Philippe Briand, Christel Geiss, Stefan Geiss, Céline Labart
In this paper, we study in the Markovian case the rate of convergence in the Wasserstein distance of an approximation of the solution to a BSDE given by a BSDE which is driven by a scaled random walk as introduced in Briand, Delyon and M{é}min (Electron. Comm. Probab. 6(2001),1-14).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
A note on convergence rate for reflected BSDEs with quadratic generators by penalization method
本文利用BMO鞅技术证明了带二次生成元的反射倒向随机微分方程与其惩罚方程的收敛速度为惩罚参数的1/2阶，并将结果应用于欧拉折线法数值逼近。
In this paper we present a weak approximation scheme for BSDEs driven by a Wiener process and an (in)finite activity Poisson random measure with drivers that are general Lipschitz functionals of the solution of the BSDE. The approximating backward stochastic difference equations (BSΔEs) are driven by random walks that weakly approximate the given Wiener process and Poisson random measure. We establish the weak convergence to the solution of the BSDE and the numerical stability of the sequence of solutions of the BSΔEs. By way of illustration we analyse explicitly a scheme with discrete step-si…
In this paper we study different algorithms for backward stochastic differential equations (BSDE in short) basing on random walk framework for 1-dimensional Brownian motion. Implicit and explicit schemes for both BSDE and reflected BSDE are introduced. Then we prove the convergence of different algorithms and present simulation results for different types of BSDEs.
Convergence rate for random walk approximations of mean field BSDEs
该研究将经典BSDE的Donsker型定理收敛速率结果拓展至平均场情形，采用冻结技术与修正Hölder连续性概念处理奇异性，结合BMO技巧得到指数型Lorentz空间尺度下积分梯度过程的收敛速率。
In this paper, we prove new convergence results improving the ones by Chassagneux, Elie and Kharroubi [Ann. Appl. Probab. 22 (2012) 971--1007] for the discrete-time approximation of multidimensional obliquely reflected BSDEs. These BSDEs, arising in the study of switching problems, were considered by Hu and Tang [Probab. Theory Related Fields 147 (2010) 89--121] and generalized by Hamadène and Zhang [Stochastic Process. Appl. 120 (2010) 403--426] and Chassagneux, Elie and Kharroubi [Electron. Commun. Probab. 16 (2011) 120--128]. Our main result is a rate of convergence obtained in the Lipschit…
We use the language of errors to handle local Dirichlet forms with square field operator (cf [2]). Let us consider, under the hypotheses of Donsker theorem, a random walk converging weakly to a Brownian motion. If in addition the random walk is supposed to be erroneous, the convergence occurs in the sense of Dirichlet forms and induces the Ornstein-Uhlenbeck structure on the Wiener space. This quite natural result uses an extension of Donsker theorem to functions with quadratic growth. As an application we prove an invariance principle for the gradient of the maximum of the Brownian path compu…
In this note, we take up the study of weak convergence for stochastic differential equations driven by a (Liouville) fractional Brownian motion $B$ with Hurst parameter $H\in(1/3,1/2)$. In the current paper, we approximate the $d$-dimensional fBm by the convolution of a rescaled random walk with Liouville's kernel. We then show that the corresponding differential equation converges in law to a fractional SDE driven by $B$.
