Riemann ^ stoch integral, learning notes

In a Riemann integral, each strip has an area-under-the-curve being either positive or negative, depending on the integrand’s sign in the strip. If the strip is “under water” then area is negative.

In stochastic integral [1], each piece is “increment   *   integrand”, where both increment and integrand values can be positive/negative. In contrast, the Riemann increment is always positive.

With Riemann, if we know integrand is entirely positive over the integration range, then the sum must be positive. This basic rule doesn’t apply to stochastic integral. In fact, we can’t draw a progression of adjacent strips as illustration of stochastic integration.

Even if the integrand is always positive, the stoch integral is often 0. For an (important) example, in a fair game or a drift-less random walk, the dB part is 50-50 positive/negative.

[1] think of the “Simple Process” defined on P82 by Greg Lawler.

On P80, Greg pointed out
* if integrand is random but the dx is “ordinary” then this is an ordinary integral
* if the dx is a coin flip, then whether integrand is random or not, this is a stoch integral

So the defining feature of a stoch integral is a random increment

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