# Tag Archives: JCM_math545_HW3_S23

## Practice with Ito Formula

Let $$B_t$$  be a standard Brownian motion. For each of the following definitions of  $$Y_t$$, find adapted stochastic process $$\mu_t$$ and $$\sigma_t$$ so that $$dY_t =\mu_t dt + \sigma_t dB_t$$

1. $$Y_t =\sin(B_t)$$
2. $$Y_t= (B_t)^p$$ for $$p>0$$
3. $$Y_t=\exp( B_t – t^2)$$
4. $$Y_t=\log(B_t)$$
5. $$Y_t= t^2 B_t$$

## Practice with Ito and Integration by parts

Define

$X_t =X_0 + \int_0^t B_s dB_s$

where $$B_t$$ is a standard Brownian Motion. Show that $$X_t$$ can also be written

$X_t=X_0 + \frac12 (B^2_t -t)$

## Covariance of Ito Integrals

Let $$f_t$$ and $$f_t$$ be two stochastic processes adapted to a filtration $$\mathcal F_t$$ such that

$\int_0^\infty \mathbf E (f_t^2) dt < \infty \qquad \text{and} \qquad \int_0^\infty \mathbf E (g_t^2) dt < \infty$

Let $$W_t$$ be a standard brownian motion  also adapted to the filtration $$\mathcal F_t$$ and define the stochastic processes

$X_t =\int_0^t f_s dW_s \qquad \text{and} \qquad Y_t=\int_0^t g_s dW_s$

Calculate the following:

1. $$\mathbf E (X_t X_s )$$
2. $$\mathbf E (X_t Y_t )$$
Hint: You know how to compute $$\mathbf E (X_t^2 )$$ and $$\mathbf E (Y_t^2 )$$. Use the fact that $$(a+b)^2 = a^2 +2ab + b^2$$ to answer the question. Simplify the result to get a compact expression for the answer.
3. Show that if $$f_t=\sin(2\pi t)$$ and $$g_t=\cos(2\pi t)$$ then $$X_1$$ and $$Y_1$$ are independent random variables.(Hint: use the result here  to deduce that $$X_1$$ and $$Y_1$$ are mean zero gaussian random variables. Now use the above results to show that the covariance of $$X_1$$ and $$Y_1$$ is zero. Combining these two facts implies that the random variables are independent.)

## Solving a class of SDEs

Let us try a systematic procedure for solving SDEs which works for a class of SDEs. Let
\begin{align*}
X(t)=a(t)\left[ x_0 + \int_0^t b(s) dB(s) \right] +c(t) \ .
\end{align*}
Assuming $$a$$, $$b$$, and $$c$$ are differentiable, use Ito’s formula to find the equation for $$dX(t)$$ of the form
\begin{align*}
dX(t)=[ F(t) X(t) + H(t)] dt + G(t)dB(t)
\end{align*}
were $$F(t)$$, $$G(t)$$, and $$H(t)$$ are some functions of time depending on $$a,b$$ and maybe their derivatives. Solve the following equations by matching the coefficients. Let $$\alpha$$, $$\gamma$$ and $$\beta$$ be fixed numbers.

Notice that
\begin{align*}
X(t)=a(t)\left[ x_0 + \int_0^t b(s) dB(s) \right] +c(t)=F(t,Y(t)) \ .
\end{align*}
where $$dY(t)=b(t) dB(t)$$. Then you can apply Ito’s formula to this definition to find $$dX(t)$$.

1. First consider
$dX_t = (-\alpha X_t + \gamma) dt + \beta dB_t$
with $$X_0 =x_0$$
. Solve this for $$t \geq 0$$
2. Now consider
$dY(t)=\frac{\beta-Y(t)}{1-t} dt + dB(t) ~,~~ 0\leq t < 1 ~,~~Y(0)=\alpha.$
Solve this for $$t\in[0,1]$$.
3. \begin{align*}
dX_t = -2 \frac{X_t}{1-t} dt + \sqrt{2 t(1-t)} dB_t ~,~~X(0)=\alpha
\end{align*}
Solve this for $$t\in[0,1]$$.

## Around the Circle

Consider the equation
\begin{align}
dX_t &= -Y_t dB_t – \frac12 X_t dt\\
dY_t &= X_t dB_t – \frac12 Y_t dt
\end{align}
Let $$(X_0,Y_0)=(x,y)$$ with $$x^2+y^2=1$$. Show that $$X_t^2 + Y_t^2 =1$$ for all $$t$$ and hence the SDE lives on the unit circle. Does this make intuitive sense ?