dZt = aZt dWt + bZt dt.

Note that this equation is linear in Zt , and it turns out that linear equations are almost

the only ones that have an explicit solution. In this case we can write down the explicit

solution and then verify that it satis¬es the SDE. The uniqueness result above (Theorem

15.1) shows that we have in fact found the solution.

Let

2

Zt = Z0 eaWt ’a t/2+bt .

We will verify that this is correct by using Ito™s formula. Let Xt = aWt ’ a2 t/2 + bt. Then

Xt is a semimartingale with martingale part aWt and X t = a2 t. Zt = eXt . By Ito™s

formula with f (x) = ex ,

t t

Xs

eXs a2 ds

1

Zt = Z0 + e dXs + 2

0 0

t t2 t

a

aZs dWs ’

= Z0 + Zs ds + bZs ds

2

0 0 0

t

a2 Zs ds

1

+ 2

0

t t

= aZs dWs + bZs ds.

0 0

This is the integrated form of the equation we wanted to solve.

There is a connection between SDEs and partial di¬erential equations. Let f be a

C 2 function. If we apply Ito™s formula,

t t

1

f (Xt ) = f (X0 ) + f (Xs )dXs + f (Xs )d X s .

2

0 0

t

σ(Xs )2 ds. If we substitute for dXs and d X s , we obtain

From (15.2) we know X =

t 0

t t

f (Xt ) = f (X0 ) + f (Xs )dWs + µ(Xs )ds

0 0

t

f (Xs )σ(Xs )2 ds

1

+ 2

0

t t

Lf (Xs )ds,

= f (X0 ) + f (Xs )dWs +

0 0

where we write

Lf (x) = 2 σ(x)2 f (x) + µ(x)f (x).

1

71

L is an example of a di¬erential operator. Since the stochastic integral with respect to a

Brownian motion is a martingale, we see from the above that

t

f (Xt ) ’ f (X0 ) ’ Lf (Xs )ds

0

is a martingale. This fact can be exploited to derive results about PDEs from SDEs and

vice versa.

Note 1. Let us illustrate the uniqueness part, and for simplicity, assume b is identically 0 and

σ is bounded.

Proof of uniqueness. If X and Y are two solutions,

t

Xt ’ Yt = [σ(Xs ) ’ σ(Ys )]dWs .

0

So

t t

2 2

E |Xs ’ Ys |2 ds,

E |Xt ’ Yt | = E |σ(Xs ) ’ σ(Ys )| ds ¤ c

0 0

using the Lipschitz hypothesis on σ. If we let g(t) = E |Xt ’ Yt |2 , we have

t

g(t) ¤ c g(s) ds.

0

t

2

(σ(Xs ))2 ds ¤ ct and similarly for E Yt2 , so

Since we are assuming σ is bounded, E Xt = E 0

g(t) ¤ ct. Then

t s

g(t) ¤ c c g(r) dr ds.

0 0

Iteration implies

g(t) ¤ Atn /n!

for each n, which implies g must be 0.

72

16. Continuous time ¬nancial models.

The most common model by far in ¬nance is one where the security price is based

on a Brownian motion. One does not want to say the price is some multiple of Brownian

motion for two reasons. First, of all, a Brownian motion can become negative, which

doesn™t make sense for stock prices. Second, if one invests $1,000 in a stock selling for $1

and it goes up to $2, one has the same pro¬t, namely, $1,000, as if one invests $1,000 in a

stock selling for $100 and it goes up to $200. It is the proportional increase one wants.

Therefore one sets ∆St /St to be the quantity related to a Brownian motion. Di¬er-

ent stocks have di¬erent volatilities σ (consider a high-tech stock versus a pharmaceutical).

In addition, one expects a mean rate of return µ on one™s investment that is positive (oth-

erwise, why not just put the money in the bank?). In fact, one expects the mean rate

of return to be higher than the risk-free interest rate r because one expects something in

return for undertaking risk.

So the model that is used is to let the stock price be modeled by the SDE

dSt /St = σdWt + µdt,

or what looks better,

dSt = σSt dWt + µSt dt. (16.1)

Fortunately this SDE is one of those that can be solved explicitly, and in fact we

gave the solution in Section 15.

Proposition 16.1. The solution to (16.1) is given by

2

St = S0 eσWt +(µ’(σ /2)t)

. (16.2)

Proof. Using Theorem 15.1 there will only be one solution, so we need to verify that St

as given in (16.2) satis¬es (16.1). We already did this, but it is important enough that we

will do it again. Let us ¬rst assume S0 = 1. Let Xt = σWt + (µ ’ (σ 2 /2)t, let f (x) = ex ,

and apply Ito™s formula. We obtain

t t

Xt X0 Xs

eXs d X

1

St = e =e + e dXs + s

2

0 0

t t

Ss (µ ’ 2 σ 2 )ds

1

=1+ Ss σdWs +

0 0

t

Ss σ 2 ds

1

+ 2

0

t t

=1+ Ss σdWs + Ss µds,

0 0

73

which is (16.1). If S0 = 0, just multiply both sides by S0 .

