Stopping Times: Information Available at the Decision Point
Summary
The document introduces stopping times using a filtration to represent information available over time. A decision time is a stopping time when whether it has occurred by time t can be determined from information available by t. It applies this idea to stock-price rules, including selling at a price threshold, acting at a fixed date, or using whichever of a threshold and a deadline comes first. These rules depend only on current or past information. Selling at the stock’s eventual maximum requires future knowledge, so it is not a stopping time.
The answer also distinguishes a rule that waits indefinitely for the investment to double from one that imposes a deadline. Without a deadline, the doubling event may never occur, so the time can be infinite. The discussion is a brief conceptual example rather than a full treatment of stopping-time theory. Its stated requirement that stopping times be finite almost surely is an extra assumption in this presentation; in standard definitions, stopping times may take the value infinity.
Key ideas
- A stopping time is determined by information available up to the time the decision is made.
- Price-threshold rules and fixed deadlines can be expressed as stopping times.
- A rule based on the stock’s eventual maximum depends on future information.
- A threshold rule without a deadline may never trigger and can have an infinite stopping time.
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Full text
# Properties of Stochastic Processes : Stopping times
# Properties of Stochastic Processes : Stopping times
Consider the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and the filteration $(\mathcal{F}_t)_{t\geq 0}$ i-e $$\mathcal{F}_t \subset \mathcal{F}_s \subset \mathcal{F}, \forall{t} < s $$
Assume that the decision to stop playing a game before or at time t is determined by the information $\mathcal{F}_t$ available at time t. Then this decision can be modelled by a random variable $\tau : \Omega \to [0,\infty]$ which satisfies $$ \{ \omega; \tau(\omega)\leq t\}\in \mathcal{F}_t$$ Two conditions must be satisfied to have random variable $\tau$ be the stopping time first $P(\tau =\infty)=0$ and second one $P(\tau < \infty)=1$
Let $\mathcal{F}_t$ be the information available until time t regarding the evolution of a stock price.
Assume the price of the stock at time t = 0 is $\\\$50$ per share. Which of the following decisions are stopping times and which are not, with the appropriate reasons thereof?
(a) Sell the stock when it reaches for the first time the price of $\\\$100$ per share;
(b) Buy the stock when it reaches for the first time the price of $\\\$10$ per share;
(c) Sell the stock at the end of the year;
(d) Sell the stock either when it reaches for the first time $\\\$80$ or at the end of the year.
(e) Keep the stock either until the initial investment doubles or until the end of the year;
(f) Sell the stock when it reaches the maximum level it will ever be;
(g) Keep the stock until the initial investment at least doubles.
My answer: Decisions given in a, b, c, d, e are stopping times. Decisions given in f and g are not stopping times.
Would you find out what would be the reasons thereof? 🤔
## Answer by Winodd Dhamnekar (score 0, accepted)
https://quant.stackexchange.com/a/80266
For all decisions given in (a), (b),(c),(d),(e), both the conditions are met for $\tau$ ,a random variable be a stopping time.
The decision in (f),requires information about future that is not contained in $\mathcal{F}_t$.Hence it is not a stopping time.
For decision given in (g),since the initial stock price is
$ S_0= \\\$ 50 $
the general theory of stock prices state,
$ P(S_t \geq 2S_0) = P(S_t \geq \\\$ 100) < 1 $
i-e there is + ve probability that stock never doubles its value. This contradicts the condition $P(\tau =\infty)=0$ Hence it is not a stopping time.
For decision in (e) , there are two conditions; the latter one has the occurring probability equal to one. i-e $P(\tau <\infty)=1$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.