arXiv research

The paper tackles pricing vulnerable options via generalized BSDEs and penalization schemes.

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…

Paper develops a new method for optimal stopping in American options.

In this paper, we study a type of reflected BSDE with a constraint and introduce a new kind of nonlinear expectation via BSDE with a constraint and prove the Doob-Meyer decomposition with respect to the super(sub)martingale introduced by this nonlinear expectation. We then apply the results to the pricing of American o…

Study on BSDEs with random time horizon, focusing on existence and properties.

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…

We demonstrate that the use of asymptotic expansion as prior knowledge in the "deep BSDE solver", which is a deep learning method for high dimensional BSDEs proposed by Weinan E, Han & Jentzen (2017), drastically reduces the loss function and accelerates the speed of convergence. We illustrate the technique and its imp…

Deep BSDE method for pricing and hedging complex financial portfolios.

Investment strategy optimization from discrete to continuous models.

In this paper, we study a new type of BSDE, where the distribution of the Y-component of the solution is required to satisfy an additional constraint, written in terms of the expectation of a loss function. This constraint is imposed at any deterministic time t and is typically weaker than the classical pointwise one a…

We introduce the concept of singular recursive utility. This leads to a kind of singular BSDE which, to the best of our knowledge, has not been studied before. We show conditions for existence and uniqueness of a solution for this kind of singular BSDE. Furthermore, we analyze the problem of maximizing the singular rec…

The aim of this paper is twofold. First, we extend the results of [33] concerning the existence and uniqueness of second-order reflected 2BSDEs to the case of two obstacles. Under some regularity assumptions on one of the barriers, similar to the ones in [10], and when the two barriers are completely separated, we prov…

The aim of this short note is to fill in a gap in our earlier paper [16] on 2BSDEs with reflections, and to explain how to correct the subsequent results in the second paper [15]. We also provide more insight on the properties of 2RBSDEs, in the light of the recent contributions [13, 23] in the so--called $G-$framework…

We first introduce the concept of $\mathscr{Y}^{g,ξ}$-submartingale systems, where the nonlinear operator $\mathscr{Y}^{g,ξ}$ corresponds to the first component of the solution of a reflected BSDE with generator $g$ and lower obstacle $ξ$. We first show that, in the case of a left-limited right-continuous obstacle, any…

We study pricing and (super)hedging for American options in an imperfect market model with default, where the imperfections are taken into account via the nonlinearity of the wealth dynamics. The payoff is given by an RCLL adapted process $(ξ_t)$. We define the {\em seller's superhedging price} of the American option a…

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…

The goal is to re-examine and extend the findings from the recent paper by Dumitrescu, Quenez and Sulem (2017) who studied game options within the nonlinear arbitrage-free pricing approach developed in El Karoui and Quenez (1997). We consider the setup introduced in Kim, Nie and Rutkowski (2018) where contracts of an A…

We consider the optimal stopping problem with non-linear $f$-expectation (induced by a BSDE) without making any regularity assumptions on the reward process $ξ$. and with general filtration. We show that the value family can be aggregated by an optional process $Y$. We characterize the process $Y$ as the $\mathcal{E}^f…

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…

We re-examine and extend the findings from the recent paper by Dumitrescu, Quenez and Sulem (2018) who studied American and game options in a particular market model using the nonlinear arbitrage-free pricing approach developed in El Karoui and Quenez (1997). In the first part, we provide a detailed study of unilateral…

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…

We consider an American contingent claim on a financial market where the buyer has additional information. Both agents (seller and buyer) observe the same prices, while the information available to them may differ due to some extra exogenous knowledge the buyer has. The buyer's information flow is modeled by an initial…

In the recent paper \cite{DESZ}, the notion of $\mathscr{Y}^{g,ξ}$-submartingale processes has been introduced. Within a jump-diffusion model, we prove here that a process $X$ which satisfies the simultaneous $\mathscr{Y}^{\mathbb{Q},g,ξ}$ -submartingale property under a suitable family of equivalent probability measur…

A deep BSDE approach tackles multi-layered xVA calculations for portfolio valuation.

Study uses G-BSDEs to decompose pricing kernels under robust G-expectation.

Develops geometric BSDEs for modeling dynamic return risk measures.

New methods solve complex financial equations.

The paper characterizes dynamic return and star-shaped risk measures via BSDEs.

(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…

We discuss a general dynamic replication approach to counterparty credit risk modeling. This leads to a fundamental jump-process backward stochastic differential equation (BSDE) for the credit risk adjusted portfolio value. We then reduce the fundamental BSDE to a continuous BSDE. Depending on the close out value conve…

This paper is concerned with the determination of credit risk premia of defaultable contingent claims by means of indifference valuation principles. Assuming exponential utility preferences we derive representations of indifference premia of credit risk in terms of solutions of Backward Stochastic Differential Equation…

Paper presents a new backward deep BSDE method for solving nonlinear FBSDE problems.

BSDEs help in financial pricing and utility maximization.

In this work we study the price-hedge issue for general defaultable contracts characterized by the presence of a contingent CSA of switching type. This is a contingent risk mitigation mechanism that allow the counterparties of a defaultable contract to switch from zero to full/perfect collateralization and switch back …

KANHedge improves hedging of high-dimensional options using learnable B-spline activation functions.

We provide a probabilistic solution of a not necessarily Markovian control problem with a state constraint by means of a Backward Stochastic Differential Equation (BSDE). The novelty of our solution approach is that the BSDE possesses a singular terminal condition. We prove that a solution of the BSDE exists, thus part…

A new deep generative model uses BSDEs for high-dimensional data generation.

Paper introduces a new method to solve complex PDEs efficiently.

Study approximates BSDEs with constraints using machine learning.

A new method solves complex financial problems using deep learning.

We consider Lipschitz-type backward stochastic differential equations (BSDEs) driven by cylindrical martingales on the space of continuous functions. We show the existence and uniqueness of the solution of such infinite-dimensional BSDEs and prove that the sequence of solutions of corresponding finite-dimensional BSDEs…

As is known, an option price is a solution to a certain partial differential equation (PDE) with terminal conditions (payoff functions). There is a close association between the solution of PDE and the solution of a backward stochastic differential equation (BSDE). We can either solve the PDE to obtain option prices or…

Study uses BSDEs to price European options in markets with multiple defaults.

Study shows convergence rates for BSDEs approximated by compound Poisson processes.

We introduce and solve a new type of quadratic backward stochastic differential equation systems defined in an infinite time horizon, called \emph{ergodic BSDE systems}. Such systems arise naturally as candidate solutions to characterize forward performance processes and their associated optimal trading strategies in a…

Novel filter uses deep BSDE for nonlinear density approximation.

Study optimal liquidation strategies with infinite horizon and regime switching.

New integration method improves BSDE-based PDE solvers.