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
\int_0^t W(s)\circ dW(t)\;.
we then define the integral buy
\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)
where \(t_k^n=k\frac{t}n\). Prove that with probability one
X_t= \int_0^t W(s)\circ dW(s)= \frac12 W(t)^2\;.
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

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

  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 ?