intuitive introduction to stochastic calculushichoi/finmath/finance_lecture_2012_ch4.pdf · reader...

Chapter 4

Intuitive Introduction toStochastic Calculus

Copyright c©2008–2012 Hyeong In Choi, All rights reserved.

In this chapter, we introduce the concept of Brownian motion andthe Ito integral. We then derive the famous Ito formula, which isthe foundation of everything we do in the subsequent part of thislecture. We end this chapter by introducing the measure changeformula, in particular, the celebrated Girsanov theorem, which willbe extensively used throughout this lecture.

The materials we have to cover in this chapter is quite vast andthus it is not possible to treat them in a mathematically rigorousway, nor is it necessary to do so, as the reader can consult manyexcellent books. What we are aiming at in this chapter is to help thereader grasp the essence of the involved mathematics in an intuitiveand accessible manner.

4.1 Brownian Motion

The Brownian motion is, roughly speaking, a limit of random walks.Fix the time interval [0, 1] and divide it into n equal sub-intervalsof length 1

n . Suppose we are tossing a fair coin n times; i.e., letX1, · · · , Xn be a sequence of I.I.D. random variables such that

Xi =

1, with probability 1/2,−1, with probability 1/2.

Then by the Central Limit Theorem,

X1 + · · ·+Xn√n

4.1. BROWNIAN MOTION 113

converges weakly to a standard Gaussian random variable Z that hasmean zero and variance 1. This situation is depicted in the standardlattice model:

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•

•

•

•

•

•

•

. . .

. . .

. . .

•

•

•

•

•

•

•

•

...

0 1n

2n

3n · · · n

n = 1

0

h

2h

3h

−h

0

h

−2h

−h

−3h

Figure 4.1: Lattice over [0, 1].

In this model, the values of the nodes start with 0 at the rootnode, and the value of the child node will increase or decrease byh = 1√

ndepending on the outcome of toss is +1 or −1. [Note in

particular that this is an additive random walk, not the multiplica-tive one discussed in Chapter 3.] Let us extend this lattice up toarbitrarily chosen time t, and form the sum

W (n)(t) =X1 + · · ·+X[nt]√

n, (4.1)

where [x] is the Gauss symbol representing the largest integer lessthan or equal to x, i.e., [x] = bxc, the floor of x. The situation isdepicted in Figure 4.2. The reason we had to take the Gauss symbol[nt] is that nt may not be an integer so that the summation can bedone only up to [nt] terms. Let W (t) be the weak limit of W (n)(t).


•

•

•

•

•

•

•

•

•

•

. . .

. . .

. . .

•

•

•

•

•

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...

0 1n

2n

3n · · · [nt]

n

0

h

2h

3h

−h

0

h

−2h

−h

−3h

Figure 4.2: Lattice over [0, t].

Then rewriting W (n)(t) by

W (n)(t) =√t

√[nt]√nt

X1 + · · ·+X[nt]√[nt]

and noting that

√[nt]√nt

converges to 1, we can conclude that W (t) ∈N(0, t), where Y ∈ N(µ, σ2) means that Y is a random variable withthe normal (Gaussian) probability density function with mean µ andvariance σ2.

√tZ is an example such that

√tZ ∈ N(0, t), too. But

W (t) is more than the standard Gaussian random variable multipliedby√t, which we will see shortly. Before we go on, we need to make

an observation.

Remark 4.1. W (t) defined thus far is a random variable in the sensethat we know its law, i.e., induced probability of R. But we have notyet established the sample space Ω nor other critical components likeF or P of the underlying probability space (Ω,F , P ). It is the topic


we will turn to after we study various immediate consequences of thedefinition.

W (t) has many nice properties. First, look at

W (n)(t)−W (n)(s) =X1 + · · ·+Xnt√

n− X1 + · · ·+Xns√

n

for 0 < s < t. (Hence, we neglect to pay attention to the possibilitythat nt or ns may not be an integer, and pretend both are. Thereason is that while we can correctly write the right hand side byusing the Gauss symbol as we have done in (4.1), it does not reallymatter after we take the limit as n→∞.) Thus,

W (n)(t)−W (n)(s) =Xns+1 + · · ·+Xnt√

n

=√t− s Xns+1 + · · ·+Xnt√

n(t− s). (4.2)

It is clear that the right hand side of (4.2) converges weakly to therandom variable ∈ N(0, t−s) while the left hand side does to W (t)−W (s).

Moreover, since X1, · · · , Xns are independent of Xns+1, · · · , Xnt,W (n)(s) is independent of W (n)(t) −W (n)(s). Therefore their weaklimits must be independent. i.e., W (s) is independent of W (t) −W (s). (Independence is preserved under weak limit.) These proper-ties characterize the Brownian motion. Before we state it, we needthe following definition.

Definition 4.2. Let I be an interval in [0,∞). A continuous timestochastic process Xt is a family of random variables for t ∈ I definedover a probability space (Ω,F , P ). In particular, for each t, Xt is arandom variable Xt : Ω → R. An information structure is a family(Ft) of sub σ-fields of F parameterized by I such that for any t1, t2 ∈I with t1 < t2, Ft1 is a sub σ-field of Ft2.

Definition 4.3. A Brownian motion is a continuous time stochasticprocess Wt defined on some probability space (Ω,F , P ) for t ∈ [0,∞)satisfying

(i) Wt has continuous sample paths, i.e., for almost all w ∈ Ω,Wt(w) is a continuous function of t,

(ii) W0 = 0,

(iii) for 0 ≤ s < t, Wt −Ws ∈ N(0, t− s),

(iv) for 0 ≤ s < t, Wt −Ws is independent of Wu for ∀u ∈ [0, s].


We have shown above that the weak limit Wt of W (n)(t) satisfiesProperties (ii), (iii) and (iv). As for Property (i), it takes some workto make sense out of the context of the current construction of Wt

presented above.Although we have shown that Wt as a weak limit of W (n)(t)

satisfies some of the desirable properties of the Brownian motion, wecannot yet say we have “contructed” Wt, because to do so requirespinning down the probability space (Ω,F , P ) on which Wt is definedas a random variable Wt : Ω → R, which is what is to be discussedbelow.

• Sample space

Define the sample space Ω to be the set of all continuous paths in Rstarting at 0. To be precise, we set

Ω = ω : ω : [0,∞)→ R is continuous and ω(0) = 0.

• Stochastic process

The Stochastic process Wt is defined in the tautological manner: i.e.,for each t ≥ 0, define Wt to be the random variable

Wt : Ω→ R

defined byWt(w) = ω(t)

for every sample point (i.e., path) ω ∈ Ω. It is also called a projectionmap, because the value of Wt at ω is simply the value of ω at time t.

• σ-field

For a Borel set B of R and t ≥ 0, define

C(t,B) = w ∈ Ω : w(t) ∈ B,

which is called a basic cylinder set. C(t,B) is seen to be the collectionof paths whose value at time t is in B. The σ-field F is defined to bethe smallest σ-field containing all basic cylinder sets for any t ≥ 0and Borel set B of R. Thus in particular, the following cylinder set

C(t1,B1)∩ · · · ∩C(tN ,BN ) = w ∈ Ω : w(t1) ∈ B1, · · · , w(tN ) ∈ BN

is a measurable subset of Ω, and furthermore, the set

∞⋂i=1

C(ti,Bi) = w ∈ Ω : w(ti) ∈ Bi, for every i = 1, 2, · · · .


is also a measurable subset of Ω for countable collection of Borel setsBi∞i=1. In other words, the event of specifying the range of valuesof Wt at countably infinitely many time t is a measurable event.

