SDE Notation: Raising quantities to the "+" and "-"

Question

I apologize in advance for the broad question. I have been searching for some time and cannot wrap my head around this notation for Stochastic Differential Equations... I have seen it in some proofs and most recently in a homework problem I am trying to tackle.

In some solutions to SDE problems, an expression is split into components, one with a "-" exponent and one with a "+" exponent... In my current homework problem the following expression is defined:

\begin{equation*} F(\omega) = (B_T - K)^+ \end{equation*}

In this context, K>0 and T>0 are constants and Bt is one-dimensional Brownian motion.. can somebody please explain to me these types of operators and how they can be used to solve SDE-related problems. My biggest problem is that I don't even know what this is called and searching my textbook has not helped.

Thanks!

Answer 1

Typically,

$$ x^+ = \max \{ x,0 \} $$

and

$$ x^- = \max \{-x,0 \}.$$

That is, $x^+$ and $x^-$ are the positive and negative parts of $x$ respectively. Note that by this definition, $x^- \ge 0$, so that we can write

$$ x = x^+ - x^- $$

and

$$ |x| = x^+ + x^- . $$