Adomian's Decomposition Method Applied to an Exponential Equation
DOI:
https://doi.org/10.52737/18291163-2025.17.10-1-17Keywords:
Stochastic Exponential, Martingale, Adomian Series, Brownian MotionAbstract
The purpose of this study is to obtain a decomposition of the solution to a backward stochastic differential equation used in the dual problem of mathematical finance. Some explicitly solvable equations considered. We convert the equation into a system of recurrent relations. By solving this system and proving convergence of the series the solution to the equation can be determined. In this study, Adomian's method was applied to solve the backward stochastic differential equation. An explicit solution was obtained for some examples.
References
G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics, Kluwer, 1989. DOI: https://doi.org/10.1007/978-94-009-2569-4
R. Bellman and G. Adomian, Partial Differential Equations, Springer Dordrecht, 1985. DOI: https://doi.org/10.1007/978-94-009-5209-6
F. Biagini, P. Guasoni and M. Pratelli, Mean variance hedging for stochastic volatility models. Math. Finance, 10 (2000), no. 2, pp. 109-123. DOI: https://doi.org/10.1111/1467-9965.00084
L. Gabet, The theoretical foundation of the Adomian method. Computers Math. Applic., 27 (1994), no. 12, pp. 41-52. DOI: https://doi.org/10.1016/0898-1221(94)90084-1
M. De Donno, P. Guasoni and M. Pratelli, Superreplication and utility maximization in large financial markets. Stoch. Process. their Appl., 115 (2005), no. 12, pp. 2006-2022. DOI: https://doi.org/10.1016/j.spa.2005.06.010
C. Dellacherie and P.A. Meyer, Probabilites et Potentiel. Theorie des martingales. Actualites Scientifiques et Industrielles Hermann, Paris, 1980.
J. Jacod, Calcule Stochastique et Problemes des Martingales. Lecture Notes in Mathematics 714, Springer, Berlin, 1979. DOI: https://doi.org/10.1007/BFb0064907
N. Kazamaki, Continuous Exponential Martingales and BMO. Lecture Notes in Mathematics 1579, Springer, Berlin, 1994. DOI: https://doi.org/10.1007/BFb0073585
M. Kobylanski, Backward stochastic differential equation and partial differential equations with quadratic growth. Ann. Probab., 28 (2000), no. 2, pp. 558-602. DOI: https://doi.org/10.1214/aop/1019160253
H-H. Kuo, Gaussian Measures in Banach Spaces. Lecture notes in Mathematics 463, Springer, Berlin, 1975. DOI: https://doi.org/10.1007/BFb0082007
M. Mania and R. Tevzadze, An exponential martingale equation. Electron. Commun. Probab., 11 (2006), pp. 206-216. DOI: https://doi.org/10.1214/ECP.v11-1220
M. Mania and R. Tevzadze, A semimartingale Bellman equation and the variance-optimal martingale measure. Georgian Math. J., 7 (2000), no. 4, pp. 765-792. DOI: https://doi.org/10.1515/GMJ.2000.765
M. Mania and R. Tevzadze, A semimartingale backward equation and the variance optimal martingale measure under general information flow. SIAM J. Control. Optim., 42 (2003), no. 5, pp. 1703-1726. DOI: https://doi.org/10.1137/S036301290240628X
R. Tevzadze, Solvability of Backward Stochastic Differential Equation with Quadratic Growth. Stoch. Proces. their Appl., 118 (2008), no. 3, pp. 503-515. DOI: https://doi.org/10.1016/j.spa.2007.05.009
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