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We study a class of non-linear Backward stochastic differential equations (BSDE) 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 ξ is allowed to take the value +∞, i.e., singular. 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 ∞ at terminal time S. Our goal is to show existence of solutions to the BSDE for a range of singular terminal values under the assumption that S is solvable. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., it is continuous at time S and attains the terminal value with probability 1. We consider three types of terminal values: 1) Markovian: i.e., ξ is of the form ξ = g(ΞS) where Ξ is a continuous Markovian diffusion process, S is a hitting time of Ξ and g is a deterministic function 2) terminal conditions of the form ξ1 = ∞· 1(τ≥) and 3) ξ2 = ∞· 1(τ≥) where τ is another stopping time. For general ξ we prove that minimal supersolution has a limit at time S provided that F is left continuous at time S. Finally, we discuss the implications of our results about Markovian terminal conditions to the solution of non-linear elliptic PDE with singular boundary conditions. [ABSTRACT FROM AUTHOR]

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 Papapantoleon et al. [Electron. J. Probab.23 (2018) EJP240] (not only Brownian-Poisson setting). We also consider Lp-data and explore the time regularity of the solution in the Itô setting, which is also new in this jump setting. [ABSTRACT FROM AUTHOR]

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 behaviour of Y close to the final time and enlarges the uniqueness result to a wider class of generators. [ABSTRACT FROM AUTHOR]

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 Papapantoleon et al. [Electron. J. Probab. 23 (2018) EJP240] (not only Brownian-Poisson setting). We also consider Lp-data and explore the time regularity of the solution in the Itˆo setting, which is also new in this jump setting. [ABSTRACT FROM AUTHOR]

In this paper, we show that the minimal solution of a backward stochastic differential equation gives a probabilistic representation of the minimal viscosity solution of an integro-partial differential equation both with a singular terminal condition. Singularity means that at the final time, the value of the solution can be equal to infinity. Different types of regularity of this viscosity solution are investigated: Sobolev, Hölder or strong regularity. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of existence and uniqueness of a solution for a backward stochastic differential equation (BSDE for short) with one reflecting barrier in the case when the terminal value, the generator and the obstacle process are Lp-integrable with p ∈ ]1, 2[. To construct the solution we use two methods: penalization and Snell envelope. As an application we broaden the class of functions for which the related obstacle partial differential equation problem has a unique viscosity solution. [ABSTRACT FROM AUTHOR]

We solve a class of BSDE with a power function f (y) = y q , q > 1 , driving its drift and with the terminal boundary condition ξ = ∞ ⋅ 1 B (m , r) c (for which q > 2 is assumed) or ξ = ∞ ⋅ 1 B (m , r) , where B (m , r) is the ball in the path space C ([ 0 , T ]) of the underlying Brownian motion centered at the constant function m and radius r. The solution involves the derivation and solution of a related heat equation in which f serves as a reaction term and which is accompanied by singular and discontinuous Dirichlet boundary conditions. Although the solution of the heat equation is discontinuous at the corners of the domain, the BSDE has continuous sample paths with the prescribed terminal value. [ABSTRACT FROM AUTHOR]

Abstract: We consider backward stochastic differential equations (BSDEs) with nonlinear generators typically of quadratic growth in the control variable. A measure solution of such a BSDE will be understood as a probability measure under which the generator is seen as vanishing, so that the classical solution can be reconstructed by a combination of the operations of conditioning and using martingale representations. For the case where the terminal condition is bounded and the generator fulfills the usual continuity and boundedness conditions, we show that measure solutions with equivalent measures just reinterpret classical ones. For the case of terminal conditions that have only exponentially bounded moments, we discuss a series of examples which show that in the case of non-uniqueness, classical solutions that fail to be measure solutions can coexist with different measure solutions. [Copyright &y& Elsevier]

We consider insurance derivatives depending on an external physical risk process, for example, a temperature in a low dimensional climate model. We assume that this process is correlated with a tradable financial asset. We derive optimal strategies for exponential utility from terminal wealth, determine the indifference prices of the derivatives, and interpret them in terms of diversification pressure. Moreover, we check the optimal investment strategies for standard admissibility criteria. Finally, we compare the static risk connected with an insurance derivative to the reduced risk due to a dynamic investment into the correlated asset. We show that dynamic hedging reduces the risk aversion in terms of entropic risk measures by a factor related to the correlation. [ABSTRACT FROM AUTHOR]