Test on the course ”Stochastic Processes”, Tampere,

1. Let {X(t), t ∈ T } be stochastically continuous on T and f : T → R be a real function. Prove or disprove the following statement:

• stochastic process Y (t) = X(t) + f(t) is stochastically continuous in those and only those points of T where f(t) is continuous.

2. Let {X(t), t ∈ R} be a stochastic process of the form: X(t, ω) = 0 for t < ξ(ω) and X(t, ω) = η(ω) for t ≥ ξ(ω) where ξ and η are independent random variables and ξ has uniform distribution on [0, 1], while η has two-points distribution P (η = ±1) = 1/2. Is it true that {X(t), t ∈ R} is the stochastic process with independent increments ? Answer must be argued.

3. Let {W (t), t ≥ 0} be a Brownian motion. Calculate the mean function and the covariance function of the following stochastic process {W (t) − tW (1), t ∈ [0, 1]} (Brownian bridge). Is it a stationary in the wide sense and strictly stationary stochastic process ? Answer must be argued.

4. Let ξ from Herglotz’s theorem has the triangle distribution with the density function:

f(x) = \{ \begin{matrix}\frac{1}{π^{2}} (π + x) , & −π < x ≤ 0 \\ \frac{1}{π^{2}} (π − x) , & 0 < x < π \\ 0, & otherwise \end{matrix} .

Calculate the covariance function corresponding to such a spectral distribution.

5. Let {X(t), t ∈ R} be a differentiable second order stationary in the wide sense stochastic process. Show that X(t) and X′(t) are uncorrelated for all t.

1. Let {X(t), t ≥ 0} be a second-order stochastic process having independent stationary increments and let X(0) = 0. Show that

i) m(t_{1} + t_{2}) = m(t_{1}) + m(t_{2}), ii) B(t_{1} + t_{2}) = B(t_{1}) + B(t_{2}), where m(t) = EX(t), B(t) = E((X(t) − m(t))^{2}.

2. Let {Y (t), t ≥ 0} be a Poisson process. Calculate the mean function and the covariance function of the stochastic process {tY (t^{−1}), t > 0} and check whether it is a stationary in the wide sense process.

3. Let {W (t), t ≥ 0} be a Brownian motion while both the functions g(t) and h(t) have continuous derivatives on [0, 1] and are integrable on [0, 1]. Calculate

E [∫_{0}^{1} g′(s)W (s)ds ∫_{0}^{1} h′(t)W (t)dt] .

1. Define a stochastic process {X(t), t ∈ R} as X(t) = ξ + t(η + t), where ξ and η are independent standard normal variables. Describe a set of all sample paths for such a process and calculate P (X(0) ≤ 0, X(1) ≤ 1).

2. Let {Y (t), t ≥ 0} be a Poisson process. Calculate the mean function and the covariance function of the stochastic process {Y (t + 1) − Y (t), t ≥ 0} and check whether it is a stationary in the wide sense and strictly stationary stochastic process.

3. Let {W (t), t ≥ 0} be a Brownian motion. Calculate the mean function and the covariance function of the process X(t) = ∫_{0}^{t} W (s)ds.