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## Friday, March 13, 2015

### stoch integral - bets on each step of a random walk

label - intuitive

Gotcha -- In ordinary integration, if we integrate from 0 to 1, then dx is always a positive "step". If the integrand is positive in the "strip", then the area is positive. Stoch integral is different. Even if integrand is always positive the strip "area" can be negative because the dW is a coin flip.

Total area is a RV with Expectation = 0.
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In Greg Lawler's first mention (P 11) of stoch integral, he models the integrand's value (over a brief interval deltaT) as a "bet" on a coin flip, or a bet on a random walk. I find this a rather intuitive, memorable, and simplified description of stoch integral.

Note the coin flip can be positive or negative and beyond our control. We can bet positive or negative. The bet can be any value. For now, don't worry about the magnitude of the random step. Just assume each step is +1/-1 like a coin flip

If the random walk has no drift (fair coin), then any way you bet on it, you are 50/50 i.e. no way to beat a martingale. Therefore, the integral is typically (expected to be) 0. Let's denote the integral as C. What about E[C2] ? Surely positive.  We need the variance rule...

Q: Does a stoch integral always have expectation equal to last revealed value of the integral?
A: Yes. It is always a local martingale. If it's bounded, then it's also a martingale.