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a b s t r a c tThis paper works out fair values of the stock loan model with automatic termination clause, cap and margin. This stock loan is treated as a generalized perpetual American option with possibly negative interest rate and some constraints. Since it helps a bank to control the risk, the banks charge lower service fees compared to stock loans without any constraints. The automatic termination clause, cap and margin are in fact a stop order set by the bank. Mathematically, it is a kind of optimal stopping problem arising from the pricing of financial products which is first revealed. We aim at establishing explicitly the value of such a loan and ranges of fair values of key parameters : this loan size, interest rate, cap, margin and fee for providing such a service and quantity of this automatic termination clause and the relationships among these parameters as well as the optimal exercise times. We present numerical results and make analysis about the model parameters and how they impact on value of stock loan.

This paper studies the retirement decision, optimal investment and consumption strategies under habit persistence for an agent with the opportunity to design the retirement time. The optimization problem is formulated as an interconnected optimal stopping and stochastic control problem (Stopping-Control Problem) in a finite time horizon. The problem contains three state variables: wealth x, habit level h and wage rate w. We aim to derive the retirement boundary of this "wealth-habit-wage" triplet (x, h, w). The complicated dual relation is proposed and proved to convert the original problem to the dual one. We obtain the retirement boundary of the dual variables based on an obstacle-type free boundary problem. Using dual relation we find the retirement boundary of primal variables and feed-back forms of optimal strategies. We show that if the so-called "de facto wealth" exceeds a critical proportion of wage, it will be optimal for the agent to choose to retire immediately. In numerical applications, we show how "de facto wealth" determines the retirement decisions and optimal strategies. Moreover, we observe discontinuity at retirement boundary: investment proportion always jumps down upon retirement, while consumption may jump up or jump down, depending on the change of marginal utility. We also find that the agent with higher standard of life tends to work longer.

This paper considers time‐inconsistent problems when control and stopping strategies are required to be made simultaneously (called stopping control problems by us). We first formulate the time‐inconsistent stopping control problems under general multidimensional controlled diffusion model and propose a formal definition of their equilibria. We show that an admissible pair false(trueû,Cfalse)$(\hat{u},C)$ of control‐stopping policy is equilibrium if and only if the auxiliary function associated with it solves the extended HJB system, providing a methodology to verify or exclude equilibrium solutions. We provide several examples to illustrate applications to mathematical finance and control theory. For a problem whose reward function endogenously depends on the current wealth, the equilibrium is explicitly obtained. For another model with a nonexponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general nonconstant equilibrium exists and is described by singular boundary value problems. This example shows that considering our combined problems is essentially different from investigating them separately. In the end, we also provide a two‐dimensional example with a hyperbolic discount.

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