PDE for Finance Notes, Spring 2011 – Section 1.
Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use in connec
tion with the NYU course PDE for Finance, G63.2706. Prepared in 2003, minor updates
made in 2011.
Links between stochastic differential equations and PDE.
A stochastic differential
equation, together with its initial condition, determines a diffusion process. We can use it
to define a deterministic function of space and time in two fundamentally different ways:
(a) by considering the expected value of some “payoff,” as a function of the initial position
and time; or
(b) by considering the probability of being in a certain state at a given time, given knowl
edge of the initial state and time.
Students of finance will be familiar with the BlackScholes PDE, which amounts to an
example of (a). Thus in studying topic (a) we will be exploring among other things the origin
of the BlackScholes PDE. The basic mathematical ideas here are the
backward Kolmogorov
equation
and the
FeynmanKac formula
.
Viewpoint (b) is different from (a), but not unrelated. It is in fact
dual
to viewpoint (a),
in a sense that we will make precise.
The evolving probability density solves a different
PDE, the
forward Kolmogorov equation
– which is actually the adjoint of the backward
Kolmogorov equation.
It is of interest to consider how and when a diffusion process crosses a barrier. This arises
in thinking subjectively about stock prices (e.g. what is the probability that IBM will reach
200 at least once in the coming year?). It is also crucial for pricing barrier options. Prob
abilistically, thinking about barriers means considering
exit times
. On the PDE side this
will lead us to consider
boundary value problems
for the backward and forward Kolmogorov
equations.
For a fairly accessible treatment of much of this material see Gardiner (the chapter on the
FokkerPlanck Equation). Parts of my notes draw from Oksendal, however the treatment
there is much more general and sophisticated so not easy to read.
Our main tool will be Ito’s formula, coupled with the fact that any Ito integral of the form
R
b
a
f dw
has expected value zero. (Equivalently:
m
(
t
) =
R
t
a
f dw
is a martingale.) Here
w
is
Brownian motion and
f
is nonanticipating. The stochastic integral is defined as the limit
of Ito sums
∑
i
f
(
t
i
)(
w
(
t
i
+1

w
(
t
i
)) as Δ
t
→
0. The sum has expected value zero because
each of its terms does:
E
[
f
(
t
i
)(
w
(
t
i
+1
)

w
(
t
i
))] =
E
[
f
(
t
i
)]
E
[
w
(
t
i
+1
)

w
(
t
i
)] = 0.
********************
Expected values and the backward Kolmogorov equation
. Here’s the most basic
version of the story. Suppose
y
(
t
) solves the scalar stochastic differential equation
dy
=
f
(
y, s
)
ds
+
g
(
y, s
)
dw,
1
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and let
u
(
x, t
) =
E
y
(
t
)=
x
[Φ(
y
(
T
))]
be the expected value of some payoff Φ at maturity time
T > t
, given that
y
(
t
) =
x
. Then
u
solves
u
t
+
f
(
x, t
)
u
x
+
1
2
g
2
(
x, t
)
u
xx
= 0 for
t < T
, with
u
(
x, T
) = Φ(
x
)
.
(1)
The proof is easy: for any function
φ
(
y, t
), Ito’s lemma gives
d
(
φ
(
y
(
s
)
, s
))
=
φ
y
dy
+
1
2
φ
yy
dydy
+
φ
s
ds
=
(
φ
s
+
fφ
y
+
1
2
g
2
φ
yy
)
dt
+
gφ
y
dw.
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 Fall '11
 Bayou
 Finance, Boundary value problem, Kolmogorov Equation

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