Riemann surface
matlab

Problem 1. Calculate the variance of the Ito integral $\int_0^T W_t d W_t$.
Hint #1: you may use without proof the Ito isometry, which says that
$$\mathbb{E}\left(\left(\int_0^T \Delta_t d W_t\right)^2\right)=\mathbb{E} \int_0^T \Delta_t^2 d t .$$
Hint #2 (alternative to Hint #1): You may start by using Ito’s formula to calculate $d\left(W^2\right)$.

Solution: $\Delta=W$ and $\int \mathbb{E}\left(W_s^2\right) d s=\int s d s=T^2 / 2$.
Alternatively,
$$d\left(W^2\right)=2 W d W+\frac{1}{2} 2 d t$$
so
$$W_T^2=\int 2 W d W+T$$
so
$$\int W d W=\frac{1}{2}\left(W_T^2-T\right)$$
and
$$\operatorname{Var} \int W d W=\operatorname{Var} \frac{1}{2}\left(W_T^2-T\right)=\frac{1}{4} \operatorname{Var}\left(W_T^2\right)=\frac{1}{4}\left(\mathbb{E}\left(W_T^4\right)-\left(E\left(W_T^2\right)\right)^2\right)$$
using the fourth moment of $N\left(0, \sigma^2\right)$ is $3 \sigma^4$,
$$=\frac{1}{4}\left(3 T^2-T^2\right)=T^2 / 2$$

Problem 2. (a) Solve the stochastic differential equation
$$d X_t=t d t-d W_t$$
(in other words, find $X_t$ ) by integrating both sides from 0 to $T$.
(b) Consider the stochastic differential equation
$$d Y_t=t d t+d W_t$$
What can be said about $X_t$ and $Y_t$, assuming $X_0=Y_0$ and the same Brownian motion $W_t$ is used to define both $X_t$ and $Y_t$ ? Justify.
(i) $X_t=Y_t$ almost surely?
(ii) $X_t$ and $Y_t$ are not equal almost surely, but they have the same probability distribution?
(iii) $X_t$ and $Y_t$ are not equal almost surely, and they also do not have the same probability distribution?
(c) Draw (sketch) what a typical random path of the solution $Y_t$ may look like in a $\left(t, Y_t\right)$ coordinate system.

SOLUTION: Take all of space, then remove the ball $x^2+y^2+$ $z^2 \leq 4$. Add back in the sphere $x^2+y^2+z^2=4$. This is the set (all of space with a spherical cavity).

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