• Wiener measure

Define the probability measure, usually called the Wiener measure,P as follows:

P (C(t,B)) = P (Wt ∈ B)

=

∫B

1√2πt

e−x2

2t dx

If we interpret the integrand as the probability density function ofthe heat (Brownian) particle (which starts from x = 0 at time t = 0)to reach x at time t, P (C(t,B)) is the probability of the Brownianparticle to be in the set B at time t. In fact, we use the followingnotation

p(t, x, y) =1√2πt

e−(x−y)2

2t . (4.3)

and

p(t, x) = p(t, 0, x) =1√2πt

e−x2

2t . (4.4)

Note that p(t, x, y) stands for the probability density function of theheat (Brownian) particle starting from x at t = 0 to reach y at timet. For the intersection of two cylinder sets, define

P (C(t1,B1) ∩ C(t2,B2))

= P (Wt1 ∈ B1,Wt2 ∈ B2)

=

∫B1

∫B2

p(t2 − t1, x1, x2)p(t1, 0, x1)dx2dx1 (4.5)

The inside integral ∫B2

p(t2 − t1, x1, x2)dx2 (4.6)

signifies the probability of the Brownian particle starting from x1 tobe in B2 at time t2 − t1, i.e., it is a conditional probability whosestarting point conditioned at x1. Then the iterated integral (4.5) isto be interpreted as the probability gotten by averaging (4.6) (w.r.t.the density function p(t1, 0, x1)) over B1. Pursuing this kind of idea,


time

R space

t1 t2

•x1

• x2

^

_

B1

^

_

B2

t2 − t1

Figure 4.3: Meaning of iterated integral.

the probability of finite intersection of cylinder set is defined by

P (C(t1,B1) ∩ · · · ∩ C(tN ,BN ))

= P (Wt1 ∈ B1, . . . ,WtN ∈ BN )

=

∫B1

∫B2

· · ·∫BN

p(tN − tN−1, xN−1, xN ) · · ·

p(t2 − t1, x1, x2)p(t1, 0, x1)dxN · · · dx2dx1.

We then have to define P on all sets in F , which is done by invokingthe Kolmogorov extension theorem. The reader may consult manyexcellent books on this subject. The measure defined in this wayis called the Wiener measure, and often times people confuse theWiener measure with the Brownian motion, as it captures most ofthe essence of the Brownian motion.

• Information Structure

Before we go on, let us introduce the following notation. Let E bea collection of random variables. We define σ(E) to be the smallestσ-field that makes every random variable in E measurable.

The information structure then is defined as

Ft = σ(Ws : s ≤ t).

Note that it is trivial to check that Fs ⊂ Ft if s < t. Note also thatFt ⊂ F as Wt ∈ F for all t.


Now let Bi∞i=1 be a countable collection of Borel sets. Supposeti∞i=1 be a sequence such that ti ≤ t. For each i,

ω ∈ Ω : ω(ti) ∈ Bi= ω ∈ Ω : Wti(ω) ∈ Bi= Wti ∈ Bi.

Since ti ≤ t,Wti ∈ Bi ∈ Fti ⊂ Ft.

Therefore it is easy to see that Ft is a σ-field generated by the familyof all such sets. However as Ft only concerns with events before or attime t, it distinguishes no event that occurs after t. For example, letB is a Boreal set. Then ω ∈ Ω : ω(s) ∈ B /∈ Ft if s > t. Pictoriallyany Ft measurable set looks like the set of paths in Figure 4.4

time

R space

t1 t2 · · · tN t^

_

B1

^

_B2

^

_

BN

Figure 4.4: Typical Ft measurable set.

Remark 4.4. It helps to visualize Ft if the reader goes back toChapter 2 and recollect how the discrete information structure wasdefined. In fact, an alert reader may discover that our continuousversion of information structure is a direct analog of the discrete one.

Remark 4.5 (Uniqueness of Brownian motion). This is the rightplace to ask whether the Brownian motion is unique. The answer is“Yes” and “No.” First of all, there are many ways of constructing(Ω,F , P ) other than that expounded above. For instance, we maydefine Ω to be the sets of path with discontinuities, and proceed


with the same construction. It then can be proved that the set ofnon-continuous paths has the (Wiener) measure zero so that in theprobabilistic sense, they don’t matter. There are many other waysof devising (Ω,F , P ) so that in this sense the Brownian motion isnot unique. However, in no matter which way we define (Ω,F , P ),the properties (i) to (iv) of Definition 4.3 are satisfied. Thus if weconcentrate out attention on the “law” of the Brownian motion, i.e.,on how Wt behaves as in Definition 4.3, they are unique.

Let us now give a more general definition of Brownian motion.

Definition 4.6. Let (Ω,F , P ) be a probability space, and let (Ft) bean information structure on it. A stochastic process Xt is a Brownianmotion if

(i) Xt ∈ Ft,

(ii) EP [|Xt|] <∞, ∀t ≥ 0,

(iii) for almost all w ∈ Ω, Xt(w) is a continuous function of t,

(iv) X0(w) = 0, ∀w ∈ Ω,

(v) for any 0 ≤ s < t, Xt −Xs ∈ N(0, t− s),

(vi) for any 0 ≤ s < t, Xt −Xs is independent of Fs.

We also need the following definition.

Definition 4.7. Let (Ω,F , P ) be a probability space and let (Ft) bean information structure on it. Let Xt be a stochastic process suchthat Xt ∈ Ft, ∀t ≥ 0. Assume also that EP [|Xt|] < ∞, ∀t ≥ 0. Wesay Xt is a martingale, if for all 0 ≤ s < t, EP [Xt|Fs] = Xs. Wesay it is a submartingale(resp. supermartingale) if EP [Xt|Fs] ≥ Xs

(resp, EP [Xt|Fs] ≤ Xs).

Remark 4.8. The martingale condition of Xt depends crucially onP as well as (Ft)t≥0. To emphasize its dependence on various en-tities, we sometimes say that Xt is a P -martingale, Ft-martingale,or (P,Ft)-martingale depending on which context we want to em-phasize. The same appellation convention applies to the Brownianmotion.

Proposition 4.9. Brownian motion is a martingale.

Proof. Since Xt ∈ N(0, t), the integrability

EP [|Xt|] =

∫ ∞−∞|x| 1√

2πte−x

2/2tdx <∞

4.2. ITO INTEGRAL 121

is easily proved. Let 0 ≤ s < t. Since Xt −Xs is independent of Fs,

EP [Xt −Xs|Fs] = EP [Xt −Xs] = 0,

where the last equality is due to the fact that Xt−Xs has mean zero.Since Xs ∈ Fs, it means that

0 = EP [Xt −Xs|Fs] = EP [Xt|Fs]− EP [Xs|Fs]= EP [Xt|Fs]−Xs.

The following proposition is also useful later.

Proposition 4.10. Suppose Xt is a Brownian motion, then X2t − t

is a martingale.

Proof. It is easy to check that X2t −t is integrable and has continuous

path as long as X does. Now

EP [X2t −X2

s |Fs] = EP [(Xt −Xs +Xs)2 −X2

s |Fs]= EP [(Xt −Xs)

2 + 2Xs(Xt −Xs)|Fs]= (t− s) + 2EP [Xs(Xt −Xs)|Fs]

(∵ Xt −Xs ∈ N(0, t− s))= (t− s) + 2XsEP [Xt −Xs|Fs] (∵ Xs ∈ Fs)

= t− s. (∵ Xt −Xs ∈ N(0, t− s))

Since Xs ∈ Fs, we have

EP (X2t |Fs)−X2

s = t− s.

ThusEP [X2

t − t|Fs] = X2s − s.

4.2 Ito Integral

For reasons that become clearer later, it is reasonable to model themovement of stock price as a stochastic process St on a fixed prob-ability space (Ω,F , P ) and an information structure (Ft)t≥0 on it.Namely, let us assume that St is a stochastic process on (Ω,F , P )such that St ∈ Ft, i.e., adapted, representing the stock price and letBt ∈ Ft be the process representing the bank account.


Typically a trader trades stocks at discrete times, say, specifiedby a partition P given by:

P : 0 = t0 < t1 < · · · < tN = T.

Let us form a self-financing portfolio in which the trading(portfoliochange) is done at t0, t1, . . . , tN . Say, let ψti be the number of stocksto be carried during the period ti ≤ t < ti+1 and φti be the number ofunits of the bank account for the same period. Then the portfolio’svalue VT at T is given by

VT = V0 +N−1∑i=0

ψti∆Sti +N−1∑i=0

φti∆Bti (4.7)

when ∆Sti = Sti+1 − Sti and ∆Bti = Bti+1 −Bti .Fix w ∈ Ω. As there is no change in the portfolio during [ti, ti+1),

ψti(w) and φti(w) are determined at ti, the values of ψt(w) and φt(w)for t ∈ [ti, ti+1) remain the same. Figure 4.5 shows their pattern:

· · · · · ·

0 = t0 t1 t2 · · · ti ti+1 · · · tN−1 tN = T

•

•

• •

Figure 4.5: Pattern of ψt(w)

In other words, we can write ψt as

ψt(w) =N−1∑j=0

etj (w)1[tj ,tj+1)(t) (4.8)

where etj is an random variable ∈ Ftj and 1[tj ,tj+1) is the indicatorfunction on R defined by

1[tj ,tj+1)(t) =

1, if t ∈ [tj , tj+1)0, otherwise.

