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

Tutorial Stochastic Processes, (Serie 4)

Problems with * are additional problems.

16.) Suppose Ω = {1, 2, 3, 4, 5, 6} with P ({i}) = 1/6 for i = 1, 2, 3, 4, 5, 6. Defin X(ω) = ω = face value on die with P (X = i) = 1/6 for i = 1, 2, 3, 4, 5, 6

and Y (ω) = \{ \begin{matrix}2 & if ω ∈ \{1, 3, 6\} \\ 3 & if ω ∈ \{2, 5\} \\ 5 & if ω = 4 \end{matrix} . Find Z = E(X|Y) !

17.) a) Let X be uniform on (0 , 1], C_{1} = (0 , 1/2], C_{2} = (1/2 , 1], and F_{2} = σ(C_{1}, C_{2}). Find V = E(X|F_{2})!

b) Remember Example 3.2 with F_{4} = σ(A_{1}, A_{2}, A_{3}, A_{4}). Here C_{1} = A_{1} ∪ A_{2}, C_{2} = A_{3} ∪ A_{4}

and Z(ω) = E(X|F_{4}) = \{ \begin{matrix}1/8 & if ω ∈ A_{1} \\ 3/8 & if ω ∈ A_{2} \\ 5/8 & if ω ∈ A_{3} \\ 7/8 & if ω ∈ A_{4} \end{matrix} . Find U = E(Z|F_{2}) !

18.) Let (W (t), t ≥ 0) be a standard Wiener Process and (N(t), t ≥ 0) a homogenuos Poisson Process mit dem Parameter λ > 0. Prove that

(a) W^{2}(t) + 2 W (t) − t, ; (b) W^{3}(t) − 3 t W (t), ; (c*) exp\{−θW (t) − \frac{1}{2}θ^{2} t\} and (d) N(t) − λt, ; (e*) (N(t) − λt)^{2} − λt, ; (f*) exp\{−θN(t) + λt(1 − e^{−θ})\}

are martingales with respect to its natural filtration

F_{t} = F(W (u), 0 ≤ u ≤ t) or F_{t} = F(N(u), 0 ≤ u ≤ t) .

(g) Show that W^{3}(t) and N(t) do not be martingales.

19.) Let (W (t), t ≥ 0) be a standard Wiener Process. Calculate (4.1) E ∫_{0}^{t} W_{s} dW_{s} = 0 and (4.2) E(∫_{0}^{T} W_{s} dW_{s})^{2} = ∫_{0}^{T} E(W_{s}^{2}) ds

20*.) Try to find a stochastic process A_{t} such that (W_{t}^{4} + A_{t} , F_{t}) is a martingale with respect to the filtration F_{s} = σ(W_{u}, 0 ≤ u ≤ s), s ≥ 0. Hint: First calculate E(((W_{t} − W_{s}) + W_{s})^{4} | F_{s}) for s < t .