<< стр. 17(всего 17)СОДЕРЖАНИЕ
Diffusion models are a standard way to represent random variables in continuous time.
The ideas are analogous to the handling of discrete-time stochastic processes. We start with
a simple shock series, Оµt in discrete time and dzt in continuous time. Then we build up more
complex models by building on this foundation.
The basic building block is a Brownian motion which is the natural generalization of a
random walk in discrete time. For a random walk

zt в€’ ztв€’1 = Оµt

the variance scales with time; var(zt+2 в€’ zt ) = 2var(zt+1 в€’ zt ). Thus, deп¬Ѓne a Brownian
motion as a process zt for which

(358)
zt+в€† в€’ zt в€ј N (0, в€†).

We have added the normal distribution to the usual deп¬Ѓnition of a random walk. As E(Оµt Оµtв€’1 ) =
0 in discrete time, increments to z for non-overlapping intervals are also independent. I use
the notation zt to denote z as a function of time, in conformity with discrete time formulas;
many people prefer to use the standard representation of a function z(t).
ItвЂ™s natural to want to look at very small time intervals. We use the notation dzt to rep-
resent zt+в€† в€’ zt for arbitrarily small time intervals в€†, and we sometimes drop the subscript
when itвЂ™s obvious weвЂ™re talking about time t. Conversely, the level of zt is the sum of its

456
SECTION 23.2 DIFFUSION MODEL

small differences, so we can write the stochastic integral
Z t
dzs .
zt в€’ z0 =
s=0

The variance of a random walk scales with time, so the standard deviation scales with the
square root of time. The standard deviation is the вЂњtypical sizeвЂќ of a movement in aв€љ normally
distributed random variable, so the вЂњtypical sizeвЂќ of zt+в€† в€’ zt in time interval в€† is в€†. This
в€љ
fact means that (zt+в€† в€’ zt ) /в€† has typical size 1/ в€†, so though the sample path of zt is
continuous, zt is not differentiable.
For this reason, itвЂ™s important to be a little careful with notation. dz, dzt or dz(t) mean
zt+в€† в€’ zt for arbitrarily small в€†. We are used to thinking about dz as the derivative of a
function, but since a Brownian motion is not a differentiable function of time, dz = dz(t) dt
dt
makes no sense.
From (23.358), itвЂ™s clear that

Et (dzt ) = 0.

Again, the notation is initially confusing вЂ“ how can you take an expectation at t of a random
variable dated t? Keep in mind, however that dzt = zt+в€† в€’ zt is the forward difference. The
variance is thus the same as the second moment, so we write it as
ВЎ 2Вў
Et dzt = dt.

It turns out that not only is the variance of dzt equal to dt, but
2
dzt = dt

for every sample path of zt . z 2 is a differentiable function of time, though z itself is not.
We can see this with the same sort of argument I used for zt itself. If x в€ј N (0, Пѓ2 ), then
var(x2 ) = 2Пѓ 4 . Thus,
ВЈ В¤
var (zt+в€† в€’ zt )2 = 2в€†4 .
в€љ
The mean of (zt+в€† в€’ zt )2 is в€†, while the standard deviation of (zt+в€† в€’ zt )2 is 2в€†2 . As
в€† shrinks, the ratio of standard deviation to mean shrinks to zero; i.e. the series becomes
deterministic.

23.2 Diffusion model

457
CHAPTER 23 CONTINUOUS TIME

I form more complicated time series processes by adding drift and diffusion terms,
dxt = Вµ(В·)dt + Пѓ(В·)dzt
I introduce some common examples,
Random walk with drift: dxt = Вµdt + Пѓdzt .
AR(1) dxt = в€’П†(x в€’ Вµ) dt + Пѓdzt
в€љ
Square root process dxt = в€’П†(x в€’ Вµ) dt + Пѓ xt dzt
dpt
Price process pt = Вµdt + Пѓdzt .

You can simulate a diffusion process by approximating it for a small time interval,
в€љ
xt+в€† в€’ xt = Вµ(В·)в€†t + Пѓ(В·) в€†t Оµt+в€† ; Оµt+в€† в€ј N (0, 1).