A stochastic process of the form (4.8) is called an elementary stochas-tic process.

Our aim in this section is to study what happens to the sum(4.7) as we let |∆P| → 0, i.e., N →∞. For this, let ζt and ξt be Ft


adapted stochastic processes on (Ω,F , P ). We assume further thatall paths of ζt and ξt are continuous (One way of assuring it is tochoose Ω to be the set of all continuous paths in the first place aswe did in the construction of the Brownian motion in Section 4.1.)Then for the partition

P : 0 = t0 < t1 < · · · < tN = T,

the sum (4.7) is written as

VT (w) = V0 +

N−1∑j=0

ζtj (w)∆Stj (w) +

N−1∑j=0

ξtj (w)∆Btj (w) (4.9)

for each w ∈ Ω. The question is: “Would the sum in (4.9) convergeto anything for each w?” Before we attempt to answer this question,let us recall several facts from mathematical analysis.

• Functions of finite (bounded) variation

Definition 4.11. Let f : [a, b] → R be a function. We say it is ofbounded variation (finite variation) if

N−1∑i=0

|f(ti+1)− f(ti)| ≤ C

for some constant C > 0 for any partition

P : a = t0 < t1 < · · · < tN = b

of [a, b]. T ba(f) defined by

T ba(f) = supP

N−1∑i=0

|f(ti+1)− f(ti)|

is called the total variation of f over [a, b], where the supremum istaken over all partitions of [a, b].

Remark 4.12. A typical example of FV(or BV) function is a contin-uously differentiable(C1) function. Then, for each interval [ti, ti+1],there exists t∗i ∈ (ti, ti+1) such that

f(ti+1)− f(ti) = f ′(t∗i )∆ti,

where ∆ti = ti+1 − ti. Thus

N−1∑i=0

|f(ti+1)− f(ti)| =N−1∑i=0

|f ′(t∗i )|∆ti,


which converges to ∫ b

a|f ′(t)|dt.

The functions of bounded variation are of course more general thanC1 functions but not much, so the readers who are not too familiarwith mathematical analysis may pretend the FV(or BV) functionsare roughly the ones which are differentiable except at a few “bad”points and the absolute value of the derivative is bounded by someconstant.

Definition 4.13. A stochastic process Xt is a FV(or BV) process iffor almost all w ∈ Ω, Xt(w) is a FV(or BV) function of t. The totalvariation process X or |X|t of Xt is defined by

|X|t(w) = supPt

N−1∑i=0

|Xti+1(w)−Xti(w)|, (4.10)

where the supremum is taken over all partitions

Pt : 0 = t0 < t1 < · · · < tN = t

of [0, t].

4.2.0.1 • Riemann-Stieltjes integral

Let us now recall the Riemann-Stieltjes integral. Suppose f : [a, b]→R is a function and let α : [a, b]→ R be a function of finite variation.The Riemann- Stieltjes sum over the partition

P : a = t0 < t1 < · · · < tN = b.

of [a, b] is defined as

N∑i=0

f(t∗i )(α(ti+1)− α(ti)) =N∑i=0

f(t∗i )∆α(ti+1),

where t∗i ∈ (ti, ti+1). If this sum converges to a limit as we take|∆P| → 0 for every choice of t∗i in (ti, ti+1), we say the Riemann-Stieltjes integral exists and we denote the limit by∫ b

af(t)dα(t) =

∫ b

afdα.

It is clear that if f is continuous and α is C1, then this Riemann-Stieltjes integral exists and is equal to∫ b

af(t)α′(t)dt.

In fact, it is equally easy to see that if f is continuous and α is ofFV, then the Riemann-Stieltjes integral exists.


• The FV processes as the integrator of the Riemann-Stieltjesintegral

A typical example of a FV process is the bank account process Bt.It is typically defined by

dBt = rtBtdt

B0 = 1, (4.11)

where rt is usually (at least in this lecture) taken to be a continuousstochastic process so that (4.11) really becomes an O.D.E.

dBt(ω) = rt(ω)Bt(ω)dt

B0(ω) = 1

for each fixed ω ∈ Ω. The solution of this separable O.D.E. is simple,i.e., Bt(w) is written as

Bt(w) = exp

∫ t

0rs(w)ds.

In other words, we write

Bt = exp

(∫ t

0rsds

). (4.12)

In case rt(w) is always constant r, Bt becomes ert, i.e., the result ofthe continuous compounding. As a result of (4.12), it is trivial to seethat Bt is a FV process, if rt is a continuous stochastic process. It isto say that for each ω ∈ Ω, Bt(ω) is a finite variation function of t.

Since ξt is assumed to be a continuous process and Bt is a FVprocess, the last sum of (4.9) is seem to converge to the Riemann-Stieltjes integral ∫ t

0ξs(w)dBs(w),

as a function of t when each ω is held fixed. Dropping w, we writeit as ∫ t

0ξsdBs. (4.13)

In this sense, (4.13) is called the pathwise integral. Let us nowturn to the first sum of (4.9):

N−1∑j=0

ζtj (w)∆Stj (w) (4.14)


Typically, our model for St involves the Brownian motion Wt. So forthe purpose of simple illustration, let us, for now, assume St is theBrownian motion Wt. In this case (4.14) becomes

N−1∑j=0

ζtj (w)∆Wtj (w). (4.15)

Unfortunately, it is well known that the Brownian motion is not ofFV. In fact, its paths have the fractal nature so that, for instance,Wt(w) is a nowhere differentiable functions of t for almost all w ∈Ω. It means that it is not to be expected for the integral (4.15) toconverge to anything for any fixed w ∈ Ω. As a way of illustration,let us look at the following sum

N−1∑i=0

Wt∗i∆Wti =

N−1∑i=0

Wt∗i(Wti+1 −Wti)

for the partition

P : 0 = t0 < t1 < · · · < tN = T,

where t∗i is an appropriately chosen number in [ti, ti+1]. If (4.15)converges in the sense of Riemann-Stieltjes integral, then for almostall w ∈ Ω,

N−1∑i=0

Wt∗i(w)∆Wti(w) (4.16)

must converge to the same number regardless of the choice of t∗i ∈[ti, ti+1]. But as we shall see, it is not the case. First, let t∗i = ti.Then

E

[N−1∑i=0


]=

N−1∑i=0

E[Wti(Wti+1 −Wti)]

=

N−1∑i=0

E[Wti ] E[(Wti+1 −Wti)]

= 0.

The second to the last equality is due to the fact that Wti is inde-


pendent of Wti+1 −Wti . Next, choose t∗i = ti+1. Then

E

[N−1∑i=0


]

=

N−1∑i=0

E[Wti+1(Wti+1 −Wti)]−N−1∑i=0

E[Wti(Wti+1 −Wti)]

(∵ the second sum is 0.)

=

N−1∑i=0

E[(Wti+1 −Wti)2]

=

N−1∑i=0

(ti+1 − ti)

= T.

Therefore it does not make sense to talk about∫ T

0Wt(w)dWt(w)

as the pathwise limit of (4.16), whereas∫ T

0ξt(w)dBt(w)

makes sense as a pathwise integral for each w ∈ Ω.

• Ito’s Approach

Ito came up with a great idea of defining the stochastic integral∫ T

0ζtdWt

not in pathwise but in a certain average sense. It involves a seriesof lemmas. We present here Ito’s original L2 approach with someminor modification.

Definition 4.14. A stochastic process φt is called an elementaryprocess if φt(w) is written as a finite sum of the form

φt(w) =

N−1∑j=0

etj (w)1[tj ,tj+1)(t)

for some partition

P : 0 = t0 < t1 < · · · < tN = T,


and random variables etj ∈ Ftj for j = 0, . . . , N − 1. For such φt,we define ∫ T

0φt(ω)dWt(ω) =

N−1∑j=0

etj (ω)∆Wtj (ω),

in the pathwise way, where ∆Wtj = Wtj+1 −Wtj.

The following lemma is the key tool this approach relies upon.

Lemma 4.15 (L2-isometry). For the elementary process φt,

E

[(∫ T

0φtdWt

)2]

= E

[∫ T

0φ2tdt

].

Proof. Let φt be of the form

φt(w) =N−1∑j=0

etj (w)1[tj ,tj+1)(t).

Then ∫ T

0φtdWt =

N−1∑j=0

etj∆Wtj ,

where ∆Wtj = Wtj+1 −Wtj . Then

(∫ T

0φtdWt

)2

=∑i6=j

0≤i,j≤N−1

etietj∆Wti∆Wtj +

N−1∑j=0

e2tj (∆Wtj )

2.

