# Tag Archives: JCM_math545_HW6_S14

## 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)

## Ito Variation of Constants

For  functions $$f(x)$$ and$$g(x)$$ and constant $$\beta>0$$,  define $$X_t$$ as the solution to the following SDE
$dX_t = – \beta X_t dt + h(X_t)dt + g(X_t) dW_t$

where $$W_t$$ is a standard Brownian Motion.

1.  Show that $$X_t$$ can be written as
$X_t = e^{-\beta t} X_0 + \int_0^{t} e^{-\beta (t-s)} h(X_s) ds + \int_0^{t} e^{-\beta (t-s)} g(X_s) dW_s$
See exercise:  Ornstein–Uhlenbeck process for guidance.
2. Assuming that $$|h(x)| < K$$  and $$|g(x)|<K$$, show that  there exists a constant $$C(X_0)$$ so that
$\mathbf E [|X_t|] < C(X_0)$
for all $$t >0$$. It might be convenient the remember the Cauchy–Schwarz inequality.
3. * Assuming that $$|h(x)| < K$$  and $$|g(x)|<K$$, show that  for any integer $$p >0$$ there exists a constant $$C(p,X_0)$$ so that
$\mathbf E [|X_t|^{2p}] < C(p,X_0)$
for all $$t >0$$. See exercise: Ito Moments for guidance.

## Ornstein–Uhlenbeck process

For $$\alpha \in \mathbf R$$ and $$\beta >0$$,  Define $$X_t$$ as the solution to the following SDE
$dX_t = – \beta X_t dt + \alpha dW_t$

where $$W_t$$ is a standard Brownian Motion.

1.  Find $$d(e^{\beta t} X_t)$$ using Ito’s Formula.
2. Use the calculation of   $$d(e^{\beta t} X_t)$$ to show that
\begin{align}  X_t = e^{-\beta t} X_0 + \alpha \int_0^t e^{-\beta(t-s)} dW_s\end{align}
3. Conclude that $$X_t$$ is Gaussian process (see exercise: Gaussian Ito Integrals ). Find its mean and variance at time $$t$$.
4. * Let $$h(t)$$ and $$g(t)$$ be  deterministic functions of time and let $$Y_t$$ solve
$dY_t = – \beta Y_t dt + h(t)dt+ \alpha g(t) dW_t$
show find a formula analogous to part 2 above for $$Y_t$$ and conclude that $$Y_t$$ is still Gaussian. Find it mean and Variance.

## 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
$dM_t =\alpha M_t dI_t = \alpha M_t \sigma_t dB_t$