- 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 \(\mathbf E\int_0^t \sigma(s,\omega)^2 ds < \infty\).
- 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. - 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.
- * 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)

# Tag Archives: JCM_math545_HW6_S14

## Homogeneous Martingales and Hermite Polynomials

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## 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.

- 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. - 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. - * 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.

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## 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.

- Find \( d(e^{\beta t} X_t)\) using Ito’s Formula.
- 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} - Conclude that \(X_t\) is Gaussian process (see exercise: Gaussian Ito Integrals ). Find its mean and variance at time \(t\).
- * 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.

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## 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 \(I(t,\omega)=\int_0^t \sigma(s,\omega) dB(s,\omega)\).

- 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. - Show that \(M_t\) satisfies the equation

\[ dM_t =\alpha M_t dI_t = \alpha M_t \sigma_t dB_t\]

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