Electronic Communications in Probability

Existence of solution to scalar BSDEs with $L\exp{\left (\!\!\sqrt {{2\over \lambda }\log {(1+L)}}\,\right )} $-integrable terminal values

Ying Hu and Shanjian Tang

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In this paper, we study a scalar linearly growing backward stochastic differential equation (BSDE) with an $L\exp{\left (\!\!\sqrt {{2\over \lambda }\log {(1+L)}}\,\right )} $-integrable terminal value. We prove that a BSDE admits a solution if the terminal value satisfies the preceding integrability condition with the positive parameter $\lambda $ being less than a critical value $\lambda _0$, which is weaker than the usual $L^p$ ($p>1$) integrability and stronger than $L\log L$ integrability. We show by a counterexample that the conventionally expected $L\log L$ integrability and even the preceding integrability for $\lambda >\lambda _0$ are not sufficient for the existence of solution to a BSDE with a linearly growing generator.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 27, 11 pp.

Received: 5 July 2017
Accepted: 14 March 2018
First available in Project Euclid: 28 April 2018

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]

backward stochastic differential equation $L\exp{\left (\!\!\sqrt {{2\over \lambda }\log {(1+L)}}\,\right )} $ integrability terminal condition dual representation

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Hu, Ying; Tang, Shanjian. Existence of solution to scalar BSDEs with $L\exp{\left (\!\!\sqrt {{2\over \lambda }\log {(1+L)}}\,\right )} $-integrable terminal values. Electron. Commun. Probab. 23 (2018), paper no. 27, 11 pp. doi:10.1214/18-ECP127. https://projecteuclid.org/euclid.ecp/1524881135

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