Introduction to SDE Part 1

1. introduction to sde
Tomohiro Koana
May 10, 2016
the University of Tokyo
Hasegawa lab

2. what is sde?
SDE stands for Stochastic Differential Equation.
A system is called stochastic when uncertainty is contained in the
values of parameters, measurements, expected input and
disturbances.
A SDE is a differential equation in which at least one term is a
alertstochastic process.
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3. stochastic process
1. A collection {X(t)|t ≥ 0} of random variables is called a
stochastic process.
2. For each point ω ∈ Ω, the mapping t → X(t, ω) is the
corresponding sample path.
When a stochastic process is observed over time in an
experiment, it corresponds to one sample path for some ω.
Another sample path will be found when you run the
experiment again.
2

4. example of sde
{
˙X(t) = b(X(t)) + B(X(t))ξ(t)
X(0) = x0
where B : Rn
→ Mn×m
and ξ(·) := m dimensional white noise.
Note that ξ(·) hasn’t been defined mathematically.
When m = n, x0 = 0, b ≡ 0, and B ≡ I, the solution to the equation is
called Wiener Process or Brownian Motion, denoted W(·).
˙W(·) = ξ(t)
3

5. brownian motion
Consider when a unit amount of ink is injected at time t = 0, at the
location x = 0. Let f(x, t) denote the density of ink particle at time t.
Initially we have
f(x, 0) = σ0
We denote the probability of the event that an ink particle moves
from x to x + y in small time τ is ρ(τ, y). Then,
f(x, t+τ) =
∫ ∞
−∞
f(x−y, t)ρ(τ, y)dy =
∫ ∞
−∞
(f−fxy+
1
2
fxxy2
+...)ρ(τ, y)dy
ρ(τ, −y) = ρ(τ, y) by symmetry. We assume that
∫ ∞
−∞
y2
ρdy = Dτ
Consequently,
f(x, t + τ) − f(x, t)
τ
=
Dfxx(x, t)
2
+ {Higher order}
4

6. brownian motion
f(x, t + τ) − f(x, t)
τ
=
Dfxx(x, t)
2
+ {Higher order}
Sending τ → 0, we obtain
ft =
D
2
fxx
This is called the diffuse equation or the heat equation. With the
initial condition,
f(x, t) =
1
(2πDt)
1
2
e− x2
2Dt = N(0, Dt)
D is a constant computed as follows:
D =
RT
NAf
where



R: gas constant
T: absolute temperature
NA: Avogadro’s constant
f: friction coefficient
5

7. a variant of brownian motion
The random walks can be considered as a variant of the Brownian
motion. Consider particle created on x = 0 at time t = 0. Every ∆t,
the particle moves by ∆x either to the left or right with probability
1/2. Let p(m, n) denote the probability that the particle is at position
m∆x at time n∆t. Then
p(m, n + 1) =
1
2
p(m − 1, n) +
1
2
p(m + 1, n)
and hence
p(m, n + 1) − p(m, n) =
1
2
(p(m + 1, n) − 2p(m, n) + p(m − 1, n))
This closely resembles what we derived in the last slide:
ft =
D
2
fxx
6

8. brownian motion definition
A real-valued stochastic process is called a Brownian motion if
1. W(0) = 0,
2. W(t) − W(s) is N(0, t − s) for ∀ 0 ≤ s ≤ t
3. ∀ 0 < t1 < t2... < tn, W(t1), W(t2) − W(t1), ... are independent.
Note that
E(W(t)) = 0, E(W2
(t)) = t
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9. brown motion lemma
Lemma
E(W(t)W(s)) = t ∧ s = min{t, s}
Proof
Assume t ≥ s ≥ 0.
E(W(t)W(s)) = E((W(s) + W(t) − W(s))W(s))
= E(W2
(s)) + E((W(t) − W(s))W(s))
= s + E(W(t) − W(s))E(W(s))
= s = t ∧ s
(1)
Here we used the fact that W(t) − W(s) and W(s) are independent.
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10. on white noise
Recall that
˙W(t) = ξ(t)
Φh(s) = E
((
W(t + h) − W(t)
h
) (
W(s + h) − W(s)
h
))
=
1
h2
(((t + h) ∧ (s + h)) − ((t + h) ∧ s) − (t ∧ (s + h)) + (t ∧ s))
(2)
Φh(s) → 0 as h → 0 and
∫
Φh(s) = 1 hold, so presumably
Φh(s) → δ(s − t). Sending h → 0, we obtain E(ξ(t)ξ(s)) = Φh(s). Thus
by heuristics
E(ξ(t)ξ(s)) = δ0(s − t)
9

11. lévy–ciesielski construction of brownian motion
The family of {hk(·)}∞
k=0 of Haar functions
h0(t) := 1 for 0 ≤ t ≤ 1
h1(t) :=
{
1 for 0 ≤ t ≤ 1/2
−1 for 1/2 ≤ t ≤ 1
If 2n
≤ k < 2n+1
, n = 1, 2, ...
hk(t) :=



2n/2
for k−2n
2n ≤ t ≤ k−2n
+1/2
2n
−2n/2
for k−2n
+1/2
2n ≤ t ≤ k−2n
+1
2n
0otherwise
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