As we add up serially uncorrelated shocks Оµt to form discrete time ARMA models, we
build on the shocks dzt to form diffusion models. I proceed by example, introducing some
popular examples in turn.
Random walk with drift. In discrete time, we model a random walk with drift as
xt = Вµ + xtв€’1 + Оµt
The obvious continuous time analogue is
dxt = Вµdt + Пѓdzt .
ItвЂ™s easy to п¬Ѓgure out the implications of this process for discrete horizons,
xt = x0 + Вµt + Пѓ(zt в€’ z0 )
or
xt = x0 + Вµt + Оµt ; Оµt ЛњN (0, Пѓ2 t).
This is a random walk with drift.
AR(1). The simplest discrete time process is an AR(1),
xt = (1 в€’ ПЃ)Вµ + ПЃxtв€’1 + Оµt
or
xt в€’ xtв€’1 = в€’(1 в€’ ПЃ)(xtв€’1 в€’ Вµ) + Оµt

The continuous time analogue is
dxt = в€’П†(xt в€’ Вµ) dt + Пѓdzt .

458
SECTION 23.2 DIFFUSION MODEL

This is known as the Ohrnstein-Uhlenbeck process. The mean or drift is

Et (dxt ) = в€’П†(xt в€’ Вµ)dt.

This force pulls x back to its steady state value Вµ, but the shocks Пѓdzt move it around.
Square root process. Like its discrete time counterpart, the continuous time AR(1) ranges
over the whole real numbers. It would be nice to have a process that was always positive, so
it could capture a price or an interest rate. An extension of the continuous time AR(1) is a
workhorse of such applications,
в€љ
dxt = в€’П†(xt в€’ Вµ) dt + Пѓ xt dzt .

Now, volatility also varies over time,

Et (dx2 ) = Пѓ2 xt dt
t

as x approaches zero, the volatility declines. At x = 0, the volatility is entirely turned off, so
x drifts up to Вµ. We will show more formally below that this behavior keeps x в‰Ґ 0 always.
Л™
This is a nice example because it is decidedly nonlinear. Its discrete time analogue
в€љ
xt = (1 в€’ ПЃ)Вµ + ПЃxtв€’1 + xt Оµt

is not a standard ARMA model, so standard linear time series tools would fail us. We could
not, for example, give a pretty equation for the distribution of xt+s for п¬Ѓnite s. It turns out
that we can do this in continuous time. Thus, one advantage of continuous time formulations
is that they give rise to a toolkit of interesting nonlinear time series models for which we have
closed form solutions.
Price processes. A modiп¬Ѓcation of the random walk with drift is the most common model
for prices. We want the return or proportional increase in price to be uncorrelated over time.
The most natural way to do this is to specify

dpt = pt Вµdt + pt Пѓdzt

or more simply
dpt
= Вµdt + Пѓdzt .
pt

Diffusion models more generally. A general picture should emerge. We form more com-
plex models of stochastic time series by changing the local mean and variance of the under-
lying Brownian motion.

dxt = Вµ(xt )dt + Пѓ(xt )dzt

More generally, we can allow the drift Вµ and diffusion to be a function of other variables and

459
CHAPTER 23 CONTINUOUS TIME

of time explicitly. We often write

dxt = Вµ(В·)dt + Пѓ(В·)dzt

processes; they are just like easily understandable discrete time processes
в€љ
(359)
xt+в€† в€’ xt = Вµ(В·)в€†t + Пѓ(В·) в€†t Оµt+в€† ; Оµt+в€† в€ј N (0, 1).

