Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466
Stochastic Processes and
Advanced Mathematical Finance
Intuitive Introduction to Diffusions
Rating
Mathematically Mature: may contain mathematics beyond calculus with
proofs.
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Section Starter Question
√
Suppose you wanted to display the function y = x with a computer plotting
program or a graphing calculator. Describe the process to choose a proper
window to display the graph.
Key Concepts
1. This section introduces the passage from discrete random walks to continuous time stochastic processes from the probability point of view and
the partial differential equation point of view.
2. To get a sensible passage from discrete random walks to a continuous
time stochastic process the step size must be inversely proportional to
the square root of the stepping rate.
Vocabulary
1. A diffusion process, or a diffusion for short, is a Markov process for
which all sample functions are continuous.
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Mathematical Ideas
Visualizing Limits of Random Walks
The question is “How should we set up the limiting process so that we can
make a continuous time limit of the discrete time random walk?” First we
consider a discovery approach to this question by asking what do we require
so that we can visualize the limiting process. Next we take a probabilistic
view using the Central Limit Theorem to justify the limiting process to pass
from a discrete probability distribution to a probability density function.
Finally, we consider the limiting process to a differential equation derived
from the difference equation that is the result of first-step analysis.
The Random Walk
Consider a random walk starting at the origin. The nth step takes the walker
to the position Tn = Y1 + · · · + Yn , the sum of n independent, identically
distributed Bernoulli random variables Yi assuming the values +1 and −1
with probabilities p and q = 1 − p respectively. Then recall that the mean of
a sum of random variables is the sum of the means:
E [Tn ] = (p − q)n
and the variance of a sum of independent random variables is the sum of the
variances:
Var [Tn ] = 4pqn.
Trying to use the mean to derive the limit
Now suppose we want to display a video of the random walk moving left and
right along the x-axis. This would be a video of the “phase line” diagram of
the random walk. Suppose we want the video to display 1 million steps and
be a reasonable length of time, say 1000 seconds, between 16 and 17 minutes.
This fixes the time scale at a rate of one step per millisecond. What should
be the window in the screen to get a good sense of the random walk? For
this question, we use a fixed unit of measurement, say centimeters, for the
width of the screen and the individual steps. Let δ be the length of the steps.
To find the window to display the random walk on the axis, we then need to
know the size of δ · Tn . Now
E [δ · Tn ] = δ · (p − q)n
3
−4
−3
−2
−1
0
1
2
3
4
Figure 1: Image of a possible random walk in phase line after an odd number
of steps.
and
Var [δ · Tn ] = δ 2 · 4pqn.
We want n to be large (about 1 million) and to see the walk on the screen
we want the expected end place to be comparable to the screen size, say 30
cm. That is,
E [δ · Tn ] = δ · (p − q)n < δ · n ≈ 30 cm
so δ must be 3 × 10−5 cm to get the end point on the screen. But then the
movement of the walk measured by the standard deviation
p
√
Var [δ · Tn ] ≤ δ · n = 3 × 10−2 cm
will be so small as to be indistinguishable. We will not see any random
variations!
Trying to use the variance to derive the limit
Let us turn the question around: We want to see the variations in many-step
random walks, so the standard deviations must be a reasonable fraction D
of the screen size
p
√
Var [δ · Tn ] ≤ δ · n ≈ D · 30 cm .
For n = 106 this is possible if δ = D · 3 × 10−2 cm . We still want to be able
to see the expected ending position which will be
E [δ · Tn ] = δ · (p − q)n = (p − q) · D · 3 × 104 cm .
To be consistent with the requirement that the ending position is on the
screen this will only be possible if (p − q) ≈ 10−3 . That is, p − q must be at
most comparable in magnitude to δ = 3 × 10−2 .
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The limiting process
Now generalize these results to visualize longer and longer walks in a fixed
amount of time. Since δ → 0 as n → ∞, then likewise (p − q) → 0, while
p + q = 1, so p → 1/2. The analytic formulation of the problem is as follows.
Let δ be the size of the individual steps, let r be the number of steps per unit
time. We ask what happens to the random walk in the limit where δ → 0,
r → ∞, and p → 1/2 in such a manner that:
(p − q) · δ · r → c
and
4pq · δ 2 · r → D.
Each of these says that we should consider symmetric (p = 1/2 = q) random
walks with step size inversely proportional to the square root of the stepping
rate.
The limiting process taking the discrete time random walk to a continuous time process is delicate. It is delicate because we are attempting to
scale in two variables, the step size or space variable and the stepping rate
or time variable, simultaneously. The variables are not independent, two
relationships connect them, one for the expected value and one for the variance. Therefore we expect that the scaling is only possible when the step size
and stepping rate have a special relationship, namely the step size inversely
proportional to the square root of the stepping rate.
Probabilistic Solution of the Limit Question
In our accelerated random walk, consider the nth step at time t = n/r and
consider the position on the line x = k · δ. Let
vk,n = P [δ · Tn = kδ]
be the probability that the nth step is at position k. We are interested in the
probability of finding the walk at given instant t and in the neighborhood of
a given point x, so we investigate the limit of vk,n as n/r → t, and k · δ → x
with the additional conditions that (p − q) · δ · r → c and 4pq · δ 2 · r → D.
Remember that the random walk can only reach an even-numbered position after an even number of steps, and an odd-numbered position after an
odd number of steps. Therefore in all cases n + k is even and (n + k)/2 is
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an integer. Likewise n − k is even and (n − k)/2 is an integer. We reach
position k at time step n if the walker takes (n + k)/2 steps to the right and
(n − k)/2 steps to the left. The mix of steps to the right and the left can
be in any order. So the walk δ · Tn reaches position kδ at step n = rt with
binomial probability
n
vk,n =
p(n+k)/2 q (n−k)/2 .
