HW2 MA5161. Due February 3

Exercise 1. Let $m:\mathbb{R}_{\ge 0} \rightarrow \mathbb{R}$ and let $R: \mathbb{R}_{\ge 0} \times \mathbb{R}_{\ge 0} \rightarrow \mathbb{R}$ be a symmetric and positive function. Show that there exists a probability space $\left( \Omega , \mathcal{F}, \mathbb{P} \right)$ and a Gaussian process $(X_t)_{t \ge 0}$ defined on it, whose mean function is $m$ and whose covariance function is $R$.

Exercise 2. Let $(X_t)_{t \ge 0}$ be a continuous process adapted to a filtration $\mathcal{F}_t$. Show that $(X_t)_{t \ge 0}$ is progressively measurable.

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