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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