Donsker's theorem in {Wasserstein}-1 distance
本文计算了随机游走与布朗运动在Wasserstein-1距离下的误差，通过Stein方程的新估计，并应用于局部时间的收敛速率。
We consider backward stochastic differential equations with drivers of quadratic growth (qgBSDE). We prove several statements concerning path regularity and stochastic smoothness of the solution processes of the qgBSDE, in particular we prove an extension of Zhang's path regularity theorem to the quadratic growth setting. We give explicit convergence rates for the difference between the solution of a qgBSDE and its truncation, filling an important gap in numerics for qgBSDE. We give an alternative proof of second order Malliavin differentiability for BSDE with drivers that are Lipschitz contin…
We study the convergence rate of a class of linear multi-step methods for BSDEs. We show that, under a sufficient condition on the coefficients, the schemes enjoy a fundamental stability property. Coupling this result to an analysis of the truncation error allows us to design approximation with arbitrary order of convergence. Contrary to the analysis performed in \cite{zhazha10}, we consider general diffusion model and BSDEs with driver depending on $z$. The class of methods we consider contains well known methods from the ODE framework as Nystrom, Milne or Adams methods. We also study a class…
We consider the empirical process G_t of a one-dimensional diffusion with finite speed measure, indexed by a collection of functions F. By the central limit theorem for diffusions, the finite-dimensional distributions of G_t converge weakly to those of a zero-mean Gaussian random process G. We prove that the weak convergence G_t\Rightarrow G takes place in \ell^{\infty}(F) if and only if the limit G exists as a tight, Borel measurable map. The proof relies on majorizing measure techniques for continuous martingales. Applications include the weak convergence of the local time density estimator …
In this paper, we study the stability and convergence of some general quadratic semimartingales. Motivated by financial applications, we study simultaneously the semimartingale and its opposite. Their characterization and integrability properties are obtained through some useful exponential submartingale inequalities. Then, a general stability result, including the strong convergence of the martingale parts in various spaces ranging from $\mathbb{H}^1$ to BMO, is derived under some mild integrability condition on the exponential of the terminal value of the semimartingale. This can be applied …
In this paper, we obtain the existence and uniqueness theorem for backward stochastic differential equation driven by G-Brownian motion (G-BSDE) under degenerate case. Moreover, we propose a new probabilistic method based on the representation theorem of G-expectation and weak convergence to obtain the regularity of fully nonlinear PDE associated to G-BSDE.
This article deals with the numerical resolution of Markovian backward stochastic differential equations (BSDEs) with drivers of quadratic growth with respect to $z$ and bounded terminal conditions. We first show some bound estimates on the process $Z$ and we specify the Zhang's path regularity theorem. Then we give a new time discretization scheme with a non uniform time net for such BSDEs and we obtain an explicit convergence rate for this scheme.
We extend the branching process based numerical algorithm of Bouchard et al. [3], that is dedicated to semilinear PDEs (or BSDEs) with Lipschitz nonlinearity, to the case where the nonlinearity involves the gradient of the solution. As in [3], this requires a localization procedure that uses a priori estimates on the true solution, so as to ensure the well-posedness of the involved Picard iteration scheme, and the global convergence of the algorithm. When, the nonlinearity depends on the gradient, the later needs to be controlled as well. This is done by using a face-lifting procedure. Converg…
Vanishing-noise asymptotics for Donsker-Varadhan rate functions on the circle
研究圆上一维扩散过程Donsker-Varadhan速率函数在噪声消失时的渐近行为，发现两种表示极限不总一致，导致无穷维空间中的连续与不连续现象，并利用Γ-收敛在大偏差原理中取极限。
This paper is dedicated to solving high-dimensional coupled FBSDEs with non-Lipschitz diffusion coefficients numerically. Under mild conditions, we provided a posterior estimate of the numerical solution that holds for any time duration. This posterior estimate validates the convergence of the recently proposed Deep BSDE method. In addition, we developed a numerical scheme based on the Deep BSDE method and presented numerical examples in financial markets to demonstrate the high performance.
This article deals with the existence and the uniqueness of solutions to quadratic and superquadratic Markovian backward stochastic differential equations (BSDEs for short) with an unbounded terminal condition. Our results are deeply linked with a strong a priori estimate on $Z$ that takes advantage of the Markovian framework. This estimate allows us to prove the existence of a viscosity solution to a semilinear parabolic partial differential equation with nonlinearity having quadratic or superquadratic growth in the gradient of the solution. This estimate also allows us to give explicit conve…
In this paper we consider the random walk approximation of the solution of a Markovian BSDE whose terminal condition is a locally H{ö}lder continuous function of the Brownian motion. We state the rate of the L 2-convergence of the approximated solution to the true one. The proof relies in part on growth and smoothness properties of the solution u of the associated PDE. Here we improve existing results by showing some properties of the second derivative of u in space.
