scholarly journals Mean square rate of convergence for random walk approximation of forward-backward SDEs

2020 ◽  
Vol 52 (3) ◽  
pp. 735-771
Author(s):  
Christel Geiss ◽  
Céline Labart ◽  
Antti Luoto
Keyword(s):  
Differential Equation ◽  
Random Walk ◽  
Rate Of Convergence ◽  
Backward Stochastic Differential Equation ◽  
Stochastic Equations ◽  
Terminal Condition ◽  
Probabilistic Methods ◽  
Mean Square ◽  
Finite Difference Equations ◽  
Backward Sdes

AbstractLet (Y, Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk $B^n$ from the underlying Brownian motion B by Skorokhod embedding, one can show $L_2$-convergence of the corresponding solutions $(Y^n,Z^n)$ to $(Y, Z).$ We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in $C^{2,\alpha}$. The proof relies on an approximative representation of $Z^n$ and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by probabilistic methods.

scholarly journals A Mean Field Game of Optimal Portfolio Liquidation

2021 ◽  
Author(s):  
Guanxing Fu ◽  
Paulwin Graewe ◽  
Ulrich Horst ◽  
Alexandre Popier
Keyword(s):  
Differential Equation ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equation ◽  
Mean Field ◽  
Uniqueness Result ◽  
Terminal Condition ◽  
Optimal Portfolio ◽  
Linear Quadratic ◽  
Mean Field Game ◽  
Terminal Values

We consider a mean field game (MFG) of optimal portfolio liquidation under asymmetric information. We prove that the solution to the MFG can be characterized in terms of a forward-backward stochastic differential equation (FBSDE) with a possibly singular terminal condition on the backward component or, equivalently, in terms of an FBSDE with a finite terminal value yet a singular driver. Extending the method of continuation to linear-quadratic FBSDEs with a singular driver, we prove that the MFG has a unique solution. Our existence and uniqueness result allows proving that the MFG with a possibly singular terminal condition can be approximated by a sequence of MFGs with finite terminal values.

OPTIMAL STOCHASTIC CONTROL PROBLEM UNDER MODEL UNCERTAINTY WITH NONENTROPY PENALTY

2017 ◽  
Vol 20 (03) ◽  
pp. 1750015
Author(s):  
WAHID FAIDI ◽  
ANIS MATOUSSI ◽  
MOHAMED MNIF
Keyword(s):  
Differential Equation ◽  
Control Problem ◽  
Stochastic Control ◽  
Model Uncertainty ◽  
General Framework ◽  
Backward Stochastic Differential Equation ◽  
Terminal Condition ◽  
Penalty Term ◽  
Stochastic Control Problem ◽  
Continuous Filtration

In this paper, a stochastic control problem under model uncertainty with general penalty term is studied. Two types of penalties are considered. The first one is of type [Formula: see text]-divergence penalty treated in the general framework of a continuous filtration. The second one called consistent time penalty is studied in the context of a Brownian filtration. In the case of consistent time penalty, we characterize the value process of our stochastic control problem as the unique solution of a class of quadratic backward stochastic differential equation with unbounded terminal condition.

scholarly journals Terminal-Dependent Statistical Inference for the Integral Form of FBSDE

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Qi Zhang
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Statistical Inference ◽  
Backward Stochastic Differential Equation ◽  
Integral Form ◽  
Estimation Procedure ◽  
Terminal Condition ◽  
Finite Sample ◽  
Inference Procedure ◽  
Key Factor

Backward Stochastic Differential Equation (BSDE) has been well studied and widely applied. The main difference from the Original Stochastic Differential Equation (OSDE) is that the BSDE is designed to depend on a terminal condition, which is a key factor in some financial and ecological circumstances. However, to the best of knowledge, the terminal-dependent statistical inference for such a model has not been explored in the existing literature. This paper is concerned with the statistical inference for the integral form of Forward-Backward Stochastic Differential Equation (FBSDE). The reason why I use its integral form rather than the differential form is that the newly proposed inference procedure inherits the terminal-dependent characteristic. In this paper the FBSDE is first rewritten as a regression version, and then a semiparametric estimation procedure is proposed. Because of the integral form, the newly proposed regression version is more complex than the classical one, and thus the inference methods are somewhat different from those designed for the OSDE. Even so, the statistical properties of the new method are similar to the classical ones. Simulations are conducted to demonstrate finite sample behaviors of the proposed estimators.

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