ASSIGNMENT

STAT433/833 Assignment 5:

Due December 19th at 10am

1 | Time Reversal [6]

Suppose that W is a standard Wiener process (SWP) and that Mt = sup0≤s≤t Ws.

a) [3] Show that for Vt = W1−t − W1 is a SWP on time interval [0, 1].

b) [3] Show that Mt − Wt and Mt have the same distribution for each t > 0.

2 | Arcsin Law [6]

Show that the probability a standard Wiener processes has no 0s on the interval (t, 1) is (2/π) arcsin √t and that the probability density function for the time of the last 0 before time 1 is 

3 | Inverse Gaussians [12]

Suppose that W is a standard Wiener process and that a > b > c > 0 and n ∈ N . Let Tc = inf(t ≥ 0 s.t. Wt = c) and define Ta, Tb, T1, Tb−a, and Tn similarly.

a) [3] Find the Laplace transform. and expected value of Tc: E exp(−λTc) and E Tc.

b) [3] Is (Ta − Tb) ⊥⊥ FTc? How does the distribution of Tb − Ta compare with that of Tb−a?

c) [3] Let λ > 0. How does the joint distribution of (Ta, Tb, Tc) compare to the joint distribution of (λTa/√λ, λTb/√λ, λTc/√λ)?

d) [3] Suppose that T(i) are IID from the same distribution as T1. What is the distribution of  What is the distribution of Tn? What is the distribution of n2T1?

4 | Recurrence and Transience [10]

a) [1] Fully describe the set of harmonic functions on an open interval (a, b) of R = R1 (d = 1!!).

b) [2] For a SWP in d = 1, and for 0 < a < x < b compute Px(Ta < Tb).

c) [2] Does a SWP in d = 1 revisit neighbourhoods of the origin with probability 1 from any initial location?

d) [1] Does a SWP in d = 1 revisit the origin with probability 1 from any initial location?

e) [2] Is the set of times where a SWP in d = 1 visits the origin unbounded?

f) [1] Would you describe the SWP in d = 1 as recurrent or Transient?

g) [1] How does the recurrence or transience of a SWP as a function of the dimension compare to component-wise random walks on Zd?

5 | Total and Quadratic variation [16]

An partition of [0, T] of cardinality n is a finite collection of points, π = (t0, t1, ..., tn) such that 0 = t0 < t1... < tn = T. The size of a partition π = (t0, t1, ..., tn) is ∥π∥ = maxk∈{1,...n} (tk − tk−1). The collection of all partitions of [0, T] is denoted Π.

For a function H : II → R define

If lim sup∥π∥→0 H(π) = lim inf∥π∥→0 H(π) we say that lim∥π∥→0 H(π) exists and is equal to that common value.

The total variation of a function f can be defined as TVT (f) = lim∥π∥→0 sadf(π) for

(sadf stands for “sum-absolute-df” and “df” refers to a change in f) The quadratic variation of a function f can be defined as [f]T = lim∥π∥→0 s(df) 2 (π) for

(s(df)2 stands for “sum-of-the-squared-dfs”)

a) [2] Show that if f is differentiable on [0, T] and f ′ is continuous on that interval then for any partition 0 = t0 < t1 < ... < tn = T it holds that

b) [2] Under the assumptions above argue that for any ϵ > 0 that there is a partition where this is within ϵ > 0 of an equality.

c) [3] Under the assumptions above show that the quadratic variation of f on [0, T], [f]T , is 0.

d) [3] For Wt standard Wiener process, let

Show that E Yn is unbounded in n and that Var(Yn) is bounded in n.

e) [3] Show that lim sup∥π∥→0 sadW(π) = +∞ with probability 1.

f) [3] Define the quadratic co-variation of functions f and g by

Show that if the quadratic variations of f and g exist, then ⟨f, g⟩T = lim∥π∥→0 s(df)(dg)(π), where

and that