The local time of Martin-L of random Brownian motion Willem L. - - PowerPoint PPT Presentation

The local time of Martin-L¨

• f random Brownian motion

Willem L. Fouch´ e and Safari Mukeru Department of Decision Sciences, University of South Africa, PO Box 392, 0003 Pretoria, South Africa CCC 2015, September 2015

2 The occupation measure of Brownian motion X : [0, 1] × Ω → R up to time t, is the random Borel measure defined by µ(t, ω, A) = λ{s ∈ [0, t] : X(s, ω) ∈ A}, A Borel in R, ω ∈ Ω. Here λ is the Lebesgue measure. L´ evy (1940 ... ) proved that for almost all ω ∈ Ω, the occupation measure µ(t, ω, .) is absolutely continuous (with respect to Lebesgue measure), that is, there exists a func- tion L(t, ω, .) : R → R, x → L(t, ω, x) such that µ(t, ω, A) =

• A

L(t, ω, x)dx, (A Borel in R). The number L(t, ω, x) is called the “local time of ω at x up to time t”. L´ evy referred to his construct as the “mesure du voisinage” suggesting to me at least that it might represent the “time that the Brownian path ω spends at the point x or infinitesimally around the point x during the time interval [0, t]”.

3 It is clear, by the Lebesgue density theorem that, for almost every x ∈ R (with respect to the Lebesgue measure), L(t, ω, x) = lim

ǫ→0+

1 2ǫλ{s ≤ t : |X(s, ω) − x| ≤ ǫ}, (1) almost surely. Trotter (1958) proved later that the occupation measure µ(t, ω, .) has continuous den- sity for almost all ω, that is, L(., ω, .) : [0, 1] × R → R (t, x) → L(t, ω, x) is continuous for almost all ω ∈ Ω. This has the implication that, for every x ∈ R, almost surely, for all t ∈ [0, 1], L(t, ω, x) = lim

ǫ→0+

1 2ǫλ{s ≤ t : |X(s, ω) − x| ≤ ǫ}. (2)

4 In 2014 , the authors (with George Davie) proved that for each complex oscillation ω, the occupation measure µ(t, ω, .) of ω up to time t is such that its Fourier transform ˆ µ(t, ω, u) = t exp(i u ω(s))ds, u ∈ R satisfies |ˆ µ(t, ω, u)|2 = O(|u|−2+ǫ), u → ∞ for all ǫ > 0. This has the implication that ˆ µ(t, ω, .) ∈ L2(R) and by a standard argument (Parseval), µ(t, ω, .) is absolutely continuous and its Radon-Nikodym derivative L(t, ω, .) is in L2(R). Hence for almost every x ∈ R (with respect to the Lebesgue measure), and every complex

• scillation ω:

L(t, ω, x) := lim

ǫ→0+

µ(x − ǫ, x + ǫ) 2ǫ = lim

ǫ→0+

1 2ǫλ{s ≤ t : |ω(s) − x| ≤ ǫ}. (3)

5 Subsequently Safari and I proved that for any complex oscillation ω and any com- putable real number x, the limit L(t, ω, x) := lim

n→+∞

1 2ǫn λ{s ≤ t : |ω(s) − x| ≤ ǫn}, where ǫn = 2−n, exists, and the function L(., ω, x) : [0, 1] − → [0, +∞), t → L(t, ω, x) is continuous. We call L(t, ω, x) the effective local time of ω at x up to time t.

6 Theorem 1 (FM 2015) For any complex oscillation ω and for any computable real num- ber a, the function φ(ω) : [0, 1] → R defined by φ(ω)(t) = t sign(ω(s) − a)dω(s) = lim

n→∞ ⌊2nt⌋

• k=1

sign(ω((k − 1)/2n) − a)(ω(k/2n) − ω((k − 1)/n)) is also a complex oscillation. Moreover, for any complex oscillation ω, there exists a unique complex oscillation ω1 such that ω = φ(ω1). Effective Ito integration as developed by Safari (2014).

7 Theorem 2 (FM 2015) For any complex oscillation ω and for any computable real num- ber a, (i) The sequence of functions fn : [0, 1]

R defined by

fn(t, ω, a) = (2ǫn)−1λ{s ≤ t : |ω(s) − a| ≤ ǫn}, ǫn = 2−n converges uniformly on [0, 1]. The limit L(t, ω, a) := lim

n→∞(2ǫn)−1λ{s ≤ t : |ω(s) − a| ≤ ǫn}

is called the (effective) local time of the complex oscillation ω at level a on [0, t].

8 (ii) Any complex oscillation satisfies Tanaka’s formula: L(t, ω, a) = |ω(t) − a| − |a| − t sign(ω(s) − a)dω(s). (4) (iii) The function ω → L(., ω, a) from C (the class of complex oscillations) into C[0, 1] is layerwise computable.

9 (iv) The local time L(t, ω) at the origin satisfies L(t, ω) = L(t, ω, 0) = max

0≤s≤t ˜

ω(s) where ˜ ω is the complex oscillation defined by ˜ ω(t) = − t sign(ω(s))dω(s). It is well-known that the maximum function of a Brownian motion is statistically similar to the local time of X at 0. The above makes this very explicit. The similarity is witnessed by the transformation ω → ˜ ω, which is in fact a layerwise computable transformation.

10 Eventually Theorem 3 (FM 2015) For any complex oscillation ω and for any computable real num- ber a, L(t, ω, a) = 2 lim

m→∞

• k∈Sm

|ω(k/2m) − a| where Sm is the subset of {1, 2, . . . , ℓ}, ℓ = ⌊t2m⌋, defined by k ∈ Sm iff sign(ω(k/2m) − a) = sign(ω((k − 1)/2m) − a). Even the almost sure version for just a = 0 of this result is new, it would appear. It would be interesting to look at this result from Nelson’s “Radically elementary probability theory” and to look more closely at its constructive and computational content within the framework presented yesterday by Sam Sanders.