Let i < j. Since ∆Wtj is independent of Ftj , so is it independent ofeti , etj and ∆Wti . Thus

E[etietj∆Wti∆Wtj ] = E[etietj∆Wti ]E[∆Wtj ] = 0.


The same conclusion is true for i > j. Thus

E

[(∫ T

0φtdWt

)2]

= E

N−1∑j=0

e2tj (∆Wtj )

2

=

N−1∑j=0

E[e2tj ]E[(∆Wtj )

2]

(∵ ∆Wtj and etj are independent.)

=N−1∑j=0

E[e2tj ](tj+1 − tj)

= E

N−1∑j=0

e2tj (tj+1 − tj)

= E

[∫ T

0φ2tdt

].

We now present the rough idea of the steps that lead to Ito inte-gral.

Step 1: Any “reasonable” (all the stochastic process we deal withare “reasonable”) stochastic process ζt is a limit of a sequence of

elementary processes φ(n)t in the sense that

limn→∞

E

[∫ T

0

(ζt − φ(n)

t

)2dt

]= 0.

Step 2: The sequence of elementary process φ(n)t converging to

ζt in the sense of Step 1 is a Cauchy sequence in the L2 sense meaningthat for any ε > 0,

E

[(∫ T

0φ

(n)t dWt −

∫ T

0φ

(m)t dWt

)2]

= E

[(∫ T

0

(φ

(n)t − φ

(m)t

)dWt

)2]

= E

[∫ T

0

(φ

(n)t − φ

(m)t

)2dt

]< ε

for sufficiently large m and n. Note that the last equality is theconsequence of the L2-isometry lemma above.


Step 3: Step 2 says that the sequence of random variables∫ T

0φ

(n)t dWt : Ω→ R

is itself a Cauchy sequence in the L2(Ω), which means that for anyε > 0

E

[(∫ T

0φ

(n)t dWt −

∫ T

0φ

(m)t dWt

)2]< ε

for sufficiently large m and n. Then from the general theory of L2-

space, this sequence∫ T

0 φ(n)t dWt converges to a limit, which is defined

to be the Ito integral∫ T

0 ζtdWt, i.e.,∫ T

0φ

(n)t dWt

L2(Ω)−−−−→∫ T

0ζtdWt.

In other words, the Ito integral∫ T

0 ζtdWt is defined this way in theaverage(L2) sense. This is the gist of Ito’s remarkable insight. Fur-thermore, varying t, we can define a new stochastic process

I(ζ)t =

∫ t

0ζsdWs.

Then if ζt is “reasonable”, the following facts hold:

Proposition 4.16.

(i) For all t ≥ 0, I(ζ)t is L2, meaning that

E[(I(ζ)t)2] <∞.

(ii) I(ζ)t has continuous sample paths for almost all w ∈ Ω.

(iii) I(ζ)t ∈ Ft and is an Ft-martingale.

Let us see why (iii) holds. First, assume ζt is an elementaryprocess of the form

ζt(w) =

N−1∑j=0

etj (w)1[tj ,tj+1)(t).

Fix 0 ≤ s < t. We may assume without loss out of generality that sis one of the partition point, say tk, for otherwise we may add s tothe partition and declare es = etk if tk ≤ s < tk+1. Then

I(ζ)t =N−1∑j=0

etj∆Wtj

=k−1∑j=0

etj∆Wtj +N−1∑j=k

etj∆Wtj .

4.3. RUDIMENT OF STOCHASTIC DIFFERENTIALEQUATIONS 131

For j > k, Ftj ⊃ Ftk . Thus

E[etj∆Wtj |Ftk ] = E[E[etj∆Wtj | Ftj ] | Ftk

].

Now etj ∈ Ftj and ∆Wtj is independent of Ftj ,

E[etj∆Wtj | Ftj ] = etjE[∆Wtj | Ftj ]= etjE[∆Wtj ]

= 0.

When j = k, etk ∈ Ftk , so that

E[etk∆Wtk | Ftk ] = etkE[∆Wtk | Ftk ]

= E[etk ] E[∆Wtk | Ftk ]

= 0.

For j ≤ k − 1, etj and ∆Wtj+1 ∈ Ftk = Fs. Thus

E[I(ζ)t|Fs] =k−1∑j=0

etj∆Wtj+1 = I(ζ)s. (4.17)

We have shown that (4.17) holds for elementary process ζt. Thenthe general statement follows by passing to the limit.

Remark 4.17. Of paramount importance in finance is the Fact (iii),i.e., the assertion that stochastic integral with respect to the Brow-nian motion is martingale.

4.3 Rudiment of Stochastic Differential Equa-tions

The ordinary differential equation is written in full generality in theform

dx(t)

dt= α(t, x(t)) (4.18)

where α is a given function with suitable continuity and differentia-bility properties, In general x(t) is an n-tuple of functions x(t) =(x1(t), . . . , xn(t)) and α(t, x(t)) is also of the form

α(t, x(t)) = (α1(t, x1(t), . . . , xn(t)), . . . , αn(t, x1(t), . . . , xn(t))).

It is used to model all kinds of natural phenomena of deterministicnature. But when there is a “noise,” one has to add a “noise” term.There are of course many ways to model the noise. The oldest and

4.3. RUDIMENT OF STOCHASTIC DIFFERENTIALEQUATIONS 132

still widely used is the so-called “white noise.” It is roughly speakinga “signal” that has certain universal noise characteristic. Formallythe white noise behaves like dWt

dt . The trouble is that it is well known

that the Brownian motion path is nowhere differentiable so that dWtdt

does not make sense.1 Forgetting the “correctness,” the ordinarydifferential equation with the “white noise” term added is written as

dx(t)

dt= α(t, x(t)) + β(t, x(t))

dWt

dt. (4.19)

For simplicity, let us now deal with the case n = 1. Multiplying dton both sides of (4.19) and writing Xt in place of x(t), we have

dXt = α(t,Xt)dt+ β(t,Xt)dWt. (4.20)

While (4.19) is only a formal expression, (4.20) can be rigorouslydefined and studied. This (4.20) is a typical form of the stochasticdifferential equation. It turns out that for most “reasonable” α andβ, (4.20) has the solution Xt as defined as the stochastic processsatisfying the following stochastic integral equation

Xt = X0 +

∫ t

0α(s,Xs)ds+

∫ t

0β(s,Xs)dWs. (4.21)

As the Brownian motion employed in (4.20) or (4.21) are notunique in general, the question concerning which Brownian motionis to be used naturally arises. Going further along this line, one mayask whether the probability setup (Ω,F , P ) and (Ft) are also givena priori or they are to be constructed as part of the solution. Themost stringent is the concept of strong solution. It presuppose all theprobability apparatus like (Ω,F , P ) and (Ft), and, most of all, theBrownian motion Wt as given, and only seeks to find the stochasticprocess Xt satisfying (4.21). On the other hand, one may presupposesome or none of the probability setup and tries to construct themissing ones as part of the solution apparatus. This kind of solutionis generally called a weak solution. For instance, the most generalapproach would be to assume nothing and try to construct (Ω,F , P )and (Ft) and Wt as well as Xt that satisfying (4.21).

Under “reasonable” conditions on α and β, it is guaranteed thatthe strong solution exists, and thus it is this strong solution that weutilize for the rest of this lecture unless stated otherwise.

The concept of uniqueness is also a bit complicated. A strongsolution Xt is called pathwise unique, if for Yt another solution sat-isfying the same initial condition, Xt(w) = Yt(w) holds for almost

1Later, it was shown to make sense in a complicated functional analysis setting,but it is beyond the scope of this lecture.

4.4. QUADRATIC VARIATION PROCESS 133

all w ∈ Ω. For weak solutions, pathwise uniqueness makes no sense.But then one may ask if its law (induced measure) is unique. In anycase, if α and β are “reasonable,” it is known that either uniquenessholds.

