Scalar dynamic risk measures for univariate positions in continuous time are commonly represented as backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of set-valued functionals in the recent literature. There are two possible extensi…
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48 results for “Backward stochastic difference equations”
Backward stochastic difference equations on lattices with application to market equilibrium analysismath.PR
Study BSΔE on lattices for asset price analysis.
problem Optimal investment and market equilibrium analysis in asset price models.
method Backward stochastic difference equations on lattices.
result Applications to optimal investment and market equilibrium analysis.
We propose a numerical algorithm for backward stochastic differential equations based on time discretization and trigonometric wavelets. This method combines the effectiveness of Fourier-based methods and the simplicity of a wavelet-based formula, resulting in an algorithm that is both accurate and easy to implement. F…
This paper presents a novel approach to numerically solve stochastic differential games for nonlinear systems. The proposed approach relies on the nonlinear Feynman-Kac theorem that establishes a connection between parabolic deterministic partial differential equations and forward-backward stochastic differential equat…
Paper proves stability of complex equations under various conditions.
problem Stability of backward stochastic differential equations with jumps.
method General framework for convergent sequences of data and solutions.
result Convergent sequence of solutions for associated data.
In this paper, we first establish the reflected backward stochastic difference equations with finite state (FS-RBSDEs for short). Then we explore the Existence and Uniqueness Theorem as well as the Comparison Theorem by "one step" method. The connections between FS-RBSDEs and optimal stopping time problems are investig…
The paper defines and implements risk-indifference pricing for American-style contingent claims.
problem Pricing American-style contingent claims under uncertainty.
method Indifference pricing using convex risk measures and stochastic volatility models, with numerical solutions via deep learning.
result Characterization of indifference prices via Backward Stochastic Differential Equations (BSDEs).
Study proves existence of equilibrium in incomplete economies with discontinuous volatility.
problem Existence of incomplete Radner equilibrium with nondegenerate endogenous volatility.
method Established existence of solution for Markovian quadratic BSDEs with discontinuous generators using unique continuation and backward uniqueness.
result Existence of incomplete Radner equilibrium with nondegenerate endogenous volatility.
We propose a new method for the numerical solution of backward stochastic differential equations (BSDEs) which finds its roots in Fourier analysis. The method consists of an Euler time discretization of the BSDE with certain conditional expectations expressed in terms of Fourier transforms and computed using the fast F…
A new deep generative model uses BSDEs for high-dimensional data generation.
problem Generating high-dimensional complex data, especially images.
method Combines BSDEs with deep neural networks for training with MMD loss.
result BSDE-Gen effectively generates high-dimensional data with stochasticity.
This paper formulates and studies a stochastic maximum principle for forward-backward stochastic Volterra integral equations (FBSVIEs in short), while the control area is assumed to be convex. Then a linear quadratic (LQ in short) problem for backward stochastic Volterra integral equations (BSVIEs in short) is present …
In this paper, we introduce a large class of convergent numerical methods, based on (linear) basis function regression technique, to approximate the solution to a forward-backward stochastic differential equation with jumps (FBSDEJ hereafter). Numerical experiment shows good applicability of the proposed method.
Paper develops methods for solving complex stochastic equations using Malliavin calculus.
problem Existence, uniqueness, and regularity of solutions to BSVIEs.
method Malliavin calculus for tackling diagonal processes and nonlinear dependence.
result Developed well-posedness results for BSVIEs, including probabilistic interpretation of PDEs and portfolio optimization.
In this introductory paper, we discuss how quantitative finance problems under some common risk factor dynamics for some common instruments and approaches can be formulated as time-continuous or time-discrete forward-backward stochastic differential equations (FBSDE) final-value or control problems, how these final val…
(Working Paper) Using a purely probabilistic argument, we prove the global well-posedness of multidimensional superquadratic backward stochastic differential equations (BSDEs) without Markovian assumption. The key technique is the interplay between the local well-posedness of fully coupled path-dependent forward backwa…
New MKABSDEs help calculate initial margins in financial contracts.
problem Calculating initial margins in financial contracts with dependencies.
method Introduced MKABSDEs, provided existence and uniqueness, applied to CVaR, used deterministic and Monte-Carlo methods for numerical approximations.
result MKABSDEs provide a new way to solve for initial margins in financial contracts.
