Model of Stochastic Process (隨機過程模型)
布朗運動
Brownian motion
Brownian motion Description
• Mathematically, Brownian motion is a Wiener
process in which the conditional probability
distribution of the particle's position change at time
t+dt, given that its position at time t is p,
• It is a normal distribution with a mean ofμdt and a
variance of σ2dt; the parameter μ is the drift
velocity, and the parameter σ2 is the power of the
noise (diffusion).
Weiner Process:
One of Brownian Motion
Weiner motion as a limit of random walks:
Weiner motion:
B(t) is called “Weiner motion,” if
1. B(0) = 0,
2. Continuous function of t
3. Independent , normally distributed increments
- independent
- normal with
Weiner motion:
probability triple
W
•w
t
Weiner motion as a limit of random walks:
• This is a special case of the Central Limit Theorem
• The proof can be done by taking its MGF
Proof:
(MGF of standard normal)
Weiner motion as a limit of random walks:
• Weiner motion B(t):
has the same distribution as B(t)
• For sufficiently large N:
Weiner motion in the limit
Covariance of Weiner motion:
• For
• Covariance of Brownian motion:
independent
Weiner motion:
probability triple
W
•w
t
Filtration generated by a Weiner motion:
• B(t) is -measurable for every t
•
are independent of
contains exactly the information learned by observing
the Weiner motion up to time t
Weiner motion:
B(t) is called “Weiner motion,” if
1. B(0) = 0,
2. Continuous function of t
3. Independent , normally distributed increments
- independent
- normal with
Discrete time models:
• Stock:
• Money market process with interest rate r:
• Portfolio value process:
- Each Dk is -measurable
- Each Xk is -measurable
Expectation of stock price:
g(x)
h(y)
g(x)
Discrete time market models:
• Stock:
• Money market process with interest rate r:
• Portfolio value process:
- Each Dk is -measurable
- Each Xk is -measurable
Weiner Motion Properties
Weiner motion starting at a non-zero point:
•
- The distribution of B(t) under is the same as
the distribution of x + B(t) under P
- Expectation of h(B(t)) under is the same as
the expectation of h(x + B(t)) under P
Weiner motion starting at a non-zero point:
• Weiner motion stating at 0:
• Weiner motion stating at x:
W
•w
x
t
defines a Brownian motion starting at x
- The distribution of B(t) under is the same as
the distribution of x + B(t) under P
• Expectation under
- Expectation of h(B(t)) under is the same as
the expectation of h(x + B(t)) under P
Markov property of Weiner :
B(s)
s s+t
B(s)
t
Proof:
independent of
由於
same distribution
因此
Strong Markov property:
x
t t+t
restart
x
t
Reflection Equality
• Proof Thm 3.7.1 :
We first consider the case m>0. We substitute into the
reflection formula (3.7.1) to obtain
if , then we are guaranteed that .
so
Transition density:
PDF that the Brownian motion
changes value from x to y in time t
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