A firm issues a convertible bond. At each subsequent time, the bondholder must decide whether to continue to hold the bond, thereby collecting coupons, or to convert it to stock. The bondholder wishes to choose a conversion strategy to maximize the bond value. Subject to some restrictions, the bond can be called by the issuing firm, which presumably acts to maximize the equity value of the firm...

We consider the problem of optimally stopping a general one-dimensional Itô diffusion X. In particular, we solve the problem that aims at maximising the performance criterion Ex [exp(− ∫ τ 0 r(Xs) ds)f (Xτ )] over all stopping times τ , where the reward function f can take only a finite number of values and has a ‘staircase’ form. This problem is partly motivated by applications to financial as...

Introduction Suppose you are determined to “prove” that green apples cause cancer. An Optional Stopping strategy (OS) is where you keep looking sampling experimental data until the observed correlation between eating green applies and cancer is significantly different from 0 (where “significantly” means that the null hypothesis is rejected by standard statistical tests). That is, you follow a r...

Much research has modeled action-stopping using the stop-signal task (SST), in which an impending response has to be stopped when an explicit stop-signal occurs. A limitation of the SST is that real-world action-stopping rarely involves explicit stop-signals. Instead, the stopping-system engages when environmental features match more complex stopping goals. For example, when stepping into the s...

We consider an optimal investment problem when a firm such as an electric power company has the operational flexibility to expand and contract capacity with fixed cost. This problem is formulated as an impulse control problem combined with optimal stopping. Consequently, we obtain optimal investment timing, optimal capacity expansion and contraction timing, and the investment value. We also sho...

We analyse optimal stopping when the economic environment changes because of learning. A primary application is optimal selling of an asset when demand is uncertain. The seller learns about the arrival rate of buyers. As time passes without a sale, the seller becomes more pessimistic about the arrival rate. When the arrival of buyers is not observed, the rate at which the seller revises her bel...

These notes are intended to accompany a Graduate course on Optimal stopping, and in places are a bit brief. They follow the book ‘Optimal Stopping and Freeboundary Problems’ by Peskir and Shiryaev, and more details can generally be found there.

We formulate a stochastic game of mean field type where the agents solve optimal stopping problems and interact through the proportion of players that have already stopped. Working with a continuum of agents, typical equilibria become functions of the common noise that all agents are exposed to, whereas idiosyncratic randomness can be eliminated by an Exact Law of Large Numbers. Under a structu...