Backward stochastic differential equations with time delayed generators - results and counterexamples

By Lukasz Delong, Peter Imkeller. 2010

88 citations recorded by the source corpus.

We deal with backward stochastic differential equations with time de layed generators. In this new type of equation, a generator at time t can depend on the values of a solution in the past, weighted with a time delay function, for instance, of the moving average type. We prove existence and uniqueness of a solution for a sufficiently small time horizon or for a suffi ciently small Lipschitz constant of a generator. We give examples of bsde with time delayed generators that have multiple solutions or that have no solutions. We show for some special class of generators that existence and uniqueness may still hold for an arbitrary time horizon and for arbitrary Lip schitz constant. This class includes linear time delayed generators which we study in more detail. We are concerned with different properties of a solution of a bsde with time delayed generator, including the inheritance of bound edness from the terminal condition, the comparison principle, the existence of a measure solution and the BMO martingale property. We give examples in which they may fail.

Subfields: Control and Systems Engineering, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/3d62415509b188026e9d77b6a596ed63e4de2c7d

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Citations in the corpus

A simple constructive approach to quadratic BSDEs with or without delay (2013, methodology): “As recently observed in [6], there exists a unique solution for functional Lipschitz BSDEs with drivers depending on the past of solution, as soon as the maturity or the Lipschitz constant of the driver is small enough.”
Counterparty Risk and Funding: A Tale of Two Puzzles (2014, methodology): “Under the pre-default CSA closeout pricing scheme and in case of a path-dependent collateralization, one ends-up with a socalled time delayed BSDE with a coefficient depending on the past of Θ̃ (see Delong and Imkeller (2010)).”
Applications of time-delayed backward stochastic differential equations to pricing, hedging and portfolio management in insurance and finance (2010, methodology): “Even for Lipschitz generator f we may have three cases, see Delong and Imkeller (2010a) and Delong (2011).”