Continuity problem for backward stochastic differential equations with singular nonmarkovian terminal conditions and deterministic terminal times

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2020-9
Ahmadi, Mahdi
In this thesis we study a class of Backward Stochastic Differential Equations (BSDE) with superlinear driver process f adapted to a filtration F = fFt; t 2 [0; T]g supporting at least a d dimensional Brownian motion and a Poisson random measure on Rm n f0g in a deterministic time interval [0; T]. The superlinearity of f allows terminal conditions that can take the value +1 with positive probability. Such terminal conditions are called “singular.” A terminal condition is said to be Markovian if it is a deterministic function of a Markov process.The first goal of the present thesis is to construct solutions to the class of BSDE we work with when they are coupled with singular non-Markovian terminal conditions. We consider the following class of terminal conditions: 1 = 1 1f 1 Tg + A 1f 1>Tg where 1 is any stopping time with a bounded density in a neighborhood of T and 2 = 1 1AT +A 1Ac T where At, t 2 [0; T] is a decreasing sequence of events adapted to the filtration F that is continuous in probability at T (equivalently, AT = f 2 > Tg where 2 is any stopping time such that P( 2 = T) = 0). In this setting we prove that the minimal supersolutions of the BSDE are in fact solutions, i.e., they are continuous at time T and attain almost surely their terminal values. Let X be a d-dimensional diffusion process driven by the Brownian motion and with strongly elliptic covariance matrix. The second goal of the present thesis is to derive density formulas for the first exit time of X from a vii time varying domain. The existence of these densities show that such exit times can be used as 1 and 2 to define the terminal conditions 1 and 2: We also discuss the implications of our results in stochastic optimal control.

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Citation Formats
M. Ahmadi, “Continuity problem for backward stochastic differential equations with singular nonmarkovian terminal conditions and deterministic terminal times,” Ph.D. - Doctoral Program, Middle East Technical University, 2020.