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 \(\mathcal{F}_t\).

# Tag Archives: JCM_math545_HW2_S23

## Stratonovich integral: A first example

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## 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

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

- 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

\[ \int_0^T \phi(t)^2 dt \] - * 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.

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## 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 \(x\).

- Explain why \[ u(x,t) = \int_{\infty}^\infty f(y)p(t,x-y)dy\]
- 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 \(c\) !) - 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 \(f\).

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## 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

- \[\lim_{N\rightarrow \infty}\mathbf E I_N^\alpha(t) \ .\]
- * \[\lim_{N\rightarrow \infty}\mathbf E \big( I_N^\alpha(t)\big)^2\]
- * 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 ?

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