Random ProcessLecture 5. Brownian Motion

Husheng Li

Min Kao Department of Electrical Engineering and Computer ScienceUniversity of Tennessee, Knoxville

Spring, 2016

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Definition

A random process W is said to be a Brownian motion (BM) if it iscontinuous and has stationary independent increments. It is called aWiener process if it is a BM with

W0 = 0, EWt = 0, VarWt = t .

A general BM is given by

Xt = X0 + at + bWt .

The covariance of a Wiener process is Cov(Ws,Wt) = min(s, t).

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Properties

Consider a Wiener process, then it has the following properties

Symmetry: −Wt is again a Wiener process.

Scaling: W = c−1/2Wct is also Wiener process (hence, it is stable withindex 2).

Time inversion, Wt = tW1/t yields a Wiener process.

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Martingale Connection

The following statements are equivalent:

W is a Wiener process.

The process exp(rWt − 1/2r 2t) is a martingale

W and W 2t − t are both martingales.

The process Mt = f ◦Wt − 12

∫ t0 dsf ′′ ◦Ws is martingale, where f is a

twice-differentiable function.

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Hitting Times

We are interested in the hitting times:

Ta(w) = inf{t > 0 : Wt(w) > a}.

Behavior at the origin: T0 = 0 almost surely.

Distribution of Ta:

P(Ta ≤ t ,Wt ∈ B) = G(t , 2a− B),

whereG(t ,B) = P(Wt ∈ B).

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Hitting Times of Points

We define

Ta− = inf{t > 0 : Wt ≥ a} = inf{t > 0 : Wt = a}.

We haveTa− = Ta,

almost surely.

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Arcsine Laws

Let X and Y be independent standard Gaussian variables, then thedistribution of A = X2

X2+Y 2 satisfies the arcsine distribution:

P(A ≤ u) =2π

arcsin√

u.

W also have arcsine law for Wiener process:

P(Wt ∈ R{0},∀t ∈ [s, u]) =2π

arcsin√√

su.

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Backward and ForwardRecurrence Times

We define Gt as the last time before t and Dt the first time after t that theparticle is at the origin:

Gt = sup{s ∈ [0, t ] : Ws = 0},

andDt = inf{u ∈ (t ,∞) : Wu = 0}

Suppose that A has the arcsine distribution. Then Gt has the samedistribution as tA and Dt has the same distribution as t/A.

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Hitting Times

We define the running maximum:

Mt(w) = maxs≤t

Ws(w).

Then we have

Ta(w) = inf{t > 0 : Mt(w) > a}, Mt(w) = inf{a > 0 : Ta(w) > t}.

We further have

P(Ta < t) = P(Mt > a) = P(|Wt | > a.)

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Wiener and Maximum

We are interested in Mt −Wt . We can obtain the probability of{Mt ∈ da,Mt −Wt ∈ db}.For a fixed t , the random variables Mt , |Wt | and Mt −Wt have the samedistribution. Moreover, |Wt | and Mt −Wt have the same law as randomprocesses.

We can also contraction M from the zeros of M −W !!!

M-W is a reflection of Wiener process.

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Path Properties

For almost all w , the path W (w) is continuous but nowheredifferentiable! On every interval, it has infinite variation.

The total variation of function f within [a, b] is denied as

maxA is a subdivision of (a,b]

∑(s,t]

∈ A|f (t)− f (s)|.

Define Vn = sum(s,t]∈An |Wt −Ws|2, where {An} is a sequence ofsubdivisions of [a, b] such that ||An|| → 0. Then, we have Vn convergesto b − a in probability.

For almost every w , the path W (w) has infinite total variation over everyinterval [a, b], with a < b.

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Holder Continuity

A function is said to be Holder continuous of order α if

|f (t)− f (s)| ≤ k |t − s|α.

For almost every w , W (w) is Holder continuous of order α for α > 1/2.In particular, for almost every w , the path is nowhere differentiable.

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Law of Iterated Logarithm

Define h(t) =√

2t log log(1/t).

Then we havelim sup

t→0

1h(t)

Wt(w) = 1,

andlim inf

t→0

1h(t)

Wt(w) = −1.

What about the case t →∞?

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