Moments of Ito Integrals for Weak Solutions a) Use the Ito isometry E b

Moments of Itˆo Integrals for Weak Solutions

a) Use the Itˆo isometry E

⎡

⎣

 b a

f(t,ω)dWt 2

⎤

⎦ =

 b a

E



f 2(t,ω)



dt to show its generalization Exercise 3.5 By transformation of two independent standard normally distributed random varables Zi ∼ N (0, 1), i = 1, 2, two new random variables are obtained by ΔW  := Z1 √
Δt, ΔY := 1 2 (Δt)
3/2 
Z1 +
1 √3 Z2 
.
Show that ΔW  and ΔY have the moments of (3.14).
Exercise 3.6 In addition to (3.14) further moments are E(ΔW) = E(ΔW3) = E(ΔW5)=0, E(ΔW2) = Δt, E(ΔW4)=3Δt2.
Assume a new random variable ΔW  satisfying P 
ΔW  = ±
√
3Δt
= 1 6 , P 
ΔW  = 0
= 2 3 and the additional random variable ΔY := 1 2 ΔWΔt . 
Show that the random variables ΔW  and ΔY have up to terms of order O(Δt3) the same moments as ΔW and ΔY .