Suppose for the moment that the interest rate r is 0. If one purchases ∆0 shares

(possibly a negative number) at time t0 , then changes the investment to ∆1 shares at time

t1 , then changes the investment to ∆2 at time t2 , etc., then one™s wealth at time t will be

Xt0 + ∆0 (St1 ’ St0 ) + ∆1 (St2 ’ St1 ) + · · · + ∆i (Sti+1 ’ Sti ). (16.3)

To see this, at time t0 one has the original wealth Xt0 . One buys ∆0 shares and the cost

is ∆0 St0 . At time t1 one sells the ∆0 shares for the price of St1 per share, and so one™s

wealth is now Xt0 + ∆0 (St1 ’ St0 ). One now pays ∆1 St1 for ∆1 shares at time t1 and

continues. The right hand side of (16.3) is the same as

t

Xt0 + ∆(s)dSs ,

t0

where we have t ≥ ti+1 and ∆(s) = ∆i for ti ¤ s < ti+1 . In other words, our wealth is

given by a stochastic integral with respect to the stock price. The requirement that the

integrand of a stochastic integral be adapted is very natural: we cannot base the number

of shares we own at time s on information that will not be available until the future.

How should we modify this when the interest rate r is not zero? Let Pt be the

present value of the stock price. So

Pt = e’rt St .

Note that P0 = S0 . When we hold ∆i shares of stock from ti to ti+1 , our pro¬t in present

days dollars will be

∆i (Pti+1 ’ Pti ).

The formula for our wealth then becomes

t

Xt0 + ∆(s)dPs .

t0

By Ito™s product formula,

dPt = e’rt dSt ’ re’rt St dt

= e’rt σSt dWt + e’rt µSt dt ’ re’rt St dt

= σPt dWt + (µ ’ r)Pt dt.

Similarly to (16.2), the solution to this SDE is

2

Pt = P0 eσWt +(µ’r’σ /2)t

. (16.4)

The continuous time model of ¬nance is that the security price is given by (16.1)

(often called geometric Brownian motion), that there are no transaction costs, but one can

trade as many shares as one wants and vary the amount held in a continuous fashion. This

clearly is not the way the market actually works, for example, stock prices are discrete,

but this model has proved to be a very good one.

74

17. Markov properties of Brownian motion.

Let Wt be a Brownian motion. Because Wt+r ’ Wt is independent of σ(Ws : s ¤ t),

then knowing the path of W up to time s gives no help in predicting Wt+r ’ Wt . In

particular, if we want to predict Wt+r and we know Wt , then knowing the path up to time

t gives no additional advantage in predicting Wt+r . Phrased another way, this says that

to predict the future, we only need to know where we are and not how we got there.

Let™s try to give a more precise description of this property, which is known as the

Markov property.

Fix r and let Zt = Wt+r ’ Wr . Clearly the map t ’ Zt is continuous since the

same is true for W . Since Zt ’ Zs = Wt+r ’ Ws+r , then the distribution of Zt ’ Zs is

normal with mean zero and variance (t + r) ’ (s + r). One can also check the other parts

of the de¬nition to show that Zt is also a Brownian motion.

Recall that a stopping time in the continuous framework is a r.v. T taking values

in [0, ∞) such that (T ¤ t) ∈ Ft for all t. To make a satisfactory theory, we need that the

Ft be right continuous (see Section 10), but this is fairly technical and we will ignore it.

If T is a stopping time, FT is the collection of events A such that A © (T > t) ∈ Ft

for all t.

Let us try to provide some motivation for this de¬nition of FT . It will be simpler to

consider the discrete time case. The analogue of FT in the discrete case is the following:

if N is a stopping time, let

FN = {A : A © (N ¤ k) ∈ Fk for all k}.

If Xk is a sequence that is adapted to the σ-¬elds Fk , that is, Xk is Fk measurable when

k = 0, 1, 2, . . ., then knowing which events in Fk have occurred allows us to calculate Xk

for each k. So a reasonable de¬nition of FN should allow us to calculate XN whenever

we know which events in FN have occurred or not. Or phrased another way, we want XN

to be FN measurable. Where did the sequence Xk come from? It could be any adapted

sequence. Therefore one de¬nition of the σ-¬eld of events occurring before time N might

be:

Consider the collection of random variables XN where Xk is a sequence adapted

to Fk . Let GN be the smallest σ-¬eld with respect to which each of these random

variables XN is measurable.

In other words, we want GN to be the σ-¬eld generated by the collection of random

variables XN for all sequences Xk that are adapted to Fk .

We show in Note 1 that FN = GN . The σ-¬eld FN is just a bit easier to work with.

Now we proceed to the strong Markov property for Brownian motion, the proof of

which is given in Note 2.

75

Proposition 17.1. If Xt is a Brownian motion and T is a bounded stopping time, then

XT +t ’ XT is a mean 0 variance t random variable and is independent of FT .

This proposition says: if you want to predict XT +t , you could do it knowing all of

FT or just knowing XT . Since XT +t ’ XT is independent of FT , the extra information

given in FT does you no good at all.

We need a way of expressing the Markov and strong Markov properties that will

generalize to other processes.

Let Wt be a Brownian motion. Consider the process Wtx = x + Wt , which is known

as Brownian motion started at x. De¬ne „¦ to be set of continuous functions on [0, ∞), let

Xt (ω) = ω(t), and let the σ-¬eld be the one generated by the Xt . De¬ne Px on („¦ , F ) by

Px (Xt1 ∈ A1 , . . . , Xtn ∈ An ) = P(Wtx ∈ A1 , . . . , Wtx ∈ An ).

1 n

What we have done is gone from one probability space „¦ with many processes Wtx to one

process Xt with many probability measures Px .