The Itô integral

Adding up the gains of a trader

Think of a trader who holds $f_i$ shares during the $i$ -th small time step. The number is chosen at the start of the step. The gain over that step is $f_i\,\Delta W_i$ , the holding times the random move. Adding all the gains and letting the steps shrink gives the Itô integral. The key rule is that $f$ is read at the start of each step, never at the end, because nobody knows the next move in advance. This is what makes the integral a fair game: its mean is $0$ .

\int_0^T f\,dW=\lim\sum_i f(t_i)\,(W_{t_{i+1}}-W_{t_i}),\qquad E\Big[\int_0^T f\,dW\Big]=0,\qquad E\Big[\Big(\int_0^T f\,dW\Big)^{2}\Big]=E\int_0^T f^{2}\,dt

(1)

The last formula is the Itô isometry. It turns the variance of a random gain into an ordinary integral of $f^{2}$ , which you can compute. Reading $f$ at the start of the step also changes some answers. For example $\int_0^T W\,dW=(W_T^{2}-T)/2$ , not $W_T^{2}/2$ as ordinary calculus would say. The missing $-T/2$ comes from the quadratic variation of the last topic.

Three shares for four yearsHold

f=3

shares all the time, so

\int_0^4 3\,dW=3W_4

. The variance of

W_4

4

, so the gain has variance

9\times4=36

. The isometry gives the same number without any random variable:

\int_0^4 3^{2}\,dt=36

Itô integral: the sum of holding times move, with the holding read at the start of each step.
Mean and isometry: the mean is $0$ and the variance is $E\int f^{2}\,dt$ .
Extra term: $\int W\,dW=(W_T^{2}-T)/2$ , with the $-T/2$ coming from quadratic variation.

Lab · what's graded

A variance from the isometry15 pts
The integral of W against W15 pts
A discrete trader's gain15 pts
A holding that grows with time15 pts
Why read the holding at the start15 pts
Match the formulas10 pts
Why the mean is zero15 pts

Adding up the gains of a trader

Lab · what's graded

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The Itô integral

Adding up the gains of a trader

Lab · what's graded

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