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Backward stochastic differential equations with non-Markovian singular terminal values. (English) Zbl 1415.60068

Summary: 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 \(\xi = \infty \cdot 1_{B(m, r)^c}\) (for which \(q > 2\) is assumed) or \(\xi = \infty \cdot 1_{B(m, r)}\), 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.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
35K57 Reaction-diffusion equations
35K67 Singular parabolic equations
60G40 Stopping times; optimal stopping problems; gambling theory
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J65 Brownian motion
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