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Continuous martingales and stochastic calculus course of the University of Oxford

Past exam of the continuous martingales and stochastic calculus course of the University of Oxford

2023

For any $h_1,\ldots,h_m\in\mathcal H$ and real $u_1,\ldots,u_m$, linearity of the finite partial sums and <convergence in L2> give
$$
\sum_{\ell=1}^mu_\ell Wh_\ell
=W\!\left(\sum_{\ell=1}^mu_\ell h_\ell\right)
\quad\text{in }L^2(\mathbb P).
$$
By part (ii), this <random variable> has the <normal distribution>
$$
\boxed{\sum_{\ell=1}^mu_\ell Wh_\ell
\sim N\!\left(0,\left\|\sum_{\ell=1}^mu_\ell h_\ell\right\|_{\mathcal H}^2\right).}
$$
A finite family is <Jointly Gaussian> precisely when every real linear combination has a possibly degenerate <normal distribution>. Thus every finite subfamily of $(Wh)_{h\in\mathcal H}$ is <Jointly Gaussian>, which is the meaning of joint Gaussianity for this collection.

Solution

Past exam of the continuous martingales and stochastic calculus course of the University of Oxford / 2023 / 1 / b / iii / Solution

Past exam of the continuous martingales and stochastic calculus course of the University of Oxford / 2023 / 1 / b / iii / Solution

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Past exam of the continuous martingales and stochastic calculus course of the University of Oxford / 2023 / 1 / b / iii / Solution

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