(n + k)/2
From the Central Limit Theorem
P [δ · Tn = k · δ] = P [(k − 1) · δ < δ · Tn < (k + 1) · δ]
#
"
δTn − (p − q)δn
(k + 1)δ − (p − q)δn
(k − 1)δ − (p − q)δn
p
p
p
<
<
=P
4pqδ 2 n
4pqδ 2 n
4pqδ 2 n
Z (k+1)δ−(p−q)δn
√
1
2
4pqδ 2 n
√ e−u /2 du
≈
(k−1)δ−(p−q)δn
2π
√
4pqδ 2 n
Z (k+1)δ
1
2
2
p
e−(z−(p−q)δn) /(2·4pqδ n) dz
=
2
2π · 4pqδ n
(k−1)δ
2δ
2
2
≈p
e−(kδ−(p−q)δn) /(2·4pqδ n)
2π · 4pqδ 2 n
2δ
2
2
e−(kδ−(p−q)δrt) /(2·4pqδ rt)
≈p
2
2π · 4pqδ rt
2δ
2
e−(x−ct) /(2·Dt) .
=√
2πDt
Similarly
1
P [a · δ < δ · Tn · δ < bδ] → √
2πDt
Z
b
exp
a
−(x − ct)2
2Dt
dt .
The integral on the right may be expressed in terms of the standard normal
cumulative distribution function.
Note that we derived the limiting approximation of the binomial distribution
2δ
−(x − ct)2
vk,n ∼ √
exp
2Dt
2πDt
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by applying the general form of the Central Limit Theorem. However, it
is possible to derive this limit directly through careful analysis. The direct
derivation is the de Moivre-Laplace Limit Theorem and it is the most basic
form of the Central Limit Theorem.
Differential Equation Solution of the Limit Question
Another method is to start from the difference equations governing the random walk, and then pass to a differential equation in the limit. Later we can
generalize the differential equation and find that the generalized equations
govern new continuous-time stochastic processes. Since differential equations
have a well-developed theory and many tools to manipulate, transform and
solve them, this method turns out to be useful.
Consider the position of the walker in the random walk at the nth and
(n + 1)st trial. Through a first step analysis the probabilities vk,n satisfy the
difference equations:
vk,n+1 = p · vk−1,n + q · vk+1,n .
In the limit as k → ∞ and n → ∞, vk,n will be the sampling of the function
v(t, x) at time intervals r, so that n = rt, and space intervals so that kδ =
x. That is, the function v(t, x) should be an approximate solution of the
difference equation:
v(t + r−1 , x) = pv(t, x − δ) + qv(t, x + δ).
We assume v(t, x) is a smooth function so that we can expand v(t, x) in a
Taylor series at any point. Using the first order approximation in the time
variable on the left, and the second-order approximation on the right in the
space variable, we get (after canceling the leading terms v(t, x) )
∂v(t, x) 1 2 ∂ 2 v(t, x)
∂v(t, x)
= (q − p) · δr
+ δ r
.
∂t
∂x
2
∂x2
In our passage to limit, the omitted terms of higher order tend to zero, so
we neglect them. The remaining coefficients are already accounted for in our
limits and so the equation becomes:
∂v(t, x)
∂v(t, x) 1 ∂ 2 v(t, x)
= −c
+ D
.
∂t
∂x
2
∂x2
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This is a special diffusion equation, more specifically, a diffusion equation
with convective or drift terms, also known as the Fokker-Planck equation for
diffusion. It is a standard problem to solve the differential equation for v(t, x)
and therefore, we can find the probability of being at a certain position at a
certain time. One can verify that
1
−[x − ct]2
v(t, x) = √
exp
2Dt
2πDt
is a solution of the diffusion equation, so we reach the same probability
distribution for v(t, x).
The diffusion equation can be immediately generalized by permitting the
coefficients c and D to depend on x, and t. Furthermore, the equation
possesses obvious analogues in higher dimensions and all these generalization
can be derived from general probabilistic postulates. We will ultimately
describe stochastic processes related to these equations as diffusions.
Sources
This section is adapted from W. Feller, in Introduction to Probability Theory
and Applications, Volume I, Chapter XIV, page 354.
Problems to Work for Understanding
1. Consider a random walk with a step to right having probability p and a
step to the left having probability q. The step length is δ. The walk is
taking r steps per minute. What is the rate of change of the expected
final position and the rate of change of the variance? What must we
require on the quantities p, q, r and δ in order to see the entire random
walk with more and more steps at a fixed size in a fixed amount of
time?
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2. Verify the limit taking to show that
vk,n
3. Show that
1
exp
∼√
2πDt
1
v(t, x) = √
exp
2πDt
−[x − ct]2
2Dt
−[x − ct]2
2Dt
.
is a solution of
∂v(t, x) 1 ∂ 2 v(t, x)
∂v(t, x)
= −c
+ D
∂t
∂x
2
∂x2
by substitution.
Reading Suggestion:
References
[1] William Feller. An Introduction to Probability Theory and Its Applications, Volume I, volume I. John Wiley and Sons, third edition, 1973. QA
273 F3712.
[2] Emmanuel Lesigne. Heads or Tails: An Introduction to Limit Theorems
in Probability, volume 28 of Student Mathematical Library. American
Mathematical Society, 2005.
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Outside Readings and Links:
1. Brownian Motion in Biology. A simulation of a random walk of a sugar
molecule in a cell.
2. Virtual Laboratories in Probability and Statistics. Search the page for
Random Walk Experiment.
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Steve Dunbar’s Home Page, http://www.math.unl.edu/~sdunbar1
Email to Steve Dunbar, sdunbar1 at unl dot edu
Last modified: Processed from LATEX source on July 27, 2016
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