APM 541: Diffusion Processes

61 Pages·2013·1.31 MB·English

by  Jay Taylor

APM 541: Stochastic Modelling in Biology
Diffusion Processes
Jay Taylor
Fall 2013
JayTaylor (ASU) APM541 Fall2013 1/61
;
BrownianMotion
Brownian Motion and Random Walks
Let (ξ :n≥0) be a collection of i.i.d. random variables with distribution
n
1
P(ξ =1)=P(ξ =−1)=
n n 2
and define the discrete-time Markov process X =(X :t ≥0) by setting X =0 and
t 0
t
X
X =X +ξ = ξ .
t t−1 t s
s=1
X is said to be a simple random walk. Since E[ξ ]=0 and Var(ξ )=1, it follows that
n n
E[X ] = 0
t
Var(X ) = t
t
for every t ≥0.
JayTaylor (ASU) APM541 Fall2013 2/61
BrownianMotion
Simple Random Walk
SimpleRandomWalk
20
15
10
5
X 0
−5
−10
−15
−20
0 10 20 30 40 50 60 70 80 90 100
time
JayTaylor (ASU) APM541 Fall2013 3/61
BrownianMotion
We can even explicitly calculate the distribution of the random variables X . Because X
t
increasesordecreasesby1ineverytimestep,itfollowsthatX willbeoddwhenevert is
t
odd and even whenever t is even. Thus, if t =2n is even, then for X =2k to be true,
2n
the process must have taken n+k steps to the right and n−k steps to the left, giving
2n !„1«2n
P(X =2k)= .
2n n+k 2
Although this result is exact, it is unwieldy when t is large. In that case, it is more
convenient to invoke the central limit theorem which tells us that X is approximately
t
normal with mean 0 and variance t, i.e.,
P(a≤X ≤b)≈Z b √1 e−x2/2tdx.
t
2πt
a
JayTaylor (ASU) APM541 Fall2013 4/61
BrownianMotion
In the previous example, we arbitrarily assumed that the random walker took steps of
size 1. Let us modify this by introducing a family of processes, say
“ ”
X((cid:15)) = X((cid:15)) :t ≥0 , indexed by a positive real number (cid:15)>0, where
t
t
X((cid:15)) =Xξ((cid:15))
t s
s=1
“ ”
and ξ((cid:15)) :n≥1 is a collection of i.i.d. random variables with
n
“ √ ” “ √ ” 1
P ξ((cid:15)) = (cid:15) =P ξ((cid:15)) =− (cid:15) = .
n n 2
√
In other words, the step size is now (cid:15). In this case we have
h i
E X((cid:15)) = 0
t
Var(X((cid:15))) = t(cid:15).
t
JayTaylor (ASU) APM541 Fall2013 5/61
BrownianMotion
Although each of the processes X((cid:15)) is a Markov jump process, when (cid:15)>0 is small, the
jumpsaresmallandsotothenakedeyethesamplepathsmayappeartobecontinuous.
SimpleRandomWalk
20
eps=1
15 eps=1/4
eps=1/16
eps=1/64
10
5
X 0
−5
−10
−15
−200 10 20 30 40 50 60 70 80 90 100
time
However, as the figure shows, as (cid:15) decreases to 0, not only do the step sizes decrease,
but all of the variables X((cid:15)) also tend to 0.
t
JayTaylor (ASU) APM541 Fall2013 6/61
BrownianMotion
To compensate for the shrinking step size, we need to inflate the time scale on which
the process is run, i.e., we will rescale time so that the process takes a greater number
of steps of smaller size. The ‘right’ rescaling can be deduced by examining the
relationship between the variance of X((cid:15)), (cid:15), and t. Since Var(X((cid:15)))=t(cid:15), if we define a
t t
new process B((cid:15)) =(B((cid:15)) :t ≥0) by setting
t
B((cid:15)) =X((cid:15))
t (cid:98)t/(cid:15)(cid:99)
then
h i
E B((cid:15)) = 0
t
Var(B((cid:15))) = t,
t
for all (cid:15)>0 and all t ≥0. With this rescaling of time, t →t/(cid:15), the variance of the
variables B((cid:15)) remains constant even as (cid:15) decreases to 0 and so it is at least plausible
t
that the processes B((cid:15)) might converge to a non-trivial limit as (cid:15)→0.
JayTaylor (ASU) APM541 Fall2013 7/61
BrownianMotion
Simple Random Walks with Diffusive Rescaling
SimpleRandomWalk
15
10
5
X 0
−5
−10 eps=1
eps=1/4
eps=1/16
eps=1/64
−15
0 10 20 30 40 50 60
time
JayTaylor (ASU) APM541 Fall2013 8/61
BrownianMotion
Apart from the picture, there are several lines of evidence that suggest that the
processes B((cid:15)) do in fact converge to a non-trivial limit as (cid:15)→0. In particular, for each
t ≥0, not only is it true that the mean and the variance of the random variables B((cid:15))
t
are constant, but in addition the central limit theorem shows that these variables
converge in distribution to a normal random variable with mean 0 and variance t:
0 1 0 1
Bt((cid:15)) ≈√t@p(cid:98)1t/(cid:15)(cid:99)(cid:98)Xt/(cid:15)(cid:99)ξ√i((cid:15)(cid:15))A =d √t@p(cid:98)1t/(cid:15)(cid:99)(cid:98)Xt/(cid:15)(cid:99)ξiA −d→ N(0,t),
i=1 i=1
as (cid:15)→0. Here we have used the fact that ξ((cid:15)) =d ξ for every (cid:15)>0. Furthermore, since
i i
every increment B((cid:15)) −B((cid:15)) is a sum of i.i.d. random variables, it follows that these also
t+s t
converge in distribution to normal random variables
B((cid:15)) −B((cid:15)) −d→ N(0,s)
t+s t
for every t,s >0, again as (cid:15)→0. This suggests that both the marginal distributions
and the increments of the limiting process, B, assuming it exists, are normal random
variables.
JayTaylor (ASU) APM541 Fall2013 9/61
BrownianMotion
In fact, the rescaled processes B((cid:15)) do converge to a limit as (cid:15) decreases to 0. This
limiting process is called Brownian motion and is described in the next definition.
Definition
A real-valued continuous-time stochastic process B =(B ;t ≥0) is called a standard
t
one-dimensional Brownian motion if
1 B0 =0 a.s.
2 B has independent stationary increments with normal distribution
B −B ∼N(0,s);
t+s t
3 B has continuous sample paths, i.e.,
“ ”
P ∀t ≥0limB =B =1.
t+h t
h→0
JayTaylor (ASU) APM541 Fall2013 10/61