Bounded solutions to backward SDE's with jumps for utility optimization and indifference hedgingmath.PR
We prove results on bounded solutions to backward stochastic equations driven by random measures. Those bounded BSDE solutions are then applied to solve different stochastic optimization problems with exponential utility in models where the underlying filtration is noncontinuous. This includes results on portfolio opti…
Paper develops a new probabilistic method for American options using entropy regularization.
problem Finding optimal stopping times for American options with entropy regularization.
method Entropy-regularized penalization scheme based on Doob-Meyer-Mertens decomposition and reflected backward stochastic differential equations.
result Explicit convergence rates and policy improvement algorithm for American options.
In this paper we propose a new kind of high order numerical scheme for backward stochastic differential equations(BSDEs). Unlike the traditional -scheme, we reduce truncation errors by taking carefully for every subinterval according to the characteristics of integrands. We give error estimates of this nonlinear…
Deep-learning method solves BSVIEs and coupled systems.
problem High-dimensional, time-inconsistent stochastic control problems.
method Trains a neural network to approximate solution fields directly.
result Non-asymptotic error bound and scalable performance.
We establish existence, uniqueness and regularity of solution results for a class of backward stochastic partial differential equations with singular terminal condition. The equation describes the value function of non-Markovian stochastic optimal control problem in which the terminal state of the controlled process is…
Backward SDEs help price XVA for OTC derivatives.
problem XVA valuation for OTC derivatives with default risk.
method Review and apply BSDEs with random horizon.
result Explicit formula for XVA correction terms.
We propose a deep neural network framework for computing prices and deltas of American options in high dimensions. The architecture of the framework is a sequence of neural networks, where each network learns the difference of the price functions between adjacent timesteps. We introduce the least squares residual of th…
We link SVEs to SPDEs and derive Kolmogorov equations for singular kernels.
problem Solving stochastic Volterra equations with singular kernels.
method Establishing connections between SVEs and SPDEs, using stochastic calculus in Hilbert spaces.
result Solutions of SVEs can be expressed in terms of backward Kolmogorov equations.
Paper solves time-inconsistent control problems with BSDEs.
problem Time-inconsistent stochastic control in continuous time.
method Probabilistic representation via BSDEs.
result Equilibrium value function resolved for inconsistent cases.
Measures financial resilience using BSDEs and their properties.
problem Measuring financial resilience in dynamic risk environments.
method Developed stochastic calculus for BSDEs with jumps, revealing resilience rate as expectation of generator.
result Resilience rate can be represented as expectation of BSDE generator, revealing properties of dynamic risk measures.
A Nonlocal Approach to The Quantum Kolmogorov Backward Equation and Links to Noncommutative Geometryq-fin.MF
The Accardi-Boukas quantum Black-Scholes equation can be used as an alternative to the classical approach to finance, and has been found to have a number of useful benefits. The quantum Kolmogorov backward equations, and associated quantum Fokker-Planck equations, that arise from this general framework, are derived usi…
We solve non-Markovian optimal switching problems in discrete time on an infinite horizon, when the decision maker is risk aware and the filtration is general, and establish existence and uniqueness of solutions for the associated reflected backward stochastic difference equations. An example application to hydropower …
Paper presents a new insurance model equation for diverse structures.
problem Handling diverse insurance models with a single equation.
method Developed a canonical model construction and stochastic backward equations.
result Comparison theorems for different models follow from the new equation.
New deep learning method solves complex BSDEs efficiently.
problem Solving high-dimensional nonlinear BSDEs.
method Reformulate as global optimization, approximate solution with deep neural network, globally minimize quadratic local loss functions.
result Demonstrated effectiveness on various high-dimensional nonlinear BSDEs, including finance applications.
The paper solves a complex control problem with stochastic elements and switching conditions.
problem Non-homogeneous stochastic LQ control with regime switching and random coefficients.
method Explicit optimal control and value obtained through two systems of backward stochastic differential equations (BSDEs). Existence and uniqueness of solutions proved using BMO martingales and contraction mapping method.
result Explicit optimal state feedback control and optimal value derived for the problem.
New methods solve complex financial equations.
problem Solving backward stochastic differential equations driven by continuous-time Markov chains.
method Multi-stage Euler-Maruyama methods and multilevel spatial discretization.
result Efficiently solved stiff Markov BSDEs.