The recently proposed numerical algorithm, deep BSDE method, has shown remarkable performance in solving high-dimensional forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). This article lays a theoretical foundation for the deep BSDE method in the general case of coupled FBSDEs. In particular, a posteriori error estimation of the solution is provided and it is proved that the error converges to zero given the universal approximation capability of neural networks. Numerical results are presented to demonstrate the accuracy of the ana…
We provide existence results and comparison principles for solutions of backward stochastic difference equations (BS$Δ$Es) and then prove convergence of these to solutions of backward stochastic differential equations (BSDEs) when the mesh size of the time-discretizaton goes to zero. The BS$Δ$Es and BSDEs are governed by drivers $f^N(t,ω,y,z)$ and $f(t,ω,y,z),$ respectively. The new feature of this paper is that they may be non-Lipschitz in z. For the convergence results it is assumed that the BS$Δ$Es are based on d-dimensional random walks $W^N$ approximating the d-dimensional Brownian motion…
Convergence of a Deep BSDE solver with jumps
该研究针对带跳解耦FBSDE及相关PIDE的深度学习数值逼近误差，拓展已有经典工作，给出有限和无限活动情形下的先验与后验误差估计。
Two discretizations of a class of locally Lipschitz Markovian backward stochastic differential equations (BSDEs) are studied. The first is the classical Euler scheme which approximates a projection of the processes Z, and the second a novel scheme based on Malliavin weights which approximates the mariginals of the process Z directly. Extending the representation theorem of Ma and Zhang leads to advanced a priori estimates and stability results for this class of BSDEs. These estimates are then used to obtain competitive convergence rates for both schemes with respect to the number of points in …
We establish a general existence and uniqueness result of $L^1$ solution for a multidimensional backward stochastic differential equation (BSDE for short) with generator $g$ satisfying a one-sided Osgood condition as well as a general growth condition in $y$, and a Lipschitz condition together with a sublinear growth condition in $z$, which improves some existing results. In particular, we put forward and prove a stability theorem of the $L^1$ solutions for the first time. A new type of $L^1$ solution is also investigated. Some delicate techniques involved in the relationship between convergen…
Convergence rates for Backward SDEs driven by Lévy processes
该研究针对Lévy过程驱动的倒向随机微分方程（BSDE），推导了其复合泊松逼近BSDE在$\boldsymbol{\rm L}^2$范数和Wasserstein距离下的收敛速度，证明该速度等于对应Lévy过程的收敛速度（由Blumenthal-Getoor指数决定），具有最优性。
Pasting of Equilibria and Donsker-type Results for Mean Field Games
该研究探究不同时间维度平均场博弈的均衡关联，证明单周期均衡存在性与单调性传播、提出多周期均衡递归拼接法，还验证离散均衡向连续均衡的收敛性并在单调性条件下给出精确收敛速率。
We propose a new probabilistic numerical scheme for fully nonlinear equation of Hamilton-Jacobi-Bellman (HJB) type associated to stochastic control problem, which is based on the Feynman-Kac representation in [12] by means of control randomization and backward stochastic differential equation with nonpositive jumps. We study a discrete time approximation for the minimal solution to this class of BSDE when the time step goes to zero, which provides both an approximation for the value function and for an optimal control in feedback form. We obtained a convergence rate without any ellipticity con…
We study the discrete-time approximation for solutions of quadratic forward back- ward stochastic differential equations (FBSDEs) driven by a Brownian motion and a jump process which could be dependent. Assuming that the generator has a quadratic growth w.r.t. the variable z and the terminal condition is bounded, we prove the convergence of the scheme when the number of time steps n goes to infinity. Our approach is based on the companion paper [15] and allows to get a convergence rate similar to that of schemes of Brownian FBSDEs.
Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problems
本文针对传统深度BSDE方法无法求解最优停时问题的局限，为后向深度BSDE方法提供严格理论支撑，推导后验误差估计与损失函数上界，并通过数值算例验证了理论一致性。
A generalized scheme for BSDEs based on derivative approximation and its error estimates
本文提出一种基于拉格朗日插值近似导数的BSDEs通用数值方案，通过调整插值样本点分布获得不同稳定性和收敛阶的方案，并给出保证收敛的样本点分布条件。
We study the quantitative stability of the solutions to Markovian quadratic reflected BSDEs with bounded terminal data. By virtue of BMO martingale and change of measure techniques, we obtain stability estimates for the variation of the solutions with different underlying forward processes. In addition, we propose a truncated discrete-time numerical scheme for quadratic reflected BSDEs, and obtain the explicit rate of convergence by applying the quantitative stability result.
