July 24th, 2019, 12:57 am
It's usually solved backwards because of the "principle of optimality", which you can google if you don't know it.
The standard finance example is determining an optimal exercise strategy for an American-style put. It's day 1 and the put expires on day N. Whatever the "strategy" on day 1, it should be optimal in light of what you are going to do on day 2, day 3, etc. How to account for all that, which at first glance seems hopelessly complicated?
Well, you know what you are going to do on expiration = day N. You'll exercise if the option is in-the-money. So, it's easiest to start by calculating what you will do on day N-1. What will you do?. Conditional on the state (say just the day N-1 stock price) you first calculate the (discounted) expected value of the option on expiration. Then, if exercising will get you more than that expectation, you exercise; otherwise you wait till expiration and take your chances on the random draw of the terminal stock price. Doing that calculation will tell you the optimal exercise policy for day N-1 (for every possible state).
Then, you step back another day and repeat. The net result at the end (having finally calculated down to day 1) is that you have indeed satisfied the "principle of optimality".
Only the terminal condition is known before solving the problem. But, you can certainly relabel the day counting as I said in previous post, so that the terminal condition becomes an initial condition. The important thing is to appreciate why the problem proceeds in the normal way, backwards from expiration. The relabeling of the day count is just for possible convenience -- say of a computer program.