How to comprehend this notation?

Question

I learned mathematical finance from Bjork's Arbitrage Theory in Continous Time, and never once did I encounter the "quadratic variation"-thingy with the angle brackets.

So now that I am reading Bergomi's book on Stochastic Volatility and I run into this monster on the first chapter, you can understand my confusion:

Please explain whats' going on here. What is an "average covariation"? I cannot find this on wikipedia. I found what a "quadratic covariation" is, but what does it mean intuitively, especially in this context?

In this context, Bergomi says that he wants to equate implied volatility the future realized volatility. Okay, so I get that the implied volatility is hat-sigma and realized volatility is sigma, and he is weighting them by the "dollar gamma" and then he takes an integral because he wants the average over the period [0, T]. Cool .... but why does he then end by taking those angle-brackets? Why not just equate the two integrals? Why is equating the "covariations" or whatever it is necessary here?

Answer 1

In physics (statistical physics), this angle bracket is used to represent average, for example, here is the notation from Van Kampen’s book:

And in stochastic calculus, the quadratic variation is usually represented by the same angle brackets. But like he noted the context should make clear which one is meant.

In the equation you have referenced, an average is meant. So the thing inside the brackets is the P&L of a path, and the angle brackets is then computing the average across the paths.