Question

(i) Compute $d W^4(t)$ and then write $W^4(T)$ as the sum of an ordinary (Lebesgue) integral with respect to time and an Itô integral. (ii) Take expectations on both sides of the formula you obtained in (i), use the fact that $\mathbb{E} W^2(t)=t$, and derive the formula $\mathbb{E} W^4(T)=3 T^2$. (iii) Use the method of (i) and (ii) to derive a formula for $\mathbb{E} W^6(T)$.

    (i) Compute $d W^4(t)$ and then write $W^4(T)$ as the sum of an ordinary (Lebesgue) integral with respect to time and an Itô integral.
(ii) Take expectations on both sides of the formula you obtained in (i), use the fact that $\mathbb{E} W^2(t)=t$, and derive the formula $\mathbb{E} W^4(T)=3 T^2$.
(iii) Use the method of (i) and (ii) to derive a formula for $\mathbb{E} W^6(T)$.

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Stochastic Calculus for Finance II : Continuous-Time Models

Stochastic Calculus for Finance II : Continuous-Time Models

Steven E. Shreve 1st Edition

Chapter 4, Problem 7

Instant Answer

Step 1

Itô's Lemma states that if $f(t, W(t))$ is a twice continuously differentiable function in $W$ and once in $t$, then \[ df(t, W(t)) = \left(\frac{\partial f}{\partial t} + \frac{\partial f}{\partial W} dW(t) + \frac{1}{2} \frac{\partial^2 f}{\partial W^2} Show more…

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(i) Compute $d W^4(t)$ and then write $W^4(T)$ as the sum of an ordinary (Lebesgue) integral with respect to time and an Itô integral. (ii) Take expectations on both sides of the formula you obtained in (i), use the fact that $\mathbb{E} W^2(t)=t$, and derive the formula $\mathbb{E} W^4(T)=3 T^2$. (iii) Use the method of (i) and (ii) to derive a formula for $\mathbb{E} W^6(T)$.

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