📂Stochastic Differential Equations

Isometric Equality of Ito

Theorem ¹

For all $f \in m^{2}[a,b]$, the following equation holds. $$E \left[ \left( \int_{a}^{b} f d W_{t} \right)^{2} \right] = E \left[ \int_{a}^{b} f^{2} dt \right]$$

Explanation

While it is correct that the power outside of the integral sign $^{2}$ crosses over, attention should also be paid to the change in the integrands $d W_{t}$ and $dt$.

Proof ¹

Strategy: Since it suffices to show for sequences $\left\{ \phi_{n} \right\}_{n \in \mathbb{N}}$ in elementary processes given the definition of Itô integration, it naturally generalizes to $\phi_{n} \to f \in m^{2}$, it is enough to think only of the elementary processes $\phi_{n}$. Let’s fix one $n_{0} \in \mathbb{N}$, and set it as $\phi := \phi_{n_{0}}$.

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$$\phi (t, \omega) := \sum_{j=0}^{k-1} e_{j} (\omega) \chi_{[t_{j}, t_{j+1})} (t) \qquad , a = t_{0} < \cdots < t_{k} = b$$ Let’s say a bounded elementary process $\phi$ appears as above.

If we set as $\Delta W_{j} := W_{t_{j+1}} - W_{t_{j}}$ then since it is a Wiener process $$E \left[ e_{i} e_{j} \Delta W_{i} \Delta W_{j} \right] = \begin{cases} 0 & , \text{if } i \ne j \\ E \left[ e_{j}^{2} \right] \cdot \left( t_{j+1} - t_{j} \right) & , \text{if } i = j \end{cases}$$ Furthermore, if $i \ne j$, then $\Delta W_{i} \perp \Delta W_{j}$ hence $$\begin{align*} E \left[ \left( \int_{a}^{b} \phi d W_{t} \right)^{2} \right] =& \sum_{i,j} E \left[ e_{i} e_{j} \Delta W_{i} \Delta W_{j} \right] \\ =& \sum_{j} E \left[ e_{j}^{2} \right] \left( t_{j+1} - t_{j} \right) \\ =& E \left[ \int_{a}^{b} \phi^{2} dt \right] \end{align*}$$

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