The paper studies a new type of stochastic differential equations for financial claims.
problem Analyzing financial claims with random payment times in uncertain markets.
method Investigates linear reflected-backward stochastic differential equations (RBSDEs) under random time events.
result Identifies sufficient conditions for the existence and estimation of solutions to these equations.
The paper studies RBSDEs with arbitrary stopping times and their solutions.
problem Existence and estimation of solutions to RBSDEs under arbitrary stopping times.
method Analyzes the conditions for the existence of solutions and estimates their norms.
result Proves the existence of solutions and provides estimation methods for arbitrary stopping times.
Study optimal switching under ambiguity in finance.
problem Optimal switching problems under ambiguity in finance.
method Use multidimensional reflected backward stochastic differential equations (RBSDEs) to characterize the optimal switching.
result Value function of optimal switching under ambiguity coincides with solutions to multidimensional RBSDEs with negative switching costs.
In that paper, we provide a new characterization of the solutions of specific reflected backward stochastic differential equations (or RBSDEs) whose driver g is convex and has quadratic growth in its second variable: this is done by introducing the extended notion of g-Snell enveloppe. Then, in a second step, we re…
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…
Study American options with asymmetric buyer information using RBSDEs.
problem Value American options with asymmetric buyer information.
method Model asymmetric information, use RBSDEs for representation.
result Provide a representation for the cost of additional buyer information.
We consider controller-stopper problems in which the controlled processes can have jumps. The global filtration is represented by the Brownian filtration, enlarged by the filtration generated by the jump process. We assume that there exists a conditional probability density function for the jump times and marks given t…
Study of RBSDEs with non-right-continuous obstacles and their optimal stopping applications.
problem Existence and uniqueness of solutions to RBSDEs with non-right-continuous obstacles.
method General theory of processes, optimal stopping theory, Itô's formula generalization.
result Existence and characterization of optimal stopping times for certain financial positions.
The paper tackles pricing vulnerable options via generalized BSDEs and penalization schemes.
problem Pricing options in a general hazard process setup.
method Establishes well-posedness and comparison theorems for generalized BSDEs and RBSDEs, studies penalization schemes.
result Well-posedness results and comparison theorems for generalized BSDEs and RBSDEs, extended penalization schemes.
In this paper, we analyze a real-valued reflected backward stochastic differential equation (RBSDE) with an unbounded obstacle and an unbounded terminal condition when its generator f has quadratic growth in the z-variable. In particular, we obtain existence, comparison, and stability results, and consider the opti…
Market microstructure model with speculators who deduce asset value from prices.
problem Modeling market microstructure with agents who deduce asset value from prices.
method Control-stopping games and coupled control-stopping problems (RBSDEs).
result Existence of a solution to the system of coupled control-stopping problems.
Modeling LOB dynamics between trades using game theory.
problem Understanding market microstructure and LOB formation.
method Continuous-time large-population game, Reflected Backward Stochastic Differential Equations (RBSDEs), fixed-point problem.
result Existence of solutions to the equilibrium problem.
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…
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.
We generalize the primal-dual methodology, which is popular in the pricing of early-exercise options, to a backward dynamic programming equation associated with time discretization schemes of (reflected) backward stochastic differential equations (BSDEs). Taking as an input some approximate solution of the backward dyn…
Study values and optimizes forestry leases under risk and uncertainty.
problem Valuing and optimizing forestry leases in the presence of catastrophe risk and parameter uncertainty.
method Stochastic bio-economic models, Kalman filter, maximum likelihood estimation, RBSDEs, Monte Carlo simulations.
result Conservative strategy is recommended due to parameter uncertainty.
Study on BSDEs with random time horizon, focusing on existence and properties.
problem Existence of solutions to BSDEs and reflected BSDEs with a random time horizon.
method Method of reduction and examination of BSDEs with lahdlaug driver.
result Existence of solutions to BSDEs and reflected BSDEs with a random time horizon.
The present work studies and analyzes general defaultable OTC contract in presence of a contingent CSA, which is a theoretical counterparty risk mitigation mechanism of switching type that allows the counterparty of a general OTC contract to switch from zero to full/perfect collateralization and switch back whenever sh…
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).
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.
Corrects gaps in earlier papers on 2BSDEs with reflections.
problem Gaps in earlier papers on 2BSDEs with reflections.
method Corrects gaps and provides insights on 2RBSDEs properties.
result Corrected gaps in earlier papers and provided insights.
This work extends set-valued risk measures to discrete time, using difference inclusions and equations.
problem Defining set-valued dynamic risk measures in discrete time.
method Investigates discrete time setting with difference inclusions and difference equations.
result Provides insights for continuous time representations of set-valued dynamic risk measures.
