Abstract
This paper establishes the existence of a unique nonnegative continuous viscosity solution to the HJB equation associated with a linear-quadratic stochastic control problem with singular terminal state constraint and possibly unbounded cost coefficients. The existence result is based on a novel comparison principle for semi-continuous viscosity sub- and supersolutions for PDEs with singular terminal value. Continuity of the viscosity solution is enough to carry out the verification argument.
Highlights
Let T ∈ (0, ∞) and let (, F, (Ft )t∈[0,T ], P) that satisfies the usual conditions and carries a Poisson process N and an independent d-dimensional standard Brownian motion W
We prove the existence of a unique continuous viscosity solution to the resulting HJB equation and give a representation of the optimal control in terms of the viscosity solution
In this paper we established a novel comparison principle for viscosity solutions to HJB equations with singular terminal conditions arising in models of optimal portfolio liquidation under market impact
Summary
Let T ∈ (0, ∞) and let ( , F , (Ft )t∈[0,T ], P) that satisfies the usual conditions and carries a Poisson process N and an independent d-dimensional standard Brownian motion W. In [6], the authors studied the general verification result for stochastic impulse control problems, assuming that a comparison principle for discontinuous viscosity solutions of the HJB equation holds. This is a very strong hypothesis that can be avoided in our case. The linear-quadratic structure of our control problem allows us to characterize the value in terms of a PDE without jumps, and the verification argument can be given in terms of the associated FBSDE after the existence of the viscosity solution has been established.
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