Convergence of the Deep BSDE method for FBSDEs with non-Lipschitz coefficients

Jiang, Yifan; Jin-Feng, Li

doi:10.3934/puqr.2021019

Cited by 11 publications

(4 citation statements)

References 16 publications

(9 reference statements)

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“…In addition, a theoretical error analysis is carried out, in which we provide convergence rates for the initial and terminal conditions of the FB-SDE under a mild assumption and strong convergence of the FBSDE under stronger assumptions. Our main result is similar to the a posteriori error bound for the deep BSDE method which was established for weakly coupled FBSDEs in [32] and later extended to non-Lipschitz coefficients (but for less general diffusion coefficients) in [42]. However, these results are unlikely to be valid for strongly coupled FBSDEs and we find several examples in which the discrete terminal condition converges while the FBSDE approximation does not.…”

supporting

confidence: 80%

Convergence of a Robust Deep FBSDE Method for Stochastic Control

Andersson

Oosterlee

2023

SIAM J. Sci. Comput.

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supporting

confidence: 80%

Convergence of a Robust Deep FBSDE Method for Stochastic Control

Andersson

Oosterlee

2023

SIAM J. Sci. Comput.

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“…In this section, we use the proposed method to solve the pricing problem of a class of bonds with interest rates subject to the Vasicek model with jumps. [16,17] In this model, short-term interest rate 𝑋𝑋 𝑡𝑡 obeys the following stochastic differential equations: After calculation, Table 3 and Table 4 show the important numerical results of solving a class of bond pricing problems with interest rates subject to the Vasicek model with jumps using Adam and NAdam optimizers respectively, including that with the change of iteration steps, mean and standard deviation of loss function 𝑌𝑌 0 , mean and standard deviation of loss function, and running time. Only the numerical results of iteration steps 𝑛𝑛 ∈ {1000,2000,3000,4000} are selected as typical examples for display.…”

Section: Bond Pricing Under the Jumping Vasicek Modelmentioning

confidence: 99%

Study on Pricing of High Dimensional Financial Derivatives Based on Deep Learning

Liu¹,

Gu²

2023

Preprint

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Many problems in the fields of finance and actuarial science can be transformed into the problem of solving backward stochastic differential equations (BSDE) and partial differential equations (PDE) with jumps, which are often difficult to solve in high-dimensional cases. To solve this problem, this paper applies the deep learning algorithm to solve a class of high-dimensional nonlinear partial differential equations with jump terms and their corresponding backward stochastic differential equations (BSDE) with jump terms. Using the nonlinear Feynman-Kac formula, the problem of solving this kind of PDE is transformed into the problem of solving the corresponding backward stochastic differential equations with jump terms, and the numerical solution problem is turned into a stochastic control problem. At the same time, the gradient and jump process of the unknown solution are separately regarded as the strategy function and they are approximated respectively by using two multilayer neural networks as function approximators. Thus, deep learning-based method is used to overcome the “curse of dimensionality” caused by high-dimensional PDE with jump, and the numerical solution is obtained. In addition, this paper proposes a new optimization algorithm based on the existing neural network random optimization algorithm, and compares the results with the traditional optimization algorithm, and achieves good results. Finally, the proposed method is applied to three practical high-dimensional problems: Hamilton-Jacobi-Bellman equation, bond pricing under the jump Vasicek model and high-dimensional option pricing model with default risk. The proposed numerical method has obtained satisfactory accuracy and efficiency. The method has important application value and practical significance in investment decision-making, option pricing, insurance and other fields.

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“…Two feedforward neural networks are established at each time step t = t n in ( 21) and (22). One is to approximate the gradient of the unknown solution, which means to approximate the function X t n → Z t n : ∇ x u(t n , X t n )σ(t n , X t n ) , and this neural network is recorded as NN θ zn (x), such that θ z n represents all parameters of this neural network and Z n indicates that this neural network is used to approximate Z t n at time t n .…”

Section: General Framework For Neural Networkmentioning

confidence: 99%

“…In this section, we use the proposed method to solve the pricing problem of a class of bonds with interest rates subject to the Vasicek model with jumps [20][21][22]. In this model, the short-term interest rate X t obeys the following stochastic differential equations:…”

Section: Bond Pricing Under the Jumping Vasicek Modelmentioning

confidence: 99%

Study of Pricing of High-Dimensional Financial Derivatives Based on Deep Learning

Liu

2023

Mathematics

View full text Add to dashboard Cite

Many problems in the fields of finance and actuarial science can be transformed into the problem of solving backward stochastic differential equations (BSDE) and partial differential equations (PDEs) with jumps, which are often difficult to solve in high-dimensional cases. To solve this problem, this paper applies the deep learning algorithm to solve a class of high-dimensional nonlinear partial differential equations with jump terms and their corresponding backward stochastic differential equations (BSDEs) with jump terms. Using the nonlinear Feynman-Kac formula, the problem of solving this kind of PDE is transformed into the problem of solving the corresponding backward stochastic differential equations with jump terms, and the numerical solution problem is turned into a stochastic control problem. At the same time, the gradient and jump process of the unknown solution are separately regarded as the strategy function, and they are approximated, respectively, by using two multilayer neural networks as function approximators. Thus, the deep learning-based method is used to overcome the “curse of dimensionality” caused by high-dimensional PDE with jump, and the numerical solution is obtained. In addition, this paper proposes a new optimization algorithm based on the existing neural network random optimization algorithm, and compares the results with the traditional optimization algorithm, and achieves good results. Finally, the proposed method is applied to three practical high-dimensional problems: Hamilton-Jacobi-Bellman equation, bond pricing under the jump Vasicek model and option pricing under the jump diffusion model. The proposed numerical method has obtained satisfactory accuracy and efficiency. The method has important application value and practical significance in investment decision-making, option pricing, insurance and other fields.

show abstract

Convergence of the Deep BSDE method for FBSDEs with non-Lipschitz coefficients

Cited by 11 publications

References 16 publications

Convergence of a Robust Deep FBSDE Method for Stochastic Control

Convergence of a Robust Deep FBSDE Method for Stochastic Control

Study on Pricing of High Dimensional Financial Derivatives Based on Deep Learning

Study of Pricing of High-Dimensional Financial Derivatives Based on Deep Learning

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