STOCHASTIC CALCULUS HOMEWORK ASSIGNMENT 3 DUE ...

STATISTICS 385: STOCHASTIC CALCULUS HOMEWORK ASSIGNMENT 3 DUE NOVEMBER 7, 2016

Problem 1. Let (X (t ))t ≥0 be a one-dimensional Lévy process such that X (1) ≥ 0 a.s. Show that (X (t ))t ≥0 is a subordinator. Problem 2. Let (X (t ))t ≥0 be a d −dimensional Lévy process. (A) Show that for every θ ∈ Rd there exists a complex number ψ(θ ) such that for every t ≥ 0, E e i 〈θ ,X (t )〉 = e t ψ(λ) , and conclude that for each λ > 0 the process M θ (t ) = exp{i 〈θ , X (t )〉−t ψ(λ)} is a martingale. (B) Use the result of (A) to show that if (X (t ))t ≥0 has continuous sample paths of locally bounded variation then for some µ ∈ Rd , X (t ) = µt

a.s. for all t ≥ 0.

HINT: Use the fact that continuous martingales with paths of locally bounded variation are constant. Problem 3. The Cauchy process is the symmetric stable process X (t ) such that the distribution of X (1) is the standard Cauchy distribution, i.e., the probability distribution with density f1 (x ) =

1 π(1 + x 2 )

relative to Lebesgue measure. Recall that if (Ut , Vt ) is a standard two-dimensional Brownian motion and τ(b ) = min{t : U (t ) = b } then the process {V (τ(b ))}b ≥0 is a Cauchy process. Let P be the Poisson point process on (0, ∞) × (0, ∞) with intensity measure C d t y −2 d y , where C > 0 is a constant. Assume that to each point (t i , yi ) ∈ P is attached a random variable ξi , where ξi takes the values ±1 with probability 1/2, and where the sequence ξi is independent of the Poisson point process P . (A) Show that for each t > 0 the random series X S (t ) =

ξi yi

(t i ,yi )∈P : t i ≤t

converges with probability one. HINT: Condition on P . (B) Show that if C > 0 is chosen correctly then the resulting stochastic process S (t ) is a standard Cauchy process. 1

Problem 4. Upcrossings and Local Time at the Max. Let Yt = M t − Wt , where M t = maxs ≤t Wt , and let τ(a ) = min{t ≥ 0 : Wt = a }. For all a , b > 0, define D (a ; t ) to be the number of upcrossings of the interval [0, a ] by the process Ys in the time interval s ∈ [0, t ], that is, D (a , t ) is the maximum integer m ≥ 0 for which there exist times 0 ≤ t 1 ≤ t 2 ≤ · · · ≤ t 2m ≤ t such that for each i ≤ m, Yt 2i +1 = 0

and

Yt 2i = a .

NOTE: Thus, D (a , t ) is the number of excursions below the current max that dip farther than a below the current max. (A) Show that D (a ; τ(b )) has the same distribution as D (a %; τ(b %)), for any % > 0. HINT: Brownian scaling. (B) Show that for any a > 0, the process {D (a ; τ(b ))}b ≥0 is a Poisson process with some intensity λ(a ) > 0. (C) Show that the intensity is λ(a ) = 1/a . HINT: For any " > 0 the probability that a standard Brownian motion hits −a before it hits +" is "/(a + "). (D) Use the preceding results together with Brownian scaling to prove that for each b > 0 lim 2−k D (2−k ; τ(b )) = b

k →∞

almost surely.

HINT: It should be easy to prove convergence in probability. To get almost sure convergence, use the fact that the Poisson distribution has exponentially decaying tails.