4.4 Quadratic Variation Process

Recall that if a stochastic process Xt is a FV processes, its totalvariation process |X|t is defined by

|X|t = lim|∆P|→0

N−1∑j=0

|∆Xtj |,

where the limit is taken over all partition Pt of [0, t]

Pt : 0 = t0 < t1 < · · · < tN = t,

and ∆Xtj = Xtj+1 −Xtj . (Note that by definition |X|0 = 0.)As we said before, the Brownian motion is not a FV-process. It

can be heuristically seen as follows: Now ∆Wtj = Wtj+1 − Wtj ∈N(0,∆tj) where ∆tj = tj+1 − tj . So

∆Wtj√∆tj∈ N(0, 1)

In other words, ∆Wtj scales like√

∆tj , so that∑

j |∆Wtj | ∼∑

j

√∆tj ,

which must diverge. On the other hand, if we take the sum of thesquares, i.e.,

∑j(∆Wtj )

2, it must behave like∑

j ∆tj = t. In fact, itcan be made a rigorous fact.

Lemma 4.18.

lim|∆P|→0

E

N−1∑j=0

(∆Wj)2 − t

2

= 0,

where ∆Wj stands for ∆Wtj .

Proof. For simplicity of notation, we denote Wtj+1 −Wtj by ∆Wj

instead of ∆Wtj .

4.4. QUADRATIC VARIATION PROCESS 134

E

[∑j

((∆Wj)

2 −∆tj)2]

= E[∑i 6=j

((∆Wj)

2 −∆tj)(

(∆Wi)2 −∆ti

)]+E[∑

j

((∆Wj)

2 −∆tj)2]

=∑i 6=j

E[(∆Wj)

2 −∆tj]E[(∆Wi)

2 −∆ti]

+∑j

E[(∆Wj)

4 − 2(∆tj)(∆Wj)2 + (∆tj)

2]

=∑j

[3(∆tj)

2 − 2(∆tj)2 + (∆tj)

2]

= 2∑j

(∆tj)2

≤ 2[∑

j

(∆tj)]

sup |∆tj | → 0

as sup |∆tj | → 0.

In the course of the proof, we have used the fact

E[(∆Wi)

4

(∆ti)2

]=

1√2π

∫ ∞−∞

x4e−x2/2dx = 3,

i,e, E[(∆Wi)

4]

= 3(∆ti)2.

This lemma suggest the following definitions.

Definition 4.19. Let Xn be a sequence of random variables on(Ω,F , P ). We say Xn converges to a random variable X on (Ω,F , P )in L2 sense if

limn→∞

E|Xn −X|2 = 0.

In this cases, we write

XnL2

−→ X.

Definition 4.20. Let Xt be a stochastic process. Fix any t > 0, fora partition Pt : 0 = t0 < t1 < · · · < tN = t of [0, t], form a sum

N−1∑j=0

(∆Xtj )2 =

N−1∑j=0

(Xtj+1 −Xtj )2.

4.5. ITO FORMULA 135

If there is a process Yt such that for any t

N−1∑j=0

(∆Xtj )2 L2

−→ Yt,

where the limit is taken for all partitions Pt of [0, t] with |∆Pt| → 0,we say that Yt is the quadratic variation process of Xt and Yt isdenoted by 〈X〉t. Note that by definition 〈X〉0 = 0.

Remark 4.21. In this language, Lemma 4.18 can be interpreted assaying that

〈W 〉t = t. (4.22)

4.5 Ito formula

Ito formula is perhaps the single most important formula in the wholeof mathematical finance. Let f(t, x) be a C2 function of two (de-terministic) variables. Let Xt be a stochastic process. Ito formuladescribes the stochastic differential equation the new process f(t,Xt)satisfies.

Let P be a partition of [t0, t] given by

P : t0 < t1 < · · · < tN = t.

Then using the Taylor theorem2 for C2 functions the increment∆f(ti, Xti) = f(ti+1, Xti+1)− f(ti, Xti) can be written as

∆f(ti, Xti) = f(ti+1, Xti+1)− f(ti, Xti)

=∂f

∂t(ti, Xti)∆ti +

∂f

∂x(ti, Xti)∆Xti

+1

2

∂2f

∂t2(ti, Xti)(∆ti)

2 + 2∂2f

∂t∂x(ti, Xti)(∆ti)(∆Xti)

+∂2f

∂x2(ti, Xti)(∆Xti)

2

+ o(|∆ti|2 + |∆Xti |2).

2Taylor theorem for C2 function is given as

f(t+ h, x+ k) = f(t, x) +∂f

∂t(t, x)h+

∂f

∂x(t, x)k

+1

2

∂2f

∂t2(t, x)h2 + 2

∂2f

∂t∂x(t, x)hk +

∂2f

∂x2(t, x)k2

+ o(|h|2 + |k|2)

where o(|h|2 + |k|2) represents the remainder satisfying

lim|h|,|k|→0

o(|h|2 + |k|2)

|h|2 + |k|2 = 0


Taking the sum, we have

f(t,Xt)− f(t0, Xt0) =N−1∑i=0

∆f(ti, Xti)

=∑i

∂f

∂t(ti, Xti)∆ti +

∑i

∂f

∂x(ti, Xti)∆Xti

+1

2

∑i

∂2f


2 (4.23)

+ 2∑i

∂2f

∂t∂x(ti, Xti)(∆ti)(∆Xti)

+∑i

∂2f


2

+ o

(|∆ti|2 + |∆Xti |2

)Let us assume that Xt satisfies the stochastic differential equation

dXt = αdt+ βdWt.

Assume also that the quadratic variation process 〈X〉t of Xt exists.Then as |∆P| → 0, it is not hard to see that the limit of each termof (4.23) are as follows.

N−1∑i=0

∂f

∂t(ti, Xti)∆ti −→

∫ t

t0

∂f

∂t(u,Xu)du

N−1∑i=0

∂f

∂x(ti, Xti)∆Xti −→

∫ t

t0

∂f

∂x(u,Xu)dXu

=

∫ t

t0

∂f

∂x(u,Xu)α(u,Xu)dt

+

∫ t

t0

∂f

∂x(u,Xu)β(u,Xu)dWu

NowN−1∑i=0

∂2f

∂t2(ti, Xti)∆ti −→

∫ t

t0

∂2f

∂t2(u,Xu)du ,

as |∆P| → 0. Therefore it is easy to see that

N−1∑i=0

∂2f


2 −→ 0 ,


as |∆P| → 0. Similarly,

N−1∑t=0

∂2f

∂t∂x(ti, Xti)(∆Xti) −→

∫ t

t0

∂2f

∂t∂x(u,Xu)dXu ,

as |∆P| → 0. Therefore

N−1∑t=0

∂2f

∂t∂x(ti, Xti)∆Xti∆ti −→ 0 ,

as |∆P| → 0. Finally, the limit of the last term of (4.23) is seen as

N−1∑i=0

∂2f


2 −→∫ t

t0

∂2f

∂x2(u,Xu)d〈X〉u.

It takes some amount of easy but tedious work to see the abovestatement is true, which is omitted here. The interested reader mayconsult many standard textbooks. It is also easy to see that the sumof the remainder term converge to zero. Thus we have the following:

Theorem 4.22 (Ito formula). Let Xt be a stochastic process satis-fying the stochastic differential equation

dXt = α(t,Xt)dt+ β(t,Xt)dWt

and let f(t, x) be a C2 function of two variables t and x. Then

f(t,Xt) = f(t0, Xt0) +

∫ t

t0

∂f

∂t(u,Xu)du

+

∫ t

t0

∂f

∂x(u,Xu)dXu +

1

2

∫ t

t0

∂2f

∂x2(u,Xu)d〈X〉u.

In stochastic differential notation, we write it symbolically as

df(t,Xt) =∂f

∂t(t,Xt)dt+

∂f

∂x(t,Xt)dXt +

1

2

∂2f

∂x2(t,Xt)d〈X〉t.

When Xt is the Brownian motion itself, we have seen that 〈W 〉t =t. Thus we have

Corollary 4.23. Symbolically,

df(t,Wt) =∂f

∂t(t,Wt)dt+

∂f

∂x(t,Wt)dWt +

1

2

∂2f

∂x2(t,Wt)dt.


In fact, the above Corollary 4.23 can be interpreted as follows.Symbolically, if we were to write df(t,Wt) up to the second orderterm, it could have been written as

df(t,Wt) =∂f

∂t(t,Wt)dt+

∂f

∂x(t,Wt)dWt +

1

2

∂2f

∂t2(t,Wt)(dt)

2

+ 2∂2f

∂t∂x(t,Wt)(dt)(dWt) +

∂2f

∂x2(t,Wt)(dWt)

2. (4.24)

Comparing this with Corollary 4.23, we can conclude that the fol-lowing relations hold symbolically.

dt = dt

dWt = dWt

(dt)(dWt) = 0, (4.25)

(dt)2 = 0,

(dWt)2 = dt.

