1 Continuous Time Processes - IPM

The Bessel function I◦ is defined as
I◦(α) = 1
2π
e
2π ◦
α cos θ dθ.
Therefore the desired transition probability
where
P [Rt ≤ b | Z◦ = (x1, x2)] =
b
˜pt(ρ, r; σ) = r
tσ2 e− r2 +ρ 2
2tσ2 I◦( rρ
◦
˜pt(ρ, r; σ)dr, (1.5.19)
).
tσ2 The Markovian property of radial Brownian motion is a consequence of the
expression for transition probabilities since they depends only on (ρ, r). From
the fact that I◦ is a solution of the differential equation
d2u 1 du
+
dz2 z dz
− u = 0,
we obtain the partial differential differential equation satisfied ˜p:
∂ ˜p
∂t
= σ2
2
which is the radial heat equation. ♠
∂2 ˜p σ ∂ ˜p
+
∂r2 2r ∂r ,
The analogue of non-symmetric random walk (E[X] = 0) is Brownian
motion with drift µ which one may define as
Z µ
t = Zt + µt
in the one dimensional case. It is a simple exercise to show
Lemma 1.5.1 Z µ
t is normally distributed with mean µt and variance tσ 2 ,
and has stationary independent increments.
In particular the lemma implies that, assuming Z µ ◦ = 0, the probability
of the set of paths that at time t are in the interval (a, b) is
1 b (u−µt)2
−
√ e 2tσ
2πtσ a
2 du.
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