Introduction to Stochastic Analysis 1

1. Find the law of 5W_{1} − W_{3} + W_{7}.

2. For which parameters a and b random variables aW_{1} − W_{2} and W_{3} + bW_{5} are independent?

3. Find the law of the random vector (W_{t_{1}}, W_{t_{2}}, …, W_{t_{n}}) for 0 < t_{1} < t_{2} < … < t_{n}.

4. Show that lim_{t→∞} \frac{W_{t}}{t} = 0 a.s.

5. Prove that the following transformation of the Wiener process are also Wiener

a) X_{t} = −W_{t} (reflection)

b) Y_{t} = c^{−1/2}W_{ct}, c > 0 (scaling of time)

c) Z_{t} = tW_{1/t} for t > 0 and Z_{0} = 0 (time inversion)

d) U_{t} = W_{T +t} − W_{T} , T ≥ 0

e) V_{t} = W_{t} for t ≤ T, V_{t} = 2W_{T} − W_{t} for t > T, where T ≥ 0.

6. Show that with probability 1 trajectories of Wiener process are unbounded.

7. Show that with probability 1 trajectories of Wiener process are not uniformly continuous on ℝ_{+}.

8. Let π_{n} = {t_{0}^{(n)}, t_{1}^{(n)}, …, t_{k_{n}}^{(n)}}, where a = t_{0}^{(n)} < t_{1}^{(n)} < … < t_{k_{n}}^{(n)} = b is a sequence of partition of the segment [a, b] and let ∥π_{n}∥ := max_{k} |t_{k}^{(n)}−t_{k−1}^{(n)}| denote the diameter of π_{n}. Prove that

S_{n} := ∑_{k=1}^{k_{n}} |W_{t_{k}^{(n)}} − W_{t_{k−1}^{(n)}}|^{2} → b − a in L^{2}(Ω, F, P), when n → ∞

if ∥π_{n}∥ → 0 and S_{n} → b − a a.s. if ∑_{n} ∥π_{n}∥ < ∞.

9. Show that with probability one trajectories of the Wiener process have unbounded variation on each nondegenerate interval.

10. Prove that with probability 1 trajectories of the Wiener process are nowhere differentiable on [0, ∞). Hints:

a) If a function f is differentiable at some point in [0, 1] then there exists M < ∞ and for sufficiently large n there exists 0 ≤ j ≤ n − 3 such that |f((j + 1)/n) − f(j/n)| ≤ M/n, |f((j + 2)/n) − f((j + 1)/n)| ≤ M/n and |f((j + 3)/n) − f((j + 2)/n)| ≤ M/n.

b) Show that Pr(|W_{(j+1)/n} − W_{j/n}| ≤ M/n, |W_{(j+2)/n} − W_{(j+1)/n}| ≤ M/n, |W_{(j+3)/n} − W_{(j+2)/n}| ≤ M/n) ≤ CMn^{−3/2} and prove that this implies nowhere differentiability of Wiener process trajectories on [0, 1].

1. Prove that if a set A ∈ B(ℝ^{T}) then there exists a countable set T_{0} ⊂ T such that if x, y ∈ ℝ^{T} and x(t) = y(t) for t ∈ T_{0} then x ∈ A ⇔ y ∈ A.

2. Let T = [a, b] a < t_{0} < b, prove, that the following sets do not belong to B(ℝ^{T}).

i) A_{1} = {x ∈ ℝ^{T} : sup_{t∈[a,b]} |x_{t}| ≤ 1}; ii) A_{2} = {x ∈ ℝ^{T} : t → x_{t} cige na [a,b] }; iii) A_{3} = {x ∈ ℝ^{T} : lim_{t→t_{0}} x_{t} = 0}; iv) A_{4} = {x ∈ ℝ^{T} : t → x_{t} cige w t_{0}}. Prove measurability of these sets under the assumption of continuity (right continuity) of trajectories, i.e. prove that all these sets after intersecting them with C(T) (RC(T) respectively) belong to B(ℝ^{T}) ∩ C(T) (B(ℝ^{T}) ∩ RC(T) resp.).

3. Let T = [a, b]. Prove that F = {A∩C(T): A ∈ B(ℝ^{T})} is the Borel σ-field (for the sup metric) on C(T).

4. Prove that there exists a process (X_{t})_{t≥0} with independent increments, starting from 0 and such that X_{t} − X_{s} has the Cauchy distribution with parameter t − s (such a process is called a Cauchy process or a 1-stable process).

5. A process X is a modification of the Wiener process. Which of the following properties are satisfied by X:

a) independence of increments,

b) stationarity of increments,

c) continuity of trajectories,

d) lim_{t→∞} \frac{X_{t}}{t} = 0 a.s.,

e) lim_{t→∞} \frac{X_{t}}{t} = 0 in probability?

6. Consider the following properties of processes:

a) continuity of trajectories;

b) stochastic continuity (i.e. X_{t} →^{ℙ} X_{s} for t → s);

c) continuity in L_{p} (i.e. 𝔼|X_{t} − X_{s}|^{p} → 0 for t → s). What implications between the following properties are satisfied?