We consider a class of backward stochastic differential equations (BSDEs) with singular terminal condition and develop a numerical scheme to approximate their solution. To this end, we extend an asymptotic development of the BSDE solution known from the power case, which arises from optimal liquidation problems, to more general generators. This expansion allows to obtain a suitable approximation of the BSDE solution close to the terminal time. Using this as a terminal condition, we analyze the error of a backward Euler implicit scheme and detail its dependence on the terminal condition.

The paper deals with homogenization and higher order approximations of solutions to nonlocal evolution equations of convolution type whose coefficients are periodic in the spatial variables and random stationary in time. We assume that the convolution kernel has finite moments up to order three. Under proper mixing assumptions, we study the limit behavior of the normalized difference between solutions of the original and the homogenized problems and show that this difference converges to the solution of a linear stochastic partial differential equation.

We consider a bivariate first hitting-time model in which durations are the crossing times of dependent compound Poisson processes with fixed thresholds. The identifiability of the model is discussed, and likelihood estimators of the model parameters are proposed. We obtain the asymptotic properties of the estimators and underline their finite sample performance with a simulation study on synthetic data. The practical applicability of our approach is demonstrated by an application using data from patients suffering from mushroom poisoning.

We develop a Malliavin calculus for nonlinear Hawkes processes in the sense of Carlen and Pardoux. This approach, based on perturbations of the jump times of the process, enables the construction of a local Dirichlet form. As an application, we establish criteria for the absolute continuity of solutions to stochastic differential equations driven by Hawkes processes. We also derive sensitivity formulas for the valuation of financial derivatives with respect to model parameters.

We analyze a class of multidimensional linear-quadratic stochastic control problems with random coefficients, motivated by multi-asset optimal trade execution. The problems feature non-diffusive controlled state dynamics and a terminal constraint that restricts the terminal state to a prescribed random linear subspace. We derive the associated Riccati backward stochastic differential equation (BSDE) and identify a suitable formalization of its singular terminal condition. Via a penalization approach, we establish existence of a minimal supersolution of the Riccati BSDE and use it to characterize both the value function and the optimal control. We analyze the asymptotic behavior of the supersolution near terminal time and discuss special cases where closed-form solutions can be obtained.

The composition of the solutions of two SDEs can be used to represent the solution of some SPDEs (method of stochastic characteristics). This representation holds, for example, in the context of consistent dynamic utilities (see Karoui and Mrad [An exact connection between two solvable SDEs and a nonlinear utility stochastic PDE, SIAM J. Financial Math., 2013;4(1):697-736] and Matoussi and Mrad [Dynamic utility and related nonlinear SPDEs driven by Levy noise. Int. J. Theor. Appl. Finance, 2022;25(1):Paper No. 2250004, 45], where the SPDE can be used to establish a numerical scheme, approximating the SPDE's solution. This allows us to avoid a complicated discretization in time and space of a nonlinear SPDE. The purpose of this paper is to study the compound Euler scheme when the SDEs are driven by a Brownian motion and a Poisson random measure with infinite Levy measure. The continuous case and the case of a finite Levy measure have been treated in Gobet and Mrad [Convergence rate of strong approximations of compound random maps, application to SPDEs. Discrete Contin. Dyn. Syst. Ser. B, 2018;23(10):4455-4476] and Mrad [Solving some stochastic partial differential equations driven by Levy noise using two SDEs. Stochastics, 2022;94(8):1265-1298]. Additional terms specific to the truncation method of the small jumps will appear in our error estimates. In many cases, an optimal choice of parameters allows us to find a standard convergence rate of order $ 1/\sqrt {n} $ 1/n. However, we provide some examples of Levy measures with much slower convergence rates.

The classical optimal trading problem is the closure of a position in an asset over a time interval; the trader maximizes the expected revenues under the constraint that the position be closed by terminal time. Since the asset price is stochastic, the liquidation constraint is too restrictive; the trader may want to relax it or slow down/stop trading depending on price behavior. We consider two additional parameters that serve these purposes within the Almgren-Chriss framework: a binary valued process that prescribes when trading takes place and a set that prescribes when full liquidation is required. The permanent price impact parameter enters the problem as the negative part of the terminal cost. A terminal cost that can take negative values implies that the BSDE associated with the value function of the control problem can explode backward in time and that existence results on solutions of BSDE with singular terminal values are not directly applicable. When liquidation costs are quadratic, the problem is convex and, under a general filtration, the minimal supersolution of the BSDE gives the value function and the optimal control. For the non-quadratic case, we assume a stochastic Markovian volatility model. These give PDE/PDE-system representations for the value functions.

We study the behavior at the terminal time of the minimal supersolution of backward stochastic differential equation with singular terminal condition by using the associated integro-partial differential equation. We prove that if there are jumps (i.e. the operator of the PDE is non local), we observe a propagation of the singularity, contrary to the continuous case (local operator). We distinguish different cases of driver and terminal condition. The Riccati case is central because for quadratic and subquadratic generators the associated solution is not continuous at the terminal time, while the solutions for stronger non linearity are continuous. Finally we study the consequence for the numerical scheme. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, we investigate the optimal design problem for asymptotically efficient estimation in the context of controlled Ornstein-Uhlenbeck processes. The study builds a connection between statistical optimal design and ergodic control theories. We focus on the estimation of an unknown drift parameter with continuous observations and the possibility of an additive admissible control. Two notions of statistical asymptotic efficiency are examined in the controlled setting. The first relies on maximizing the Fisher information with respect to controls on finite horizons, while the second is based on ergodic (infinite-horizon) controls. We establish that both definitions lead to the same asymptotic variance in the case of deterministic controls, and also provide results for stochastic controls.

