Category Archives: Exponential Martingale

Homogeneous Martingales and Hermite Polynomials

1. Let $$f(x,y):\mathbb{R}^2 \rightarrow \mathbb{R}$$ be a twice differentiable function in both $$x$$ and $$y$$. Let $$M(t)$$ be defined by $M(t)=\int_0^t \sigma(s,\omega) dB(s,\omega)$. Assume that $$\sigma(t,\omega)$$ is adapted and that $$\mathbf{E} M^2 < \infty$$ for all $$t$$ a.s. .(Here $$B(t)$$ is standard Brownian Motion.) Let $$[M]_t$$ be the quadratic variation process of $$M(t)$$. What equation does $$f$$ have to satisfy so that $$Y(t)=f(M(t),[M]_t)$$ is again a martingale if we assume that $$\mathbf E\int_0^t \sigma(s,\omega)^2 ds < \infty$$.
2. Set
\begin{align*}
f_n(x,y) = \sum_{0 \leq m \leq \lfloor n/2 \rfloor} C_{n,m} x^{n-2m}y^m
\end{align*}
here $$\lfloor n/2 \rfloor$$ is the largest integer less than or equal to $$n/2$$. Set $$C_{n,0}=1$$ for all $$n$$. Then find a recurrence relation for $$C_{n,m+1}$$ in terms of $$C_{n,m}$$, so that $$Y(t)=f_n(B(t),t)$$ will be a martingale.Write out explicitly $$f_1(B(t),t), \cdots, f_4(B(t),t)$$ as defined in the previous item.
3. Again let $$M(t)=\int_0^t \sigma(s,\omega) dB(s,\omega)$$ with $$|\sigma(t,\omega)| < K$$ almost surely. Show that $$f_n(M(t),[M]_t)$$ is again a martingale where $$[M]_t$$ is the quadratic variation of $$M(t)$$ and $$f_n$$ is the function found above.
4. * Do you recognize the recursion relation you obtained above for $$f_n$$ as being associated to a famous recursion relation ? (Hint: Look at the title of the problem)

Exponential Martingale

Let $$\sigma(t,\omega)$$ be adapted to the filtration generated by a standard Brownian Motion $$B_t$$ such that $$|\sigma(x,\omega)| < K$$ for some  bound  $$K$$ . Let $$I(t,\omega)=\int_0^t \sigma(s,\omega) dB(s,\omega)$$.

1. Show that
$M_t=\exp\big\{\alpha I(t)-\frac{\alpha^2}{2}\int_0^t \sigma^2(s)ds \big\}$
is a martingale. It is called the  exponential martingale.
2. Show that $$M_t$$ satisfies the equation
$dM_t =\alpha M_t dI_t = \alpha M_t \sigma_t dB_t$

Exponential Martingale Bound

Let $$\sigma(t,\omega)$$ be nonanticipating with $$|\sigma(x,\omega)| < M$$ for some  bound  $$M$$ . Let $$I(t,\omega)=\int_0^t \sigma(s,\omega) dB(s,\omega)$$. Use the exponential martingale $\exp\big\{\alpha I(t)-\frac{\alpha^2}{2}\int_0^t \sigma^2(s)ds \big\}$ (see the problem here)  and the Kolmogorov-Doob inequality to get the estimate
$P\Big\{ \sup_{0\leq t\leq T}|I(t)| \geq \lambda \Big\}\leq 2 \exp\left\{\frac{-\lambda^2}{2M^2 T}\right\}$
First express the event of interest in terms of the exponential martingale, then use the Kolmogorov-Doob inequality and after this choose the parameter $$\alpha$$ to get the best bound.