We study a doubly reflected backward stochastic differential equation (BSDE) with integrable parameters and the related Dynkin game. When the lower obstacle L and the upper obstacle U of the equation are completely separated, we construct a unique solution of the doubly reflected BSDE by pasting local solutions and…
A method for risk valuation using backward stochastic differential equations.
problem Risk evaluation in financial markets.
method Dual representation and stochastic control problem conversion, followed by dynamic programming.
result Piecewise-constant dual control provides a good approximation for risk valuation.
Paper solves complex control problems using novel SDEs.
problem Solving stochastic differential games for nonlinear systems.
method Uses Deep Forward-Backward SDEs with neural networks.
result Numerical solution validated on two example systems.
Paper introduces a new method for solving complex stochastic equations.
problem Solving forward-backward stochastic differential equations with jumps.
method Linear basis function regression technique.
result The proposed method is convergent and effective as shown by numerical experiments.
New high-order scheme reduces BSDE truncation errors.
problem Numerical solution of backward stochastic differential equations (BSDEs).
method Proposes a new θ-scheme with careful θ selection for every subinterval. result Error estimates and verification of scheme order.
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.
Proves global well-posedness for superquadratic BSDEs without Markovian assumption.
problem Global well-posedness of multidimensional superquadratic BSDEs without Markovian assumption.
method Interplay between local well-posedness of FBSDEs and backward iterations of superquadratic BSDEs.
result Global well-posedness of superquadratic BSDEs proved.
New method decomposes submartingale systems for BSDEs with weak constraints.
problem Tackles decomposition of submartingale systems for BSDEs with weak constraints.
method Introduces Yg,ξ-submartingale systems and proves a Mertens decomposition using an original approach. result Proves a Mertens decomposition for Yg,ξ-submartingale systems. In this paper we study Backward Stochastic Differential Equations with two reflecting right continuous with left limits obstacles (or barriers) when the noise is given by Brownian motion and a Poisson random measure mutually independent. The jumps of the obstacle processes could be either predictable or inaccessible. W…
New algorithm solves high-dimensional PDEs and BSDEs using neural networks.
problem Solving high-dimensional PDEs and BSDEs efficiently and accurately.
method Analogy with reinforcement learning, neural network approximation of policy function.
result Efficiency and accuracy demonstrated in solving 100-dimensional equations.
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.
Investment strategy optimization from discrete to continuous models.
problem Optimizing investment strategies and stopping times in both continuous and discrete settings.
method Characterized value functions via quadratic reflected BSDEs for continuous case, discretized BSDEs for discrete case, and derived uniform convergence rates.
result Uniform convergence and rate from discrete to continuous quadratic reflected 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.
New algorithm enhances generative modeling for bounded domains.
problem Ad-hoc thresholding techniques for boundary enforcement in diffusion models.
method Reflected Schrödinger Bridge algorithm for entropy-regularized optimal transport.
result Generative modeling in diverse bounded domains with optimal transport properties.
A new method for solving complex financial equations.
problem Solving complex financial equations with nested conditional expectations.
method Pathwise iteration for backward SDEs.
result Computes and iteratively improves upper and lower bounds on the true solution.
Wavelets improve accuracy in solving backward SDEs.
problem Solving backward stochastic differential equations (SDEs) with high accuracy and simplicity.
method Time discretization combined with trigonometric wavelets, enhanced by antireflective boundary technique.
result Improved numerical algorithm for SDEs with enhanced accuracy and ease of implementation.
This paper uses recent results on continuous-time finite-horizon optimal switching problems with negative switching costs to prove the existence of a saddle point in an optimal stopping (Dynkin) game. Sufficient conditions for the game's value to be continuous with respect to the time horizon are obtained using recent …
Time-inconsistent game solved with differential equations.
problem Time inconsistency in stochastic linear-quadratic games.
method Defined and derived equilibrium strategies via FBSDEs.
result Explicit equilibrium strategy found for 1D deterministic case.
Deep neural networks solve high-dimensional PDEs without explicit grids.
problem Solving high-dimensional PDEs using classical methods is computationally infeasible.
method Approximate solution with a deep neural network trained via FBSDEs.
result Deep learning can solve high-dimensional PDEs efficiently.
New deep learning solver for high-dimensional derivative pricing.
problem High-dimensional derivatives pricing problems.
method Combines deep learning with least square regression for backward SDE solving.
result Accurate and efficient pricing of complex derivatives.
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.
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 10−3, demonstrating efficiency. 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.
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.
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 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…