Recall that the quadratic variation process 〈X〉t is obtained asthe limit

N−1∑i=0

(∆Xti)2 −→ 〈X〉t − 〈X〉t0 .

In other words, 〈X〉t is written symbolically

〈X〉t =

∫ t

t0

(dXu)2,

or in symbolic notation

d〈X〉t = (dXt)2. (4.26)

Thus if Xt satisfies

dXt = α(t,Xt)dt+ β(t,Xt)dWt,

then d〈X〉t must be symbolically

d〈X〉t = (dXt)2

= α(t,Xt)2(dt)2 + 2α(t,Xt)β(t,Xt)(dt)(dWt) + β(t,Xt)

2(dWt)2.

Therefore, applying (4.25), we have

d〈X〉t = (dXt)2

= β(t,Xt)2dt. (4.27)

4.6. MULTI-DIMENSIONAL ITO FORMULA 139

Corollary 4.24. Let Xt be a stochastic process satisfying

dXt = α(t,Xt)dt+ β(t,Xt)dWt,

and let f(x) be a C2 function of one variable x. Then

df(Xt) = f ′(Xt)dXt +1

2f ′′(Xt)(dXt)

2

= f ′(Xt)dXt +1

2f ′′(Xt)β(t,Xt)

2dt.

The proof of the above Corollary 4.24 is a straightforward appli-cation of the Ito formula, (4.25),(4.26), and (4.27).

Example 4.25 (Geometric Brownian motioin). Let St be a stochas-tic process satisfying

dSt = St(µdt+ σdWt)

where µ and σ are given constants. Then letting f(x) = log x, andapply Corollary 4.24, we have

d(logSt) =1

StdSt −

1

2

1

S2t

(dSt)2

= µdt+ σdWt −1

2σ2dt

=(µ− 1

2σ2)dt+ σdWt.

Upon integrating, we have

logSt − logS0 =

∫ t

0

(µ− 1

2σ2)du+ σ

∫ t

0dWu

=(µ− 1

2σ2)t+ σWt.

Therefore

St = S0 exp

[(µ− 1

2σ2)t+ σWt

].

This is exactly what we got in Section 3.4 as the limit of the mul-tiplicative random walk, except that

√tZ there is now replaced with

Wt.

4.6 Multi-dimensional Ito formula

Let us now consider how to extend Ito formula for the case with morethan one stochastic processes. To do that, we need the followingconcept.


Definition 4.26. Let Xt and Yt be stochastic processes. The quadraticvariation process 〈X,Y 〉t of Xt and Yt is the process defined as theL2-limit, if exists, of∑

j

(Xtj+1 −Xtj )(Ytj+1 − Ytj )

as |∆P| → 0. In particular, 〈X〉t = 〈X,X〉t.

The above definition formally says that

〈X,Y 〉t =

∫ t

0dXudYu,

i.e., symbolically,d〈X,Y 〉t = dXtdYt.

For instance, suppose Xt and Yt satisfies

dXt = αdt+ βdWt

dYt = γdt+ δdWt.(4.28)

Then

d〈X,Y 〉t = dXtdYt

= βδdt.

Note also that

〈X + Y 〉t = 〈X + Y,X + Y 〉t= 〈X,X〉t + 2〈X,Y 〉t + 〈Y, Y 〉t= 〈X〉t + 2〈X,Y 〉t + 〈Y 〉t. (4.29)

Thus writing out 〈X − Y 〉t similarly and subtracting it from (4.29),we have

〈X,Y 〉t =1

4〈X + Y 〉t − 〈X − Y 〉t . (4.30)

Now let us describe the multi-dimensional Ito formula. Let f(t, x1, · ··, xn) be a C2 function of n+ 1 deterministic variables t, x1, . . . , xn.Let Xi

t be a stochastic process for i = 1, 2, . . . , n. Then the sec-ond order Taylor polynomial 3 of df(t,X1

t , . . . , Xnt ) can be written

3The Taylor theorem for C2 function f(y0, . . . , yn) is given by

f(y0 + h0, y1 + h1, . . . , yn + hn)

= f(y0, y1, . . . , yn) +

n∑i=0

∂f

∂yihi +

1

2

n∑i=0

n∑j=0

∂2f

∂yi∂yjhihj + o(|h|2).

where |h| =√∑n

i=0 h2i


formally as

df(t,X1t , . . . , X

nt )

=∂f

∂t(t,X1

t , . . . , Xnt )dt+

n∑i=1

∂f

∂xi(t,X1

t , . . . , Xnt )dXi

t

+1

2

∂2f

∂t2(t,X1

t , . . . , Xnt )(dt)2 + 2

n∑i=1

∂2f

∂t∂xi(t,X1

t , . . . , Xnt )dtdXi

t

+n∑i=1

n∑j=1

∂2f

∂xi∂xj(t,X1

t , . . . , Xnt )dXi

tdXjt

.

If Xit is of the form

dXit = αidt+ βidWt,

the formal relation (4.25) tells that dtdXit = 0 and (dt)2 = 0.

Also, the above quadratic variation formula says that dXitdX

jt =

d〈Xi, Xj〉t. Therefore we have the following

Theorem 4.27 (Multi-dimensional Ito formula). Let f(t, x1, . . . , xn)be a C2 function of n+ 1 variables t, x1, . . . , xn, and let X1

t , . . . , Xnt

be stochastic process for which dXit , d〈Xi, Xj〉t makes sense4, then,

symbolically,

df(t,X1t , . . . , X

nt )

=∂f

∂t(t,X1

t , . . . , Xnt )dt+

n∑i=1

∂f

∂xi(t,X1

t , . . . , Xnt )dXi

t

+1

2

n∑i=1

n∑j=1

∂2f

∂xi∂xj(t,X1

t , . . . , Xnt )d〈Xi, Xj〉t.

A typical case the above theorem is useful is when f(t, x1, x2) =f(x1, x2). Say, let Xt and Yt be stochastic processes given by (4.28).Then

df(Xt, Yt) =∂f

∂x(Xt, Yt)dXt +

∂f

∂y(Xt, Yt)dYt

+1

2

∂2f

∂x2(Xt, Yt)(dXt)

2 +∂2f

∂x∂y(Xt, Yt)(dXt)(dYt)

+1

2

∂2f

∂y2(Xt, Yt)(dYt)

2. (4.31)

4Such processes are generally called semi-martingales. The process of the formdXt = αdt+ βdWt is a typical example of semi-martingales.

4.7. MARTINGALE AND BROWNIAN MOTION 142

If f(x, y) = xy, then (4.31) is reduced to

d(XtYt) = YtdXt +XtdYt + dXtdYt,

which given the following

Proposition 4.28 (Integration by parts formula).∫ t

0XudYu = XtYt −X0Y0 − 〈X,Y 〉t −

∫ t

0YudXu.

4.7 Martingale and Brownian Motion

Martingales and Brownian motion are closely intertwined. For in-stance, let Xt be a stochastic process obtained as a stochastic integralof the form

Xt = X0 +

∫ t

0ζudWu. (4.32)

Then by Proposition 4.16, Xt is a martingale. If (4.32) is writtensymbolically, we have

dXt = ζtdWt. (4.33)

Proposition 4.16 in a nutshell says that any process Xt of the form(4.33) is a martingale. If Xt satisfies a more general form like

dXt = ξtdt+ ζtdWt,

Xt is no longer martingale. It is usually called a semi-martingale.In other words, the difference between the martingale and the semi-martingale is the presence of the term ξtdt, which is usually calledthe drift term. The following Theorem 4.31 says that martingalesare essentially Brownian motion in the sense that it can be made aBrownian under suitable framework changes.

The three theorems presented below are useful results which wequote without giving here any proof or justification.

Theorem 4.29. Let Xt be a continuous L2-martingale. If 〈X〉t = t,then Xt is a Brownian motion.

Theorem 4.30. Let Xt be a continuous L2-martingale defined on(Ω, F , P ). Then there is a Brownian motion Wt on a extended prob-

ability space (Ω, F , P ) such that Xt = W〈X〉t.

Theorem 4.31 (Martingale Representation Theorem). Let Mt bea continuous L2-martingale. Then there is a predictable5 process φtsuch that

Mt = M0 +

∫ t

0φs dWs.

5For our purpose, we may assume it is a continuous process.

4.7. MARTINGALE AND BROWNIAN MOTION 143

In particular, dMt = φt dWt.

