# Tag Archives: JCM_math545_HW2_S23

## Stratonovich integral: A first example

Let us denote the Stratonovich integral of a standard Brownian motion $$W(t)$$ with respect to itself by
\begin{align*}
\int_0^t W(s)\circ dW(t)\;.
\end{align*}
we then define the integral buy
\begin{align*}
\int_0^t W(s)\circ dW(t) = \lim_{n \rightarrow \infty}
\sum_k\frac12\big(W(t_{k+1}^n)+W(t_{k}^n)\big)\big(W(t_{k+1}^n) -W(t_{k}^n)\big)
\end{align*}
where $$t_k^n=k\frac{t}n$$. Prove that with probability one
\begin{align*}
X_t= \int_0^t W(s)\circ dW(s)= \frac12 W(t)^2\;.
\end{align*}
Observe that this is what one would have if one used standard (as opposed to Ito) calculus. Calculate $$\mathbf E [ X_t | \mathcal{F}_s]$$ for $$s < t$$ where $$\mathcal{F}_t$$ is the $$\sigma$$-algebra generated by the Brownian motion. Is $$X_t$$ a martingale with respect to $$\mathcal{F}_t$$.

## Gaussian Ito Integrals

In this problem, we will show that the Ito integral of a deterministic function is a Gaussian Random Variable.

Let $$\phi$$ be deterministic elementary functions. In other words there exists a  sequence of  real numbers $$\{c_k : k=1,2,\dots,N\}$$ so that

$\sum_{k=1}^\infty c_k^2 < \infty$

and there exists a partition

$0=t_0 < t_1< t_2 <\cdots<t_N=T$

so that

$\phi(t) = \sum_{k=1}^N c_k \mathbf{1}_{[t_{k-1},t_k)}(t)$

1. Show that if $$W(t)$$ is a standard brownian motion then the Ito integral
$\int_0^T \phi(t) dW(t)$
is a Gaussian random variable with mean zero and variance
$\int_0^T \phi(t)^2 dt$
2. * Let $$f\colon [0,T] \rightarrow \mathbf R$$ be a deterministic function such that
$\int_0^T f(t)^2 dt < \infty$
Then it can be shown that there exists a sequence of  deterministic elementary functions $$\phi_n$$ as above such that
$\int_0^T (f(t)-\phi_n(t))^2 dt \rightarrow 0\qquad\text{as}\qquad n \rightarrow \infty$
Assuming this fact, let $$\psi_n$$ be the characteristic function  of the random variable
$\int_0^T \phi_n(t) dW(t)$
Show that for all $$\lambda \in \mathbf R$$, show that
$\lim_{n \rightarrow \infty} \psi_n(\lambda) = \exp \Big( -\frac{\lambda^2}2 \big( \int_0^T f(t)^2 dt \big) \Big)$
Then use the the convergence result here to conclude that
$\int_0^T f(t) dW(t)$
is a Gaussian Random Variable with mean zero and variance
$\int_0^T f(t)^2 dt$
by identifying the limit of the characteristic functions above.

Note: When Probabilistic say the “characteristic function” of a random distribution they just mean the Fourier transform of the random variable. See here.

## Diffusion and Brownian motion

Let $$B_t$$ be a standard Brownian Motion  starting from zero and define

$p(t,x) = \frac1{\sqrt{2\pi t}}e^{-\frac{x^2}{2t} }$

Given any $$x \in \mathbf R$$, define $$X_t=x + B_t$$ . Of course $$X_t$$ is just a Brownian Motion stating from $$x$$ at time 0. Fixing a smooth, bounded, compactly supported function $$f:\mathbf R \rightarrow \mathbf R$$, we define the function $$u(x,t)$$ by

$u(x,t) = \mathbf E_x f(X_t)$

where we have decorated the expectation with the subscript $$x$$ to remind us that we are starting from the point $$x$$.

1. Explain why $u(x,t) = \int_{\infty}^\infty f(y)p(t,x-y)dy$
2. Show by direct calculation using the formula from the previous question that for $$t>0$$, $$u(x,t)$$ satisfies the diffusion equation
$\frac{\partial u}{\partial t}= c\frac{\partial^2 u}{\partial x^2}$
for some constant $$c$$. (Find the correct $$c$$ !)
3. Again using the formula from part 1), show that
$\lim_{t \rightarrow 0} u(t,x) = f(x)$
and hence the initial condition for the diffusion equation is $$f$$.

## Ito to Stratonovich

Let’s think about different ways to make sense of $\int_0^t W(s)dW(s)$ were $$W(t)$$ is a standard Brownian motion. Fix any $$\alpha \in [0,1]$$define

\begin{equation*}
I_N^\alpha(t)=\sum_{j=0}^{N-1} W(t_j^\alpha)[W(t_{j+1})-W(t_j)]
\end{equation*}
were $$t_j=\frac{j t}N$$ and $$t_j^\alpha=\alpha t_j + (1-\alpha)t_{j+1}$$.
Calculate

1. $\lim_{N\rightarrow \infty}\mathbf E I_N^\alpha(t) \ .$
2. * $\lim_{N\rightarrow \infty}\mathbf E \big( I_N^\alpha(t)\big)^2$
3. * For which choice of $$\alpha$$ is $$I_N^\alpha(t)$$ a martingale ?

What choice of $$\alpha$$ is the standard It\^o integral ? What choice is the Stratonovich integral ?