We prove that the solution of the backward stochastic differential equation with terminal singularity has a Malliavin derivative, which is the limit of the derivative of the approximating sequence. We also provide the asymptotic behavior of this derivative close to the terminal time. We apply this result to the regularity of the related partial differential equation and to the sensitivity of the liquidation problem.

We study the limit behavior of the solution of a backward stochastic differential equation when the terminal condition is singular, that is it can be equal to infinity with a positive probability. In the Markovian setting, Malliavin’s calculus enables us to prove continuity if a balance condition between the growth w.r.t. y and the growth w.r.t. z of the generator is satisfied. As far as we know, this condition is new. We apply our result to liquidity problem in finance and to the solution of some semi-linear partial differential equation ; the imposed assumption is also new in the literature on PDE.

We consider Cauchy problem for a divergence form second order parabolic operator with rapidly oscillating coefficients that are periodic in spatial variables and random stationary ergodic in time. As was proved in Zhikov et al. (Mat Obshch 45:182–236, 1982) and Kleptsyna and Piatnitski (Homogenization and applications to material sciences. GAKUTO Internat Ser Math Sci Appl vol 9, pp 241–255. Gakkōtosho, Tokyo, 1995) in this case the homogenized operator is deterministic. The paper focuses on the diffusion approximation of solutions in the case of non-diffusive scaling, when the oscillation in spatial variables is faster than that in temporal variable. Our goal is to study the asymptotic behaviour of the normalized difference between solutions of the original and the homogenized problems.

We study a class of nonlinear BSDEs with a superlinear driver process f adapted to a filtration F and over a random time interval [[0, S]] where S is a stopping time of F. The terminal condition $\xi$ is allowed to take the value +$\infty$, i.e., singular. Our goal is to show existence of solutions to the BSDE in this setting. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., attains the terminal values with probability 1. We consider three types of terminal values: 1) Markovian: i.e., $\xi$ is of the form $\xi$ = g($\Xi$ S) where $\Xi$ is a continuous Markovian diffusion process and S is a hitting time of $\Xi$ and g is a deterministic function 2) terminal conditions of the form $\xi$ = $\infty$ $\times$ 1 {$\tau$ $\le$S} and 3) $\xi$ 2 = $\infty$ $\times$ 1 {$\tau$ >S} where $\tau$ is another stopping time. For general $\xi$ we prove the minimal supersolution is continuous at time S provided that F is left continuous at time S. We call a stopping time S solvable with respect to a given BSDE and filtration if the BSDE has a minimal supersolution with terminal value $\infty$ at terminal time S. The concept of solvability plays a key role in many of the arguments. Finally, we discuss implications of our results on the Markovian terminal conditions to solution of nonlinear elliptic PDE with singular boundary conditions.

We consider a class of Backward Stochastic Differential Equations with superlinear driver process f adapted to a filtration supporting at least a d dimensional Brownian motion and a Poisson random measure on R-m \ {0}. We consider the following class of terminal conditions: xi(1) = infinity.1({tau 1 <= T}) where tau(1) is any stopping time with a bounded density in a neighborhood of T and xi(2) = infinity.1(AT) where A(t), t is an element of [0; T] is a decreasing sequence of events adapted to the filtration F-t that is continuous in probability at T (equivalently, A(T) = {tau(2) > T} where tau(2) is any stopping time such that P(tau(2) = T) = 0). In this setting we prove that the minimal supersolutions of the BSDE are in fact solutions, i.e., they attain almost surely their terminal values. We note that the first exit time from a time varying domain of a d-dimensional diffusion process driven by the Brownian motion with strongly elliptic covariance matrix does have a continuous density. Therefore such exit times can be used as tau(1) and tau(2) to define the terminal conditions xi(1) and xi(2). The proof of existence of the density is based on the classical Green's functions for the associated PDE.

In this paper, we study backward stochastic Volterra integral equations introduced in Lin [Stochastic Anal. Appl.20(2002) 165–183] and Yong [Stochastic Process. Appl.116(2006) 779–795] and extend the existence, uniqueness or comparison results for general filtration as in Papapantoleonet al.[Electron. J. Probab.23(2018) EJP240] (not only Brownian-Poisson setting). We also considerLp-data and explore the time regularity of the solution in the Itô setting, which is also new in this jump setting.

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.

In this paper, we provide a one-to-one correspondence between the solution Y of a BSDE with singular terminal condition and the solution H of a BSDE with singular generator. This result provides the precise asymptotic behavior of Y close to the final time and enlarges the uniqueness result to a wider class of generators.

We consider a variant of the basic problem of the calculus of variations, where the Lagrangian is convex and subject to randomness adapted to a Brownian filtration. We solve the problem by reducing it, via a limiting argument, to an unconstrained control problem that consists in finding an absolutely continuous process minimizing the expected sum of the Lagrangian and the deviation of the terminal state from a given target position. Using the Pontryagin maximum principle, we characterize a solution of the unconstrained control problem in terms of a fully coupled forward-backward stochastic differential equation (FBSDE). We use the method of decoupling fields for proving that the FBSDE has a unique solution. We exploit a mono-tonicity property of the decoupling field for solving the original constrained problem and characterize its solution in terms of an FBSDE with a free backward part.

We use the functional Itô calculus to prove that the solution of a BSDE with singular terminal condition verifies at the terminal time: $\liminf _{t\to T} Y(t) = \xi = Y(T)$. Hence, we extend known results for a non-Markovian terminal condition.

We consider a Cauchy problem for a divergence form second order parabolic operator with rapidly oscillating coefficients that are periodic in spatial variables and random stationary ergodic in time. As was already proved, in this case the homogenized operator is deterministic. We obtain the leading terms of the asymptotic expansion of the solution, these terms being deterministic functions, and show that a properly renormalized difference between the solution and the said leading terms converges to a solution of some SPDE.