions, the approximation solution for the backward fuzzy doubly stochastic differential equations converges to the exact solution by using mean square error, and prove the existence and uniqueness of approximations solutions to BFDSDEs.
B. A. Bodo, M. E. Thompson, and T. E. Unny, “A review on stochastic differential equations for applications in hydrology,” Stochastic Hydrology and Hydraulics, vol. 1, no. 2, pp. 81–100, 1987.
X. Mao, Stochastic differential equations and applications. Elsevier, 2007.
J. C. Cortés, L. Jódar, and L. Villafuerte, “Numerical solution of random differential equations: a mean square approach,” Math Comput Model, vol. 45, no. 7–8, pp. 757–765, 2007.
K. Nouri and H. Ranjbar, “Mean square convergence of the numerical solution of random differential equations,” Mediterranean Journal of Mathematics, vol. 12, no. 3, pp. 1123–1140, 2015.
O. Elbarrimi and Y. Ouknine, “Approximation of solutions of mean-field stochastic differential equations,” Stochastics and Dynamics, vol. 21, no. 01, p. 2150003, 2021.
E. Pardoux and S. Peng, “Adapted solution of a backward stochastic differential equation,” Syst Control Lett, vol. 14, no. 1, pp. 55–61, 1990.
S. Peng, “A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation,” Stochastics: An International Journal of Probability and Stochastic Processes, vol. 38, no. 2, pp. 119–134, 1992.
Z. Cao and J.-A. Yan, “A comparison theorem for solutions of backward stochastic differential equations,” Advance in Mathematics, vol. 28, no. 4, pp. 304–308, 1999.
J. Liu and J. Ren, “Comparison theorem for solutions of backward stochastic differential equations with continuous coefficient,” Stat Probab Lett, vol. 56, no. 1, pp. 93–100, 2002.
W. Zhao, L. Chen, and S. Peng, “A new kind of accurate numerical method for backward stochastic differential equations,” SIAM Journal on Scientific Computing, vol. 28, no. 4, pp. 1563–1581, 2006.
S. Falah and J. Liu, “Numerical Convergence of Backward Stochastic Differential Equation with non-Lipschitz Coefficients,” International Journal of Mathematical Analysis, vol. 9, no. 44, pp. 2181–2188, 2015.
J. Ma, P. Protter, J. San Martín, and S. Torres, “Numerical method for backward stochastic differential equations,” Annals of Applied Probability, pp. 302–316, 2002.
É. Pardoux and S. Peng, “Backward doubly stochastic differential equations and systems of quasilinear SPDEs,” Probab Theory Relat Fields, vol. 98, no. 2, pp. 209–227, 1994.
A. Aman, “A numerical scheme for backward doubly stochastic differential equations,” Bernoulli, vol. 19, no. 1, pp. 93–114, 2013.
J.-M. Owo, “Backward doubly stochastic differential equations with stochastic Lipschitz condition,” Stat Probab Lett, vol. 96, pp. 75–84, 2015.
A. Matoussi and W. Sabbagh, “Numerical Computation for Backward Doubly SDEs with random terminal time,” Monte Carlo Methods Appl, vol. 22, no. 3, pp. 229–258, 2016.
F. Bao, Y. Cao, and H. Zhang, “Solving Backward Doubly Stochastic Differential Equations through Splitting Schemes,” arXiv preprint arXiv:2103.08632, 2021.
M. Stojakovic, “Fuzzy random variables, expectation, and martingales,” J Math Anal Appl, vol. 184, no. 3, pp. 594–606, 1994.
J. H. Kim, “On fuzzy stochastic differential equations,” Journal of the Korean Mathematical Society, vol. 42, no. 1, pp. 153–169, 2005.
M. T. Malinowski, “Strong solutions to stochastic fuzzy differential equations of Itô type,” Math Comput Model, vol. 55, no. 3–4, pp. 918–928, 2012.
S. Nayak and S. Chakraverty, “Numerical solution of fuzzy stochastic differential equation,” Journal of Intelligent & Fuzzy Systems, vol. 31, no. 1, pp. 555–563, 2016.
M. T. Malinowski, “Fuzzy stochastic differential equations of decreasing fuzziness: approximate solutions,” Journal of Intelligent & Fuzzy Systems, vol. 29, no. 3, pp. 1087–1107, 2015.
S. Falah and J. Liu, “Euler-Maruyama approximation of backward doubly stochastic differential delay equations,” International Journal of Applied, vol. 5, no. 3, pp. 146–151, 2016.
How to Cite
Copyright (c) 2023 Hassan alrwazq, Falah Hassan Sarhan
This work is licensed under a Creative Commons Attribution 4.0 International License.