I'm reading the textbook Stochastic Calculus for Finance: Volume I by Steven Shreve, and on page 101 (incase somebody has it on hand) I am unsure of the notation.
There is a I where the top and bottom horizontal lines are connected by two vertical lines, instead of just one. There is then a subscript saying {τ<=N}, denoting that this mystery operator applies to all tau less than or equal to N.
The I looks very similar to the E meaning expectation and P meaning probability measure, in that there is a doplication of vertical lines to make the letter look hollow.
What is this mystery operator?
Best Answer
I what you are asking about in fact is an indicator function. Consider a set $A\subset \mathbb{R}^n$, then the indicator function for the set $A$ as a function of $x\in\mathbb{R}^n$, is defined as
$$ \mathbb{1}_{A}\left(x\right) := \begin{cases} 1,\qquad \forall x\in A,\\ 0,\qquad \forall x\not\in A. \end{cases} $$
This is very useful in many areas of mathematics, and not least within stochastic calculus which is used in financial mathematics.