Sequential-offer stopping problems

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Problem 12.9.19 from Grimmett G.R., Stirzaker D.R. "Probability and Random Processes" [GrSt20]More problems in Section 12.9 "Problems"

(a)(b)(c)

A businessman has a redundant piece of equipment which he advertises for sale, inviting "offers over $£ 1000$ ". He anticipates that, each week for the foreseeable future, he will be approached by one prospective purchaser, the offers made in week $0,1, \ldots$ being $£ 1000 X_{0}, £ 1000 X_{1}, \ldots$ , where $X_{0}, X_{1}, \ldots$ are independent random variables with a common density function $f$ and finite mean. Storage of the equipment costs $£ 1000$ c per week and the prevailing rate of interest is $\alpha ( > 0)$ per week.

(a)

Explain why a sensible strategy for the businessman is to sell in the week $T$ , where $T$ is a stopping time chosen so as to maximize

\mu (T) = \mathbb {E}\left[(1+\alpha )^{-T} X_{T}-\sum _{n = 1}^{T}(1+\alpha )^{-n} c\right]

Show that this problem is equivalent to maximizing $\mathbb {E}\left[(1+\alpha )^{-T} Z_{T}\right]$ where $Z_{n} = X_{n}+c / \alpha$ .

(b)

Show that there exists a unique positive real number $\gamma$ with the property that

\alpha \gamma = \int _{\gamma }^{\infty } \mathbb {P}\left(Z_{n} > y\right) d y

and that, for this value of $\gamma$ , the sequence $V_{n} = (1+\alpha )^{-n} \max \left\{ Z_{n}, \gamma \right\}$ constitutes a supermartingale. Deduce that the optimal strategy for the businessman is to set a target price $\tau$ (which you should specify in terms of $\gamma$ ) and sell the first time he is offered at least this price.

(c)

In the case when $f(x) = 2 x^{-3}$ for $x \geq 1$ , and $c = \alpha = \frac{1}{90}$ , find his target price and the expected number of weeks he will have to wait before selling.

References

[GrSt20] Grimmett G.R., Stirzaker D.R. Probability and Random Processes. 4th ed., Oxford University Press, 2020.