Evans L. - Introduction to Stochastic Differential Equations Ver. 1.2 (Berkeley lecture notes).pdf

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AN INTRODUCTION TO STOCHASTIC
DIFFERENTIAL EQUATIONS
VERSION 1.2
Lawrence C. Evans
Department of Mathematics
UC Berkeley
Chapter 1: Introduction
Chapter 2: A crash course in basic probability theory
Chapter 3: Brownian motion and “white noise”
Chapter 4: Stochastic integrals, Ito’s formula
Chapter 5: Stochastic differential equations
Chapter 6: Applications
Exercises
Appendices
References
1
PREFACE
These are an evolvingset of notes for Mathematics 195 at UC Berkeley. This course
is for advanced undergraduate math majors and surveys without too many precise details
random differential equations and some applications.
Stochastic differential equations is usually, and justly, regarded as a graduate level
subject. A really careful treatment assumes the students’ familiarity with probability
theory, measure theory, ordinary differential equations, and perhaps partial differential
equations as well. This is all too much to expect of undergrads.
But white noise, Brownian motion and the random calculus are wonderful topics, too
good for undergraduates to miss out on.
Therefore as an experiment I tried to design these lectures so that strong students
could follow most of the theory, at the cost of some omission of detail and precision. I for
instance downplayed most measure theoretic issues, but did emphasize the intuitive idea of
σ –algebras as “containing information”. Similarly, I “prove” many formulas by confirming
them in easy cases (for simple random variables or for step functions), and then just stating
that by approximation these rules hold in general. I also did not reproduce in class some
of the more complicated proofs provided in these notes, although I did try to explain the
guiding ideas.
My thanks especially to Lisa Goldberg, who several years ago presented the class with
several lectures on financial applications, and to Fraydoun Rezakhanlou, who has taught
from these notes and added several improvements. I am also grateful to Jonathan Weare
for several computer simulations illustratingthe text.
2
CHAPTER 1: INTRODUCTION
A. MOTIVATION
Fix a point x 0 ∈ R
n and consider then the ordinary differential equation:
x ( t )= b ( x ( t )) ( t> 0)
x (0) = x 0 ,
( ODE )
where b :
R
n
→ R
n is a given, smooth vector field and the solution is the trajectory
x (
):[0 ,
∞
)
→ R
n .
Trajectory of the differential equation
Notation. x ( t )isthe state of the system at time t
≥
0, x ( t ):= dt x ( t ).
In many applications, however, the experimentally measured trajectories of systems
modeled by (ODE) do not in fact behave as predicted:
Sample path of the stochastic differential equation
Hence it seems reasonable to modify (ODE), somehow to include the possibility of random
effects disturbingthe system. A formal way to do so is to write:
(1)
˙ X ( t )= b ( X ( t )) + B ( X ( t )) ξ ( t ) t> 0)
X (0) = x 0 ,
where B :
R
n
→ M
n
×
m (= space of n
×
m matrices) and
ξ (
):= m -dimensional “white noise”.
This approach presents us with these mathematical problems:
•
Define the “white noise” ξ (
·
) in a rigorous way.
3
·
·
134559621.002.png 134559621.003.png 134559621.004.png
•
Define what it means for X (
·
) to solve (1).
Show (1) has a solution, discuss uniqueness, asymptotic behavior, dependence upon
x 0 , b , B , etc.
•
I . The solution of
(1) in this settingturns out to be the n -dimensional Wiener process ,or Brownian motion ,
denoted W (
B. SOME HEURISTICS
Let us first study (1) in the case m = n , x 0 =0, b
≡
0, and B
≡
·
). Thus we may symbolically write
˙ W (
·
)= ξ (
·
) ,
thereby assertingthat “white noise” is the time derivative of the Wiener process .
Now return to the general case of the equation (1), write
dt instead of the dot:
d X ( t )
dt
= b ( X ( t )) + B ( X ( t )) d W ( t )
dt
,
and finally multiply by “ dt ”:
( SDE )
d X ( t )= b ( X ( t )) dt + B ( X ( t )) d W ( t )
X (0) = x 0 .
This expression, properly interpreted, is a stochastic differential equation . We say that
X (
·
b ( X ( s )) ds + t
0
B ( X ( s )) d W for all times t> 0 .
Now we must:
•
Define the stochastic integral t
·
): See Chapter 3.
•
0
···
d W : See Chapter 4.
•
Show (2) has a solution, etc.: See Chapter 5.
And once all this is accomplished, there will still remain these modeling problems:
•
Does (SDE) truly model the physical situation?
•
) in (1) “really” white noise, or is it rather some ensemble of smooth,
but highly oscillatory functions? See Chapter 6.
·
As we will see later these questions are subtle, and different answers can yield completely
different solutions of (SDE). Part of the trouble is the strange form of the chain rule in
the stochastic calculus:
C. ITO’S FORMULA
Assume n = 1 and X (
·
) solves the SDE
(3)
dX = b ( X ) dt + dW.
4
d
) solves (SDE) provided
(2) X ( t )= x 0 + t
0
Construct W (
Is the term ξ (
 
Suppose next that u :
R → R
is a given smooth function. We ask: what stochastic
differential equation does
Y ( t ):= u ( X ( t )) ( t
≥
0)
solve? Offhand, we would guess from (3) that
dY = u dX = u bdt + u dW,
accordingto the usual chain rule, where =
dx . This is wrong, however ! In fact, as we
d
will see,
(4)
dW
≈
( dt ) 1 / 2
in some sense. Consequently if we compute dY and keep all terms of order dt or ( dt ) 2 ,we
obtain
2 u ( dX ) 2 + ...
= u ( b dt + dW
from (3)
)+ 1
2 u ( bdt + dW ) 2 + ...
= u b + 1
2 u dt + u dW +
{
terms of order ( dt ) 3 / 2 and higher
}
.
Here we used the “fact” that ( dW ) 2 = dt , which follows from (4). Hence
dY = u b + 1
2 u dt + u dW,
with the extra term “ 2 u dt ” not present in ordinary calculus.
A major goal of these notes is to provide a rigorous interpretation for calculations like
these, involvingstochastic differentials.
Example 1. Accordingto Ito’s formula, the solution of the stochastic differential equation
dY = YdW,
Y (0)=1
is
Y ( t ):= e W ( t ) −
t
2 ,
and not what might seem the obvious guess, namely Y ( t ):= e W ( t ) .
5
dY = u dX + 1
134559621.001.png
 

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