<filename>files/teaching/M34F22_MathematicalFinance/2016-2017/m3f22prob9.tex
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\ni m3f33prob9 \\
\begin{center}
{\bf M3F22 PROBLEMS 9.  8.12.2017} 
\end{center}
\ni Q1 {\it Gaussian distributions}.  \\
\i With the multivariate normal (multinormal, multivariate Gaussian) distribution as in V.2 W8, show that all linear combinations $\sum_1^n a_i X_i$ of a multinormal random vector are normal (in one dimension).  [This is actually the best way to {\it define} the multinormal.] \\
\i Deduce that It\^o integrals $\int_0^t f(s) dB_s$ with $f$ continuous and deterministic are normally (Gaussian) distributed. \\

\ni Q2 {\it Ornstein-Uhlenbeck process}.  \\
\i For $V = (V_t)$ the solution to the Ornstein-Uhlenbeck SDE $(OU)$\\
$$
dV = - \beta V dt + c dB:                              \eqno(OU)
$$
(i) By using the It\^o isometry, or otherwise, show that $V_t$ has distribution 
$$
V_t \sim N(0, {\s}^2 (1 - e^{-2 \b t})/(2 \b)).
$$

(ii) By (i) and independence of Brownian increments, or otherwise, show that the covariance is
$$
cov(V_t, V_{t+u}) = {\s}^2 e^{-2 \b u} (1 - 2e^{-2 \b t}) \qquad (u \geq 0).
$$
(iii) Show that $V$ is Gaussian and Markov. \\
(iv) Show that $V_t$ converges in distribution as $t \to \infty$, and find the limit distribution. 

\hfil NHB \break

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