We introduce a novel numerical approach for a class of stochastic dynamic programs which arise as discretizations of backward stochastic differential equations or semi-linear partial differential equations. Solving such dynamic programs numerically requires the approximation of nested conditional expectations, i.e., it…
We propose a new forward-backward stochastic differential equation solver for high-dimensional derivatives pricing problems by combining deep learning solver with least square regression technique widely used in the least square Monte Carlo method for the valuation of American options. Our numerical experiments demonst…
This paper is dedicated to the construction of high-order (in both space and time) finite-difference schemes for both forward and backward PDEs and PIDEs, such that option prices obtained by solving both the forward and backward equations are consistent. This approach is partly inspired by Andreasen & Huge, 2011 who re…
FBSJNN solves PIDEs and FBSDEJs with deep learning, offering theoretical and numerical efficiency.
problem Solving Partial Integro-Differential Equations and Forward-Backward Stochastic Differential Equations with Jumps.
method FBSJNN framework using a single neural network for both solution approximation and non-local integral.
result FBSJNN achieves numerical solutions with a relative error of , demonstrating efficiency.
Deep learning model solves high-dimensional PDEs using Actor-Critic approach.
problem Solving high-dimensional nonlinear PDEs efficiently.
method Reformulated PDE into BSDE system, inspired by Actor-Critic algorithm for deep RL.
result Improved model with fewer parameters, faster convergence, and less hyperparameter tuning.
The paper develops methods to price options under rough volatility models using BSPDEs.
problem Pricing options in models with non-Markovian dynamics.
method Backward stochastic partial differential equations (BSPDEs) and deep learning for numerical approximations.
result Existence and uniqueness of weak solutions for general nonlinear BSPDEs.
A Dual Method For Backward Stochastic Differential Equations with Application to Risk Valuationmath.OC
We propose a numerical recipe for risk evaluation defined by a backward stochastic differential equation. Using dual representation of the risk measure, we convert the risk valuation to a stochastic control problem where the control is a certain Radon-Nikodym derivative process. By exploring the maximum principle, we s…
We introduce two simple models of forward-backward stochastic differential equations with a singular terminal condition and we explain how and why they appear naturally as models for the valuation of CO2 emission allowances. Single phase cap-and-trade schemes lead readily to terminal conditions given by indicator funct…
Uniform diffusion approximation for SGD in non-convex settings.
problem Finite-time diffusion approximation for SGD.
method Establishing uniform-in-time diffusion approximation with strong convexity and mild conditions.
result Uniform-in-time diffusion approximation of SGD without convexity of each loss function.
We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss functi…
We consider a general time-inconsistent stochastic linear-quadratic differential game. The time-inconsistency arises from the presence of quadratic terms of the expected state as well as state-dependent term in the objective functionals. We define an equilibrium strategy, which is different from the classical one, and …
Tensor Train Diffusion: Leveraging Low-Rank Structures for High-Dimensional Score-Based Samplingstat.ML
Efficiently samples complex distributions using tensor train format.
problem Sampling from high-dimensional complex probability densities efficiently.
method Integrates tensor train format with backward stochastic differential equations (BSDEs) for fast, robust, and accurate sampling.
result Improved efficiency in sampling from challenging target distributions.
Reflected backward stochastic differential equations and a class of non linear dynamic pricing ruleq-fin.PR
In that paper, we provide a new characterization of the solutions of specific reflected backward stochastic differential equations (or RBSDEs) whose driver is convex and has quadratic growth in its second variable: this is done by introducing the extended notion of -Snell enveloppe. Then, in a second step, we re…
This paper develops a new methodology for studying continuous-time Nash equilibrium in a financial market with asymmetrically informed agents. This approach allows us to lift the restriction of risk neutrality imposed on market makers by the current literature. It turns out that, when the market makers are risk averse,…
New method reveals insights about stochastic optimization methods using modified equations.
problem Understanding the qualitative behavior of stochastic optimization algorithms.
method Developed a class of stochastic differential equations to approximate the dynamics of stochastic optimization methods.
result Mean-square stability of the modified equation provides qualitative insights about stochastic coordinate descent.
New method for dynamic valuation in markets with random endowments.
problem Dynamic valuation in markets with random endowments.
method Developed new FBSDE systems and established optimality conditions.
result Established necessary and sufficient conditions for optimality.