House-selling is one of the typical tasks of the Optimal Stopping problems. Offers come in daily for an asset, such as a house, that you wish to sell. Let Xi denote the amount of the offer received on day i. X1,X2,... are independent random variables, according uniform distribution on the interval (0...1). Each offer costs an amount C>0 to observe. When you receive an offer Xi, you must decide whether accept it or to wait for a better offer. The reward sequence depends on whether or not recall of past observations is allowed. If you may not recall past offers, then Di(X1,...,Xi)=Xi – i*C. If you are allowed to recall past offers, then Di(X1,...,Xi)=max(X1,...,Xi) – i*C. These tasks may be extended to infinite horizon (i is unlimited). So, there 4 different task statements :
- without recall, infinite horizon
- without recall, finite horizon
- with recall, infinite horizon
- with recall, finite horizon
First three tasks are quite simple, but I was unable to prove solution of the last task (in strict form, although I found a solution). If anyone knows her solution, please write it or send an article (link to the article) where it is written. Thank you in advance.
Hello, I would suggest to first understand the problem statement. The bid prices cannot be uniformly distributed on the interval from 0 to 1. The bid prices have a typical kind of distribution (log-normal). You can study it at once. This will affect the solutions you get. I see no need to formalise the price range to the interval 0,1. That's really my personal non-binding opinion
M. B. Laskin , Hello and thanks a lot for your answer. From practical point of view, you are fully right. Certainly, we can use Monte-Carlo simulation and to get solution for any distributions and any concrete values of N and C. But I want to get some analytical solution. For other (first three) conditions there are known analytical results - see, e.g. "Thomas S. Ferguson. Optimal Stopping and Applications". Assumption "uniform distribution [0...1]" is good and generally accepted initial point for these problems.
Hello, Sergey! Sorry, I don't understand what is the problem to use standard approach for such tasks - dynamic programming ?
Dear Alex, you are fully right - it is possible. But I want and I am going to get analytical expressions for thresholds - as for other 3 cases.
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