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Register a new userA free boundary problem arising from a multi-state regime-switching stock trading model
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In this paper, we study a free boundary problem, which arises from an optimal trading problem of a stock that is driven by a uncertain market status process. The free boundary problem is a variational inequality system of three functions with a degenerate operator. The main contribution of this paper is that we not only prove all the four switching free boundaries are no-overlapping, monotonic and $C^{infty}$-smooth, but also completely determine their relative localities and provide the optimal trading strategies for the stock trading problem.
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We introduce and solve a new type of quadratic backward stochastic differential equation systems defined in an infinite time horizon, called emph{ergodic BSDE systems}. Such systems arise naturally as candidate solutions to characterize forward performance processes and their associated optimal trading strategies in a regime switching market. In addition, we develop a connection between the solution of the ergodic BSDE system and the long-term growth rate of classical utility maximization problems, and use the ergodic BSDE system to study the large time behavior of PDE systems with quadratic growth Hamiltonians.
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We consider the vectorial analogue of the thin free boundary problem introduced in cite{CRS} as a realization of a nonlocal version of the classical Bernoulli problem. We study optimal regularity, nondegeneracy, and density properties of local minimizers. Via a blow-up analysis based on a Weiss type monotonicity formula, we show that the free boundary is the union of a regular and a singular part. Finally we use a viscosity approach to prove $C^{1,alpha}$ regularity of the regular part of the free boundary.
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