Convergence and qualitative properties of modified explicit schemes for BSDEs with polynomial growth
本文提出一个框架，系统分析修正显式方案的可积性、收敛性和定性性质，展示其与隐式方案相同收敛速度但计算成本更低。
Deep Operator BSDE: a Numerical Scheme to Approximate Solution Operators
本文提出基于Wiener chaos分解和经典Euler方案的数值方法，用于近似BSDE的解算子，并在更严格的条件下证明了收敛性及收敛速率，通过神经网络实现并验证了方法的准确性。
Deep BSDE Solver on Bounded Domains Part I: General Loss Rate
本文针对有界域上具有随机无界时间范围的BSDE，提出深度BSDE损失泛函的分支形式，推导其针对随机加权修改形式的一般收敛率，该率由离散时间步长和通用逼近距离表示。
A theorem of Donsker asserts that the empirical process converges in distribution to the Brownian bridge. The aim of this paper is to provide a new and simple proof of this fact.
In this paper, we study a class of Type-II backward stochastic Volterra integral equations (BSVIEs). For the adapted M-solutions, we obtain two approximation results, namely, a BSDE approximation and a numerical approximation. The BSDE approximation means that the solution of a finite system of backward stochastic differential equations (BSDEs) converges to the adapted M-solution of the original equation. As a consequence of the BSDE approximation, we obtain an estimate for the $L^2$-time regularity of the adapted M-solutions of Type-II BSVIEs. For the numerical approximation, we provide a bac…
Numerical approximation of Markovian BSDEs in infinite horizon and elliptic PDEs
本文研究了无限时间 horizon的反向随机微分方程，设计了高效的数值方案。通过马尔可夫导数建立了BSDE解的表示，并证明了皮卡方案的收缩性。开发了三种数值方案，其中前两种基于固定点论证，采用收缩性假设，后一种在更大z-利普希茨依赖下表现良好。
Non-asymptotic convergence rates for mean-field games: weak formulation and McKean-Vlasov BSDEs
本文采用弱公式化，通过新型McKean-Vlasov倒向随机微分方程表征平均场均衡，在弱假设下获得存在唯一性，并为N人博弈的价值函数和纳什均衡提供非渐近收敛速率，同时推广到闭环均衡和主方程。
In this paper, we quantify the rate of convergence between the distribution of number of zeros of random trigonometric polynomials (RTP) with i.i.d. centered random coefficients and the number of zeros of a stationary centered Gaussian process G, whose covariance function is given by the sinc function. First, we find the convergence of the RTP towards G in the Wasserstein-1 distance, which in turn is a consequence of Donsker Theorem. Then, we use this result to derive the rate of convergence between their respective number of zeros. Since the number of real zeros of the RTP is not a continuous…
Weak approximation of second-order BSDEs
本文研究了当连续驱动鞅被离散时间鞅逼近时第二阶反向随机微分方程的弱逼近问题，通过反向随机微分方程的鲁棒性性质和离散时间反向随机微分方程解的紧性，建立了该类方程的收敛性结果，并给出了具体数值方案。
The convolution method for the numerical solution of forward-backward stochastic differential equations (FBSDEs), introduced in [21], uses a uniform space grid. In this paper we utilize a tree-like spatial discretization that approximates the BSDE on the tree, so that no spatial interpolation procedure is necessary. In addition to suppressing extrapolation error, leading to a globally convergent numerical solution for the FBSDE, we provide explicit convergence rates. On this alternative grid the conditional expectations involved in the time discretization of the BSDE are computed using Fourier…
An Euler scheme for BSDEs via the Wiener chaos decomposition
提出基于Wiener混沌分解的Euler格式实现BSDE数值求解，无需前向-后向马尔可夫结构，适用于任意平方可积终端条件，并给出收敛性分析与数值验证。
Motivated by a theorem of Barbour, we revisit some of the classical limit theorems in probability from the viewpoint of the Stein method. We setup the framework to bound Wasserstein distances between some distributions on infinite dimensional spaces. We show that the convergence rate for the Poisson approximation of the Brownian motion is as expected proportional to $λ^{-1/2}$ where $λ$ is the intensity of the Poisson process. We also exhibit the speed of convergence for the Donsker Theorem and for the linear interpolation of the Brownian motion. By iterating the procedure, we give Edgeworth e…
Quadratic G-BSDEs for bond pricing with endogenous short-rate feedback
本文在G-期望框架下研究带内生短期利率反馈的稳健债券定价，通过二次G-BSDE建立解的存在唯一性及指数收敛，并应用于逆利率设计与长期限反馈规则构造。
In this paper, we propose a deep learning based numerical scheme for strongly coupled FBSDEs, stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs, fails for a simple linear-quadratic control problem, and motivate why t…