In fact, when analytical methods fail us, we can п¬Ѓgure out how diffusion models work by
simulating the discretized version (23.359) for a п¬Ѓne time interval в€†.
The local mean of a diffusion model is

Et (dxt ) = Вµ(В·)dt

and the local variance is

dx2 = Et (dx2 ) = Пѓ2 (В·)dt
t t

Variance is equal to second moment because means scale linearly with time interval в€†, so
mean squared scales with в€†2 , while the second moment scales with в€†.
Stochastic integrals. For many purposes, simply understanding the differential represen-
tation of a process is sufп¬Ѓcient. However, we often want to understand the random variable
xt at longer horizons. For example, we might want to know the distribution of xt+s given
information at time t.
Conceptually, what we want to do is to think of a diffusion model as a stochastic differ-
ential equation and solve it forward through time to obtain the п¬Ѓnite-time random variable
xt+s . Putting some arguments in for Вµ and Пѓ for concreteness, we can think of evaluating the
integral
Zt Zt Zt
xt в€’ x0 = dxs = Вµ(xs , s, ..)ds + Пѓ(xs , s, ..)dzs .
0 0 0
Rt
We have already seen how zt = z0 + 0 dzs generates the random variable zt в€ј N (0, t),
so you can see how expressions like this one generate random variables xt . The objective of
solving a stochastic differential equation is thus to п¬Ѓnd the distribution of x at some future
date, or at least some characterizations of that distribution such as conditional mean, variance
etc. Some authors dislike the differential characterization and always write processes in terms
of stochastic integrals. I return to how one might solve an integral of this sort below.

23.3 ItoвЂ™s lemma

460
SECTION 23.3 ITOвЂ™S LEMMA

Do second order Taylor expansions, keep only dz, dt,and dz 2 = dt terms.
1
= f 0 (x)dx + f 00 (x)dx2
dy
2
Вµ В¶
1 00
f 0 (x)Вµx + f (x)Пѓ 2 dt + f 0 (x)Пѓ x dz.
dy = x
2

You often have a diffusion representation for one variable, say

dxt = Вµx (В·)dt + Пѓ x (В·)dzt .

Then you deп¬Ѓne a new variable in terms of the old one,

(360)
yt = f (xt ).

Naturally, you want a diffusion representation for yt . ItoвЂ™s lemma tells you how to get it. It
says,
в€љ
Use a second order Taylor expansion, and think of dz as dt; thus as в€†t в†’ 0 keep
terms dz, dt, and dz 2 = dt, but terms dtdz, dt2 and higher go to zero.

1 d2 f(x) 2
df (x)
dy = dx + dx
2 dx2
dx
Expanding the second term,

dx2 = [Вµx dt + Пѓx dz]2 = Вµ2 dt2 + Пѓ 2 dz 2 + 2Вµx Пѓ x dtdz.
x x

Now apply the rule dt2 = 0, dz 2 = dt and dtdz = 0. Thus,

dx2 = Пѓ2 dt
x

Substituting for dx and dx2 ,

1 d2 f (x) 2
df(x)
dy = (Вµx dt + Пѓx dz) + Пѓ dt
2В¶ dx2 x
dx
Вµ
1 d2 f (x) 2
df(x) df (x)
= Вµx + Пѓx dt + Пѓx dz
2
dx 2 dx dx

Thus, ItoвЂ™s lemma.
Вµ В¶
1 d2 f(x) 2
df (x) df (x)
dy = Вµx (В·) + Пѓ x (В·) dt + Пѓx (В·)dz
2 dx2
dx dx

461
CHAPTER 23 CONTINUOUS TIME

The surprise here is the second term in the drift. Intuitively, this term captures a вЂњJensenвЂ™s
inequalityвЂќ effect. If a is a mean zero random variable and b = a2 = f (a), then the mean of
b is higher than the mean of a. The more variance of a, and the more concave the function,
the higher the mean of b.

23.4 Problems

1. Find the diffusion followed by the log price,
y = ln(p).
2. Find the diffusion followed by xy.
3. Suppose y = f (x, t) Find the diffusion representation for y. (Follow the obvious
multivariate extension of ItoвЂ™s lemma.)
4. Suppose y = f(x, w), with both x, w diffusions. Find the diffusion representation for y.
Denote the correlation between dzx and dzw by ПЃ.

462

 << стр. 17(всего 17)СОДЕРЖАНИЕ