 Solution

ID: past-exam-of-the-continuous-martingales-and-stochastic-calculus-course-of-the-university-of-oxford/2023/1/b/iii/solution

For any

h_{1}, \dots, h_{m} \in H

and real

u_{1}, \dots, u_{m}

\sum_{ℓ = 1}^{m} u_{ℓ} W h_{ℓ} = W (\sum_{ℓ = 1}^{m} u_{ℓ} h_{ℓ}) in L^{2} (P) .

ℓ = 1 \sum m u_{ℓ} W h_{ℓ} \sim N 0, ℓ = 1 \sum m u_{ℓ} h_{ℓ}_{H}^{2} .

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 H}

is Jointly Gaussian, which is the meaning of joint Gaussianity for this collection.

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