Example 4.32. [Ornstein-Uhlenbeck Process] Let Xt be the stochas-tic process such that

dXt = σ dWt + β(a−Xt) dt,

where σ, β, and α are constants. Check

d[eβt(Xt − a)] = β eβt(Xt − a) dt+ eβtdXt + β eβt dt dXt

= eβt(β(Xt − a) dt+ dXt)

= eβtσ dWt.

Therefore,

eβt(Xt − a) = X0 − a+

∫ t

0eβs σ dWs.

Hence,

Xt = a+ e−βt(X0 − a) + e−βt∫ t

0eβsσdWs. (4.34)

Another way to look at this is via Martingale property. Let

Mt =

∫ t

0eβsdWs,

then Mt is a Martingale and

dMt = eβt dWt.

Let us find Mt. Observe that

d〈M〉t = dMt · dMt = e2βt dt.

Thus

〈M〉t =

∫ t

0e2βs ds

=e2βt − 1

2β.

By Theorem 4.30, there is a Brownian motion Wt such that

Mt = W〈M〉t = W (〈M〉t) = W(e2βt − 1

2β

).

By (4.34)

eβt(Xt − a)− (X0 − a) = σMt

= σW(e2βt − 1

2β

).

4.8. GIRSANOV THEOREM 144

Therefore

Xt = a+ e−βt (X0 − a) + σe−βtW(e2βt − 1

2β

).

This shows more clearly that Xt is a random perturbation σe−βtW(e2βt−1

2β

)of function a + e−βt(X0 − a) that converges to a if β > 0. (To becompletely rigorous, one needs to know the growth properties of thepaths of the Brownian motion.)

4.8 Girsanov Theorem

The Girsanov theorem and the related facts presented in this sectionare one of the most essential tools in finance. We first present theGirsanov theorem by way of exponential martingale to produce a con-sistent family of probability measures; later in this section, we reversethe argument by starting with two equivalent probability measures,then deriving the martingale. Both approaches are equivalent.

4.8.1 Girsanov Theorem via Exponential Martingales

Let (Ω,F , P ) be a probability space and let (Ft)t≥0 be an informationstructure on it. Let us fix the maximal time horizon T , where T is apositive number ≤ ∞. If T is finite, FT is the usual σ-field as partof the information structure. If T =∞, then, we define

FT = F∞ = σ

⋃s≥0

Fs

,

i.e., F∞ is the smallest σ-field generated by⋃s≥0

Fs.

Exponential Martingale

Let Wt be a P -Brownian motion. By P -Brownian motion, we meanan adapted process Wt ∈ Ft that satisfies all the conditions of Defi-nition 4.3 in which the probability measure is P . If a new measure Qon (Ω,F) is used, Wt may no longer satisfies those conditions, thusis not a Brownian motion as longer.

We want to define an Ft-martingaleMt with respect to P measureby solving the following stochastic differential equation:

dMt = −γtMtdWt (4.35)

M0 ≡ 1


for 0 ≤ t ≤ T , where γt is an Ft-adapted process satisfying suitablecontinuity and integrability conditions. If we disregard the integra-bility condition and proceed formally, (4.35) is easy to solve. Namely,since

d(logMt) =dMt

Mt− 1

2

(dMt)2

(Mt)2

= −γtdWt −1

2|γt|2dt.

Thus, upon integrating this and using the condition M0 ≡ 1, it istrivial to see that

Mt = exp

(−∫ t

0γsdWs −

1

2

∫ t

0|γs|2ds

). (4.36)

The point here is whether this integral inside the exponential actuallyexists as a finite quantity almost surely. It is well-known that it isthe case if

EP

[exp

(1

2

∫ T

0|γs|2ds

)]<∞. (4.37)

We will not go into the detailed here as the interested reader canconsult many standard textbooks. The condition (4.37) is calledthe “Novikov condition.” We will assume that γt is always chosenin such a way that it holds throughout the lecture. It is trivial butcrucial to note that Mt is a martingale with respect to P measureand Mt ∈ Ft. We call Mt an “exponential martingale” because ofthe way it is written in (4.36).

Measure

Define a measure Qt on Ft by

dQt = Mt dP,

i.e., for A ∈ Ft define

Qt(A) =

∫AMtdP.

It means that for a random variable X ∈ Ft,

EQt [X] =

∫ΩXdQt =

∫ΩXMtdP.


Now let us check the total measure of Qt

Qt(Ω) =

∫ΩdQt =

∫ΩMt dP

=

∫ΩEP [Mt | F0] dP

(∵ Ω ∈ F0 and use the definition

of conditional expectation.)

=

∫ΩM0 dP (∵ Mt is a P -martingale.)

=

∫ΩdP

= P (Ω)

= 1.

Also (4.36) implies that Mt is positive. Therefore, Qt is a new prob-ability measure on (Ω,Ft). Now let A ∈ Fs, for any s < t, then usingthe similar arguments, we have∫

AdQt =

∫AMt dP

=

∫AEP [Mt | Fs] dP

=

∫AMs dP

=

∫AdQs.

Thus Qt and Qs coincide on Fs. i.e., Qtt is a consistent family ofprobability measures. We now define Q = QT , i.e.,

dQ = MT dP.

Remark 4.33. Let t ≤ T and let A ∈ Ft. Assume X is a randomvariable such that X ∈ Ft. Then consistency of Qt0≤t≤T impliesthat ∫

AXdQ =

∫AXdQT =

∫AXdQt.

In other words, ∫AXMTdP =

∫AXMtdP.


Construction of a new Brownian motion with respectto Q

Define a new stochastic process

Wt = Wt +

∫ t

0γsds.

Thus in the notation of stochastic differential,

dWt = dWt + γtdt.

It is clear that Wt ∈ Ft. We now show that Wt is a Q-Brownianmotion, i.e., Wt satisfies all the conditions of Definition 4.3 when(Ω,F , P ) is replaced with (Ω,FT , Q).

Claim 1 Wt is a Q-martingale.

Step 1: MtWt is a P -martingale.

Proof.

d(Mt Wt) = Mt dWt + Wt dMt + dMt dWt

= Mt (dWt + γt dt) + Wt (−γtMt dWt)− γtMt dt

= Mt (1− γt Wt) dWt.

Therefore MtWt =∫ t

0 Mu(1 − γuWu)dWu. Therefore MtWt is P -martingale by (iii) of Proposition 4.16. (See also the comment in thebeginning of Section 4.7.)

Step 2: EQ[Wt | Fs] = Ws.

Proof. Let s < t and A ∈ Fs. Then∫AWt dQ =

∫AWtMt dP

=

∫AEP [WtMt | Fs] dP

=

∫AWsMsdP (∵ WtMt is a P -Martingale.)

=

∫AWsdQ. (∵ MsdP coincides with MtdP on Fs.)

Therefore by the definition of conditional expectation,

EQ[Wt | Fs] = Ws.


Claim 2 Wt is a Q-Brownian motion.

Proof. Note thatd〈W 〉t = (dWt)

2 = dt,

i.e.,〈W 〉t = t.

Hence by Theorem 4.29, Wt is a Q-Brownian motion.

To summarize, we have

Theorem 4.34 (Girsanov Theorem). Let Wt be a Brownian motionwith respect to a probability measure P . Let Mt be a continuousL2-martingale satisfying

dMt = −γtMtdWt,

M0 ≡ 1,

where γt satisfies the Novikov condition

EP

[exp

(1

2

∫ T

0|γs|2ds

)]<∞.

Define a new measure Q by

dQ = MTdP.

Let Wt be a new stochastic process defined by

dWt = dWt + γt dt,

W0 = 0.

Then Wt is a Brownian motion with respect to the new measure Q.

4.8.2 Changing measures and drifts

Let (Ω,F , P ) be a probability space and let (Ft)t≥0 be an informationstructure. Let Xt be a stochastic process satisfying

dXt = αtdt+ βtdWt, (4.38)

where αt and βt are Ft-adapted processes and Wt is a P -Brownianmotion. Let ξt be an Ft-adapted process. The Girsanov theoremgives a means to change the drift αt to ξt by changing the measure,hence the Brownian motion.


Let Q be the new probability measure on FT and let Wt bethe corresponding Q-Brownian motion as constructed in Subsection4.8.1. Then

dXt = αtdt+ βt(dWt − γtdt)= (αt − βtγt)dt+ βtdWt.

Thus set γt so thatαt − βtγt = ξt

is satisfied, i.e.,

γt =αt − ξtβt

.

