Determining the optimal response of Feynman-Kak stochastic-financial equation based on Jacobi and Irfoil expansion

Subject Areas :

Sayed Mohammad AlaviSheshamd ¹ (Department of Business Administration, Islamic Azad University, Central Tehran, Tehran, Iran)
Shadan SediqBehzadi ² (Department of Mathematics and Statistics, Islamic Azad University, Qazvin, Qazvin, Iran)

Received: 2022-07-05 Accepted : 2022-09-07 Published : 2022-11-11

Keywords: Iran, Capital Market, Keywords: Financial risk, Financial sector, Real sector,

Abstract :

AbstractIn this paper, we solve the Feynman-Kac equation using the collocation method with Jacobi and Airfoil bases. This partial differential equation is one of the most important and widely used random equations in financial mathematics. Due to the increasing demand in applied sciences such as financial mathematics, economics and complexity in modeling, data analysis and calculation, significant efforts have been made in search of better mathematical models to obtain approximate solutions to the modeled equations in recent years. It is well established that many of the systems encountered in the new era cannot be represented by ordinary differential equations in the traditional way or by the model of random differential equations. This equation offers a solution for quadratic partial differential equations and stochastic differential equations. Applications of this formula in the field of random control, financial mathematics, risk analysis and related fields can be mentioned. In this paper, by applying numerical methods to the Feynman-Kac equation, nonlinear devices are obtained that can be solved by numerical methods for solving nonlinear devices, such as Newton's iterative method.

References:

فهرست منابع
محمدرضا نیکبخت ، علی قاسمی، محمد ایمانی، تاثیر خطای پیش بینی سود مدیریت بر پایداری اجزای نقدی و تعهدی سود و ارزشیابی بیش از حد سهام، پژوهش های حسابداری مالی و حسابرسی. دوره 12 ،شماره 46 ،مرداد 1399 ،صفحه 26-1
سید کاظم ابراهیمی، علی بهرامی نسب، صدیقه پروانه، تاثیر رقابت در بازار محصول بر ریسک پذیری سرمایه گذاران، پژوهش های حسابداری مالی و حسابرسی. دوره 10، شماره 40  ، بهمن 1397 ، صفحه 186-171
مهدی دسینه، یداله تاری وردی، فرزانه حیدر پور،تأثیرمعیارهای مبتنی بر حسابداری ویژگی های سود بر ریسک نامطلوب سود، پژوهش های حسابداری مالی و حسابرسی. دوره 11، شمار41، فروردین1398 ، صفحه 153-76

A. LE Aavil, N. Oudjane, F. Russo, Monte-Carlo Algorithms for Forward Feynman-Kac type representation for semilinear nonconservative Partial Differential Equations, Preprint hal, 2017.
A. Lejay, H. M. Gonzalez, A forward-backward probabilistic algorithm for the incompressible  Navier-Stokes equations, hal, 2019.
Ch. Beck, S. Becker, P. Cheridito, Deep splitting method for parabolic PDE Nanyang Technological University, Singapore, 2019.
Crepey .S, Financial Modeling, Springer-Verlag, 2013.
E. C. Waymire, Probability & incompressible Navier-Stokes equations: An overview of
some recent developments, Probab. Surv,5 (2005) 1–32.
F. Antonelli, Backward-forward stochastic differential equations, Ann. Appl, 8 (1993) 777–793.
H. Pham, Feynman-Kac representation of fully nonlinear PDEs and applications.
Acta Mathematica Vietnamica ,4 (2015)  255–269.
J. Ma, J. Yong, Forward-backward stochastic differential equations and their applications,
Lecture Notes in Mathematics, Springer-Verlag, 1999.
J. Sun1, D. Nie1, W. Deng, Error estimates for backward fractional Feynman-Kac
equation with non-smooth initial data, Math Na, 2019.
M. Ossiander, A probabilistic representation of solutions of the incompressible Navier-
Stokes equations in R3, Probab. Theory Relat, 2(2005)  267–298.
Michael.J,  Stochastic Calculus and Financial Applications,  Springer New York, 2001. 
P. Cheridito, H. M. Soner, N. Touzi, N. Victoir, Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs, Pure Appl. Math, 3(2007) 1081–1110.
R. N. Bhattacharya, L. Chen, S. Dobson, R. B. Guenther, C. Orum, M. Ossiander,Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations, Trans. Amer. Math, 2(2003) 5003–5040.
R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Mod.Phys, 3(1948) 367–387.
S. Benachour, B. Roynette, P. Vallois, Branching process associated with 2d-Navier Stokes equation, Rev. Mat. Iberoam, 5 (2001) 331–373.
Wazwaz. A. M, A First Course in Integral Equations, World Scientific, 2015.
Wazwaz A. M, Linear and Nonlinear Integral Equations,  Methods and Applications Springer, 2011.
X. Mao ,  Stochastic Differential Equations and Their Applications, Horwood Chichester, 1997.
Y. G , Z. C , Hybrid Switching Diffusions , Properties and Applications Springer, New York, 2010.
Y. Le Jan, A. S. Sznitman, Stochastic cascades and 3-dimensional Navier-Stokes equations,Probab, Theory Relat, 1(1997) 343–366.
Y. Zhou , W. Cai , Numerical Solution of the Robin Problem of Laplace Equations with a Feynman-Kac Formula and Reflecting Brownian Motions, 2019.