Stopping time

In the stochastics of the concept of stopping time refers to a special type of random variables that are defined on the filtered probability space. Stop times are not only of importance for the theory of stochastic processes ( for example, in the localization of process classes ), but also of practical importance, for example the problem of the optimal exercise time for American options.

Definition

Be a filtered probability space, ie a probability space with a filtration. A non-negative random variable is called stopping time if

A stop time can be interpreted as the waiting time that elapses until a certain random event occurs. If, as usual, indicates the presence of the filtering information at different times, means the above condition so that at any time should be known whether the event has already occurred or not.

Stopped process

For a stopping time T and a given process can be easily a new process, the process stopped, as

Define. In this case, refers to the minimum and, as well, the indicator function is set A. Clearly speaking, the process runs as long as defined by, until the stop time is reached. From this point, the process is stopped and reserves for all future times the value of the stop time.

For stopped processes can, for example, the optional sampling theorem proving, they are also of importance for the definition of local properties for stochastic processes.

Examples

• A gambler begins at t = 0 with an initial investment of € 10 to play; while he attended every minute of a game ( the sake of simplicity itself takes no time), in which he wins a € 50 - percent probability, and otherwise a Euro loses ( the player's account balance is then a martingale ). The waiting time until the player has lost all his money, then as a stopping time with respect to the natural filtration of the experiment: at any time the player knows whether he is bankrupt already or not. In contrast, the waiting time would be up to the moment of his penultimate game does not stop time: at the moment, since you completed his penultimate game, you do not know yet that the next game will be the last.

• The meeting time ( hitting time) of a Wiener process with drift μ to level a is distributed according to an inverse Gaussian distribution. The density is

• A generalization of the simple example from above: is a real-valued, adapted càdlàg process ( ie a stochastic process whose paths are all right-continuous and left-sided convergent ) and a closed set, then the meeting time of X in A, defined as

Calculation rules

There were and stopping times with respect to a filtration. Then we have

• The minimum is one - stop time.
• The maximum is one - stop time.
• Is a time to stop.
• Is a stopping time, where is a fixed constant.
• Is a stopping time
• Is a time to stop.
• Is a time to stop.
• Is a time to stop.