Integrability of exponential process and its application to backward stochastic differential equations

2018 ◽  
Vol 30 (4) ◽  
pp. 335-365
Author(s):  
Bujar Gashi ◽  
Jiajie Li
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Optimal Investment ◽  
Backward Stochastic Differential Equations ◽  
Type Condition ◽  
Unbounded Coefficients ◽  
Exponential Process ◽  
Market Completeness ◽  
Integrability Problem ◽  
Weaker Conditions

Abstract We consider the integrability problem of an exponential process with unbounded coefficients. The integrability is established under weaker conditions of Kazamaki type, which complements the results of Yong obtained under a Novikov type condition. As applications, we consider the solvability of linear backward stochastic differential equations (BSDEs) and market completeness, the solvability of a Riccati BSDE and optimal investment, all in the setting of unbounded coefficients.

scholarly journals Backward stochastic differential equations with unbounded generators

2019 ◽  
Vol 19 (01) ◽  
pp. 1950008 ◽  
Author(s):  
Bujar Gashi ◽  
Jiajie Li
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Unique Solution ◽  
Comparison Theorem ◽  
Linear Growth ◽  
Sufficient Conditions ◽  
Backward Stochastic Differential Equations ◽  
Type Condition ◽  
Solution Pair ◽  
Picard Iterations

In this paper, we consider two classes of backward stochastic differential equations (BSDEs). First, under a Lipschitz-type condition on the generator of the equation, which can also be unbounded, we give sufficient conditions for the existence of a unique solution pair. The method of proof is that of Picard iterations and the resulting conditions are new. We also prove a comparison theorem. Second, under the linear growth and continuity assumptions on the possibly unbounded generator, we prove the existence of the solution pair. This class of equations is more general than the existing ones.

scholarly journals Systems of BSDES with oblique reflection and related optimal switching problems

2019 ◽  
Vol 19 (04) ◽  
pp. 1950030
Author(s):  
Mateusz Topolewski
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Backward Stochastic Differential Equations ◽  
Type Condition ◽  
Optimal Switching ◽  
Existence Of A Solution ◽  
Oblique Reflection ◽  
Probability Spaces

We consider systems of backward stochastic differential equations with càdlàg upper barrier [Formula: see text] and oblique reflection from below driven by an increasing continuous function [Formula: see text]. Our equations are defined on general probability spaces with a filtration satisfying merely the usual assumptions of right continuity and completeness. We assume that the pair [Formula: see text] satisfies a Mokobodzki-type condition. We prove the existence of a solution for integrable terminal conditions and integrable quasi-monotone generators. Applications to the optimal switching problem are given.

DISSIPATIVE BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS IN INFINITE DIMENSIONS

2006 ◽  
Vol 09 (01) ◽  
pp. 155-168 ◽  
Author(s):  
FULVIA CONFORTOLA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hilbert Spaces ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equations ◽  
Uniqueness Result ◽  
Infinite Dimensions ◽  
Partial Differential

We prove an existence and uniqueness result for a class of backward stochastic differential equations (BSDE) with dissipative drift in Hilbert spaces. We also give examples of stochastic partial differential equations which can be solved with our result.

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