We study the convergence of a class of Runge-Kutta type schemes for backward stochastic differential equations (BSDEs) in a Markovian framework. The schemes belonging to the class under consideration benefit from a certain stability property. As a consequence, the overall rate of the convergence of these schemes is controlled by their local truncation error. The schemes are categorized by the number of intermediate stages implemented between consecutive partition time instances. We show that the order of the schemes matches the number $p$ of intermediate stages for $p\le3$. Moreover, we show t…
This paper investigates the finite horizon risk-sensitive portfolio optimization in a regime-switching credit market with physical and information-induced default contagion. It is assumed that the underlying regime-switching process has countable states and is unobservable. The stochastic control problem is formulated under partial observations of asset prices and sequential default events. By establishing a martingale representation theorem based on incomplete and phasing out filtration, we connect the control problem to a quadratic BSDE with jumps, in which the driver term is non-standard an…
Third-moment cancellation and predictable representation in finite-state BSDE approximation
该研究针对有限状态BSDE近似，证明了特定增量下的一阶收敛性，揭示了表示缺陷的下界，指出三阶张量带来了精度与可预测表示间的权衡关系。
Backward stochastic differential equations (BSDEs) appear in numeruous applications. Classical approximation methods suffer from the curse of dimensionality and deep learning-based approximation methods are not known to converge to the BSDE solution. Recently, Hutzenthaler et al. (arXiv:2108.10602) introduced a new approximation method for BSDEs whose forward diffusion is Brownian motion and proved that this method converges with essentially optimal rate without suffering from the curse of dimensionality. The central object of this article is to extend this result to general forward diffusions…
W-BSDEs with Common Noise: Well-posedness, Characterization, and Consequences
本文研究带共同噪声的Wasserstein空间倒向随机微分方程（W-BSDE），通过特征方程建立适定性，构造解并应用于McKean-Vlasov控制与平均场博弈，证明其作为有限维BSDE的平均场极限。
We propose a reformulation of the convergence theorem of monotone numerical schemes introduced by Zhang and Zhuo for viscosity solutions of path-dependent PDEs, which extends the seminal work of Barles and Souganidis on the viscosity solution of PDE. We prove the convergence theorem under conditions similar to those of the classical theorem in the work of Barles and Souganidis. These conditions are satisfied, to the best of our knowledge, by all classical monotone numerical schemes in the context of stochastic control theory. In particular, the paper provides a unified approach to prove the co…
Convergence in Density of Splitting AVF Scheme for Stochastic Langevin Equation
本文研究随机Langevin方程分裂AVF格式数值解的密度函数，证明其存在性与光滑性，并利用Malliavin微积分得出密度收敛阶等于最优强收敛阶。
We consider two implicit approximation schemes of the one-dimensional supercooled Stefan problem and prove their convergence, even in the presence of finite time blow-ups. All proofs are based on a probabilistic reformulation recently considered in the literature. The first scheme is a version of the time-stepping scheme studied in V. Kaushansky, C. Reisinger, M. Shkolnikov, and Z. Q. Song, arXiv:2010.05281, 2020, but here the flux over the free boundary and its velocity are coupled implicitly. Moreover, we extend the analysis to more general driving processes than Brownian motion. The second …
The Wasserstein Space of Stochastic Processes in Continuous Time
本文统一了多种适应弱拓扑定义，提出用适应Wasserstein距离度量连续时间随机过程，并证明其完备化空间具有Polish等优良性质。
Statistical Convergence Rates of Optimal Transport Map Estimation between General Distributions
本文针对现有最优传输映射估计收敛速率结果假设过严的问题，提出两种放宽假设的估计器并建立收敛速率，推导新庞加莱型不等式提升速率，还开发了基于神经网络的可扩展算法并验证其有效性与鲁棒性。
The Modified MSA, a Gradient Flow and Convergence
该研究发现改进型逐次逼近法（MSA）是梯度流的隐式欧拉格式，证明其迭代插值收敛到梯度流，并推导了非凸、凸及强凸情形下的收敛速率，借助BMO鞅理论解决了哈密顿量性质分析的技术难点。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。