Convergence of Recent Multistep Schemes for a Forward-Backward Stochastic Differential Equation

Yang, Jie; Zhao, Weidong

doi:10.4208/eajam.280515.211015a

Cited by 12 publications

(2 citation statements)

References 22 publications

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“…Wang et al studied several linear quadratic optimal control problems of the forward and backward stochastic differential equations of the mean field [3]; this control problem is different from the existing literature on the optimal control of the mean field stochastic system, and it has more applications; a closed-form optimal solution can be obtained under detailed conditions. Yang and Zhao applied it to the special type of forward and backward stochastic differential equations (FB-SDE) that appeared in finance and stochastic control and provided convergence analysis for the recently proposed multistep scheme [4], but this method is more difficult to solve. Tardelli uses reverse stochastic differential equations (BSDE) and filtering techniques to study stochastic control problems [5], but this method has certain limitations; it requires a sequence of functions that converge to a value function.…”

Section: Proposed Methodsmentioning

confidence: 99%

“…Samuelson extended his model and used the principle of stochastic dynamic programming to obtain an optimal investment strategy. Later, after the research and promotion of many researchers, the optimal investment strategy model was 4 Wireless Communications and Mobile Computing gradually improved and widely used in the financial investment market. In addition, according to whether the investment portfolio changes dynamically over time, the investment portfolio can be divided into a one-period portfolio model and a multiperiod portfolio model.…”

Section: Optimal Investment Strategymentioning

confidence: 99%

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[Retracted] Dynamic Index Optimal Investment Strategy Based on Stochastic Differential Equations in Financial Market Options

Zhang

2021

Wireless Communications and Mobile Computing

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With the gradual development and improvement of the financial market, financial derivatives such as futures and options have also become the objects of competition in the financial market. Therefore, how to make the most favorable and optimized investment and consumption when options are included? It has become a problem facing investors. Aiming at the optimal investment problem of investors, this paper studies the calculation of an optimal investment strategy in stochastic differential equations in financial market options on the basis of fuzzy theory. Now, stochastic calculus has become an important branch of stochastic analysis, finance, control, and other fields. The study of introducing stochastic differential equations is mainly to solve the stochastic control problem, which is the principle of the stochastic maximum. In finance, some hedging or pricing problems of contingent rights can eventually be transformed into a series of stochastic differential equations. Based on the historical data of five aspects of bank deposits, bonds, funds, stocks, and real estate of four listed insurance companies, the paper analyzes the optimization strategy of the capital investment of listed insurance companies based on the investment yield of the domestic investment market. According to the final results, the historical proportion of bank deposits of the superior company is 27%, and the optimal proportion given by the model is 25%; the total proportion of funds and stocks is 15%, and the optimal proportion of funds analyzed in the model is 20% and the optimal proportion of stocks is 10%. Therefore, the final results show that the investment efficiency of listed insurance companies can actually increase investment in stocks and funds and reduce the proportion of bank deposits to obtain greater investment returns.

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Section: Proposed Methodsmentioning

confidence: 99%

Section: Optimal Investment Strategymentioning

confidence: 99%

[Retracted] Dynamic Index Optimal Investment Strategy Based on Stochastic Differential Equations in Financial Market Options

Zhang

2021

Wireless Communications and Mobile Computing

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Multistep Schemes for Forward Backward Stochastic Differential Equations with Jumps

Zhao

Zhou

2016

J Sci Comput

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Convergence error estimates of the Crank-Nicolson scheme for solving decoupled FBSDEs

Yang

Zhao

2017

Sci. China Math.

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The Crank-Nicolson (short for C-N) scheme for solving backward stochastic differential equation (BSDE), driven by Brownian motions, was first developed by the authors W. Zhao, L. Chen and S. Peng [SIAM J. Sci. Comput., 28 (2006), 1563-1581, and numerical experiments showed that the accuracy of this C-N scheme was of second order for solving BSDE. This C-N scheme was extended to solve decoupled forwardbackward stochastic differential equations (FBSDEs) by W. Zhao, Y. Li and Y. Fu [Sci. China. Math., 57 (2014), 665-686], and it was numerically shown that the accuracy of the extended C-N scheme was also of second order. To our best knowledge, among all one-step (two-time level) numerical schemes with second-order accuracy for solving BSDE or FBSDEs, such as the ones in the above two papers and the one developed by the authors D. Crisan and K. Manolarakis [Ann. Appl. Probab., 24, 2 (2014), 652-678], the C-N scheme is the simplest one in applications. The theoretical proofs of second-order error estimates reported in the literature for these schemes for solving decoupled FBSDEs did not include the C-N scheme. The purpose of this work is to theoretically analyze the error estimate of the C-N scheme for solving decoupled FBSDEs. Based on the Taylor and Itô-Taylor expansions, the Malliavin calculus theory (e.g., the multiple Malliavin integration-by-parts formula), and our new truncation error cancelation techniques, we rigorously prove that the strong convergence rate of the C-N scheme is of second order for solving decoupled FBSDEs, which fills the gap between the second-order numerical and theoretical analysis of the C-N scheme.

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Convergence of Recent Multistep Schemes for a Forward-Backward Stochastic Differential Equation

Cited by 12 publications

References 22 publications

[Retracted] Dynamic Index Optimal Investment Strategy Based on Stochastic Differential Equations in Financial Market Options

[Retracted] Dynamic Index Optimal Investment Strategy Based on Stochastic Differential Equations in Financial Market Options

Multistep Schemes for Forward Backward Stochastic Differential Equations with Jumps

Convergence error estimates of the Crank-Nicolson scheme for solving decoupled FBSDEs

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