Therefore, as long as βt 6= 0 almost surely and the Novikov conditionis satisfied for γt, we have

dXt = ξtdt+ βtdWt.

In particular, if ξt = 0, we have

dXt = βtdWt,

which implies the following result.

Theorem 4.35. Any semi-martingale Xt satisfying

dXt = αtdt+ βtdWt

becomes a martingale with respect to a suitably chosen measure Q.

4.8.3 Converse Approach

Let P and Q be probability measures on (Ω,F).

Definition 4.36.

(i) We say Q is absolutely continuous with respect to P , writtenQ P , if for any A ∈ F , P (A) = 0 implies Q(A) = 0.

(ii) We say P and Q are equivalent if they are absolutely contin-uous with respect to each other; in other words, for A ∈ F ,Q(A) = 0 if and only if P (A) = 0.

Remark 4.37. It is clear that two measures P and Q introduced inSubsection 4.8.1 are equivalent.

The following theorem is well-known in measure theory.


Theorem 4.38 (Radon-Nikodym). Let µ and ν be measures on(Ω,F). Assume ν µ. Then there exists an F-measurable functionf (i.e., f ∈ F) such that dν = fdµ; in other words, for any A ∈ F ,

ν(A) =

∫Afdµ.

We say f the Radon-Nikodym derivative of ν with respect to µ anddenote it by

f =dν

dµ.

(f is defined almost everywhere with respect to µ.)

Suppose now that (Ω,F , P ) is a probability space and (Ft) aninformation structure. Assume Q is a probability measure on (Ω,F)that is equivalent to P . Let dQ

dP be the Radon-Nikodym derivative

of Q with respect to P . Since dPdQ = 1/dQdP also exists and P and Q

are probability measures, dQdP must be positive almost everywhere.

As P and Q are probability measures, it is easily seen that dQdP is L1

with respect to P . Since the total measure is finite L1-ness impliesL2-ness, i.e., dQ

dP is L2. Define Mt by

Mt = EP

[dQ

dP| Ft

]. (4.39)

Then Mt is Ft-adapted and is a P -martingale. Note that Mt is posi-tive a.s., as dQ

dP is positive a.s. We also assume that Mt is continuous.(This continuity has something to do with “continuity” of Ft, whichwe would not be specific here.) Then by the martingale represen-tation theorem, Theorem 4.31, there exists a predictable (in fact,continuous) process γt such that

dMt = −γtMtdWt, (4.40)

where Wt is a P -Brownian motion. (In fact, by Theorem 4.31, thereexists φt such that dMt = φtdWt, and define γt = − φt

Mt.)

Since F0 = ∅,Ω is a trivial σ-field, M0 must be a constant.Therefore

M0 =

∫ΩM0dP =

∫ΩEP

[dQ

dP

∣∣F0

]dP

=

∫Ω

dQ

dPdP

=

∫ΩdQ

= 1.


Therefore M0 ≡ 1 a.s.Fix the maximal time horizon T ≤ ∞, now let A ∈ Ft. Then∫

AdQ =

∫A

dQ

dPdP =

∫AEP

[dQ

dP| Ft

]dP =

∫AMtdP.

In other words, on Ft

dQ = MtdP. (4.41)

Therefore, by (4.40) and (4.41), the machinery of Subsection 4.8.1can be applied.

It is also worthwhile to keep in mind that

dQ = MTdP

on Ft, and this Q can be applied consistently to any Ft to result inthe coincidence

Q = QT

on FT . In closing, let us mention the following useful facts.

Theorem 4.39 (Change of Measure Formula). Let Xt be a randomvariable such that Xt ∈ Ft with the same notation and assumptionsas above, the following holds for s ≤ t.

(i) EQ[Xt] = EP [XtMt],

(ii) EQ[Xt | Fs] =EP [XtMt | Fs]EP [Mt | Fs]

.

Proof.

(i)

∫ΩXtdQ =

∫ΩXtMtdP , as dQ = dQt = MtdP on Ft.

(ii) It suffices to prove that MsEQ[Xt | Fs] = EP [XtMt | Fs]. LetA ∈ Fs. Then∫

AMsEQ[Xt | Fs]dP

=

∫AEQ[Xt | Fs]dQs (∵ dQs = MsdP on Fs)

=

∫AEQ[Xt | Fs]dQ (∵ dQs = dQ on Fs)

=

∫AXtdQ (∵ by definition of conditional expectation)

=

∫AXtMtdP (∵ dQ = dQt = MtdP on Ft > Fs)


Therefore by definition of conditional expectation, we have

MsEQ[Xt | Fs] = EP [XtMt | Fs].

EXERCISES 153

Exercises

4.1. Let Wt be a 1-dimensional Brownian motion. Show that thefollowing process are martingales.

(a) Xt = e12t cosWt.

(b) Mt = exp(−βWt − β2t/2) where β ≥ 0.

4.2. Define stochastic processes Mt and Xt by

dMt = Mt(a dt+ b dWt)

dXt = Xt(c dt+ d dWt)

where a, b, c, d are constants with a+ c+ bd = 0, b+ d = −1.

(a) Write down the stochastic differential equation MtXt satisfies,i.e., express d(MtXt) in terms of the quantities given above.

(b) Find the explicit formula for MtXt when M0X0 = 7.

4.3. Find the closed formula for

∫ t

0WudWu.

4.4. Answer the following questions.

(a) Show that ∫ t

0f(u)dWu = f(t)Wt −

∫ t

0Wudf(u).

where f(t) is a deterministic C1 function of t.

(b) Explain why

∫ t

0f(u)dWu can be evaluated in a pathwise man-

ner.

4.5. Let Z ∈ N(0, 1).

(a) Find the mean and variance of Xt =√tZ.

(b) Is Xt a Brownian motion?

4.6. Show that Zt = W 3t − 3tWt is a martingale.

4.7. Define stochastic processes Xt and Yt by

dXt = Xt(adt+ bdWt),

dYt = Yt(αdt+ βdWt),

where a, b, α and β are constant.

EXERCISES 154

(a) Find the SDE Xt/Yt satisfies.

(b) Assuming X0 = 1 = Y0, find the explicit formula for Xt/Yt.

4.8. Let Wt be a Brownian motion and let Xt be a continuousbounded stochastic process.

(a) Show that E

[∫ t

0XsdWs

]= 0.

(b) Show that E

[∫ t

0Xsds

]=

∫ t

0E[Xs]ds.

4.9. Let Xt be a stochastic process. It is known that

E

[∫ t

0XsdWs

]= 0 and E

[∫ t

0Xsds

]=

∫ t

0E(Xs)ds.

Define Zt = W 4t . Show

(a) Zt = 6

∫ t

0W 2s ds+ 4

∫ t

0W 3s dWs.

(b) Compute E(W 4t ).

4.10. Suppose u(x, t) is a C2 function satisfying

1

2

∂2u

∂x2(x, t) +

∂u

∂t(x, t) = 0,

for all x ∈ R and t ≥ 0. Let Wt be a Brownian motion. Show thatZt = u(Wt, t) is a martingale.

4.11. Let Xt and Yt be stochastic processes given as

dXt = Xt(µ1dt+ σ1dWt)

dYt = Yt(µ2dt+ σ2dWt),

where Wt is a Brownian motion and µ1, µ2, σ1, σ2 are constants. LetZt =

√XtYt. Write down the stochastic differential equation Zt

satisfies. (Your answer may not involve Xt or Yt explicitly.)

4.12. Let Wt be a Brownian motion. Show that the following pro-cesses are martingales.

(a) (20pts) Xt = W 3t − 3tWt.

(b) (20pts) Yt = e12t cosWt.

EXERCISES 155

4.13. Let (Ω,F , P ) be a probability space and let (Ft) be an infor-mation structure. Suppose Wt is a P -Brownian motion. Define anew stochastic process Xt by solving

dXt = adt+ bdWt

X0 = 0

where a and b are positive constants. Let T be the fixed time horizon.We want to find a new probability measure Q on (Ω,FT ) with respectto which Xt become a martingale. Answer that the following seriesof questions.

(a) Let Wt be a Q-Brownian motion. It is known that Wt satisfies

dWt = dWt + γtdt

for some process γt. Find γt so that Xt is a Q-martingale.

(b) The new measure Q is of the form dQ = MTdP , where Mt is astochastic process satisfying the following

dMt = ζtdWt

M0 = 1.

Find ζt.

(c) Compute MT .

4.14. Find a deterministic function ϕ(t) that makes Zt = W 3t +

ϕ(t)Wt a martingale.

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