Formalizing Price Thresholds as Stopping-Time Events

Summary

The document asks how to express two events involving an adapted stock-price process: reaching a fixed price level and reaching the process’s overall maximum. It correctly frames the issue through the stopping-time condition that whether a stopping time has occurred by time t must be knowable from information available at t. The author’s proposed notation compares the current price with a level or with a maximum taken over the entire future, and asks whether those events belong to the information set at time t.

No answer or formal solution is included, so the document does not establish the precise hitting-time definitions or the assumptions needed for the maximum example. Its value is as a concise prompt about the difference between events observable from current and past data and events that depend on future prices. The notation also leaves open whether the process is discrete or continuous and how equality or crossing of the level is treated.

Key ideas

• A stopping time is defined by whether its occurrence by time t is measurable using information available at t.
• A fixed price threshold can be expressed through a first hitting time of the adapted price process.
• The process’s maximum over all future times depends on information unavailable at the present time.
• The document poses the formalization question but supplies no answer or complete definitions.
• Time domain and whether threshold crossing or exact equality is intended affect the formal statement.

Tags

# Mathematical writing, stopping time event


# Mathematical writing, stopping time event












Given that I am not a mathematician, I'd like some help to write in the correct way this two events.

I know that $\begin{Bmatrix} {\tau(\omega)\leq t} \end{Bmatrix} \in \Im_t$ is the definition of stopping time, and I have this two examples:

- the price of a share reached $20$ (this is a stopping time).

- the price of a share reached its maximum (this is not a stopping time).

Knowing that the price of the share is $\Im_t$-adapted process $\begin{Bmatrix} {S_t} \end{Bmatrix}_{t \in [0,+\infty)}$, how can I "formalize" the examples above?

My attempts:

- $A=\begin{Bmatrix} S_t=20 \end{Bmatrix} \in \Im_t$;

- $B=\begin{Bmatrix} S_t=\operatorname{max}\begin{Bmatrix} S_t \end{Bmatrix}_{t \in [0,+\infty)} \end{Bmatrix} \in \Im_t$.

But I suspect they are wrong. Could you help me?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.