Численный алгоритм решения полностью нелинейных параболических уравнений на основе прямых-обратных стохастических дифференциальных уравнений и нейронных сетей
Журнал: Известия Самарского научного центра Российской академии наук @izvestiya-ssc
Рубрика: Информатика, вычислительная техника и управление
Статья в выпуске: 4 т.26, 2024 года.
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В статье рассматривается численный метод решения задачи Коши для полностью нелинейного параболического уравнения. Рассматриваемое уравнение сводится к системе квазилинейных параболических уравнений. Для этой системы построено вероятностное представление решения, основанное на решении системы прямого-обратного стохастических дифференциальных уравнений (ПОСДУ). Решение ПОСДУ сводится к решению оптимизационной задачи, которая численно решается с помощью нейронной сети. Рассмотрен пример применения данного метода для уравнения, описывающего цену оптимального портфеля на рынке Блэка-Шоулса. Численное решение было апробировано на специальных видах функции полезности, для которых существует точное решение.
Похожие статьи в разделе Вычислительная математика. Численный анализ
Короткий адрес: https://sciup.org/148330105
IDS: 148330105 | УДК: 519.633.2 | DOI: 10.37313/1990-5378-2024-26-4-161-169
A numerical algorithm for solving fully nonlinear parabolic equations based on forward-backward stochactic differencial equations and neural networks
The article presents a numerical method for solving the Cauchy problem for a fully nonlinear parabolic equation. Considered equation is reduced to a system of quasilinear parabolic equations. A probabilistic representation of this system solution was constructed based on the solution of the system of forward-backward stochastic differential equations (FBSDE). The FBSDE solution is reduced to solving an optimization problem solved numerically using a neural network. An example of applying this method to an equation describing the price of an optimal portfolio in the Black-Scholes market is considered. The numerical solution was tested while choosing utility functions for which exact solutions exist.
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