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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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