Fakultät für Mathematik, Institut für Mathematische Stochastik Prof. G. Christoph

Tutorial Stochastic Processes (Series 3, 2013/2014)

12.) Let W (t) be a standard Wiener process. Using the fact, that a Gaussian process is determined by its expectation and covariance functions, prove: the following processes are standard Wiener processes, too W_{1}(t) = −W (t), t ∈ [0, ∞) ,

W_{2}(t) = c · W (t · c^{−2}), c > 0, t ≥ 0 ,

W_{3}(t) = t · W (1/t) for t > 0, (W_{3}(0) = 0), t ∈ [0, ∞) and

W_{4}(t) = W (h − t) − W (h), h fixed, t ∈ [0, h].

13.) Let W (t) be a Wiener process, L > 0 a constant and A ∼ N(μ, τ^{2}) be a random variable independent of {W (t)}_{t∈[0,∞)}. Calculate mean vector and covariance function of the vectors (X_{k}(s), X_{k}(t)), s < t, where

a) X_{1}(t) = W (t + L) − W (L), b) X_{2}(t) = W (t + L) − W (t)

c) X_{3}(t) = A·t+W (t), d) X_{4}(t) = \{ \begin{matrix}(1 − t)W (\frac{t}{1 − t}) & , 0 ≤ t < 1 \\ 0 & , t ≥ 1 . \end{matrix}

14.) Let (W_{t}, t ≥ 0) be a standard Wiener Prozess, then B(t) = W_{t} − t W_{1} for 0 ≤ t ≤ 1 is a Brownian bridge.

a) Prove E B(t) = 0 and Cov_{B}(t, s) = min{t, s} − t s.

b) Prove, that the Process X_{t} = (1 + t) B(\frac{t}{1+t}) for 0 ≤ t < ∞ is a Wiener process, too.

15. Let W_{t}, t ≥ 0 be a standard Wiener process. Define X_{t} = 4 W_{t} − t W_{4}, 0 ≤ t ≤ 4. Calculate Cov(X_{s}, X_{t}) for 0 ≤ s, t ≤ 4.