Continuous Stochastic
Calculus with
Applications to Finance
APPLIED MATHEMATICS
Editor: R.J. Knops
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3 Material Inhomogeneities in Elasticity G. Maugin (1993)
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Library of Congress Cataloging-in-Publication Data

Meyer, Michael (Michael J.)
Continuous stochastic calculus with applications to finance / Michael Meyer.
p. cm (Applied mathematics ; 17)
Includes bibliographical references and index.
ISBN 1-58488-234-4 (alk. paper)
1. Finance Mathematical models. 2. Stochastic analysis. I. Title. II. Series.
HG173 .M49 2000
332

′.01′

5118—dc21 00-064361
referred to as 4.a.2. Displayed equations are numbered (0), (1), (2) etc. Thus
II.3.b.eq.(5) refers to equation (5) of subsection b of section 3 of Chapter II. This
same equation would be referred to as 3.b.eq.(5) from within Chapter II and as (5)
from within the subsection wherein it occurs.
Very little is new or original and much of the material is standard and can be
found in many books. The following sources have been used:
[Ca,Cb] I.5.b.1, I.5.b.2, I.7.b.0, I.7.b.1;
[CRS] I.2.b, I.4.a.2, I.4.b.0;
[CW] III.2.e.0, III.3.e.1, III.2.e.3;
vi Preface
[DD] II.1.a.6, II.2.a.1, II.2.a.2;
[DF] IV.3.e;
[DT] I.8.a.6, II.2.e.7, II.2.e.9, III.4.b.3, III.5.b.2;
[J] III.3.c.4, IV.3.c.3, IV.3.c.4, IV.3.d, IV.5.e, IV.5.h;
[K] II.1.a, II.1.b;
[KS] I.9.d, III.4.c.5, III.4.d.0, III.5.a.3, III.5.c.4, III.5.f.1, IV.1.c.3;
[MR] IV.4.d.0, IV.5.g, IV.5.j;
[RY] I.9.b, I.9.c, III.2.a.2, III.2.d.5.
vii
To
my mother

Table of Contents ix
TABLE OF CONTENTS
Chapter I Martingale Theory
Preliminaries 1
1. Convergence of Random Variables 2
1.a Forms of convergence 2
1.b Norm convergence and uniform integrability 3
2. Conditioning 8

9.a Square integrable martingales 73
9.b Quadratic variation 74
9.c Quadratic variation and L
2
-bounded martingales 86
9.d Quadratic variation and L
1
-bounded martingales 88
10. The Covariation Process 90
10.a Definition and elementary properties 90
10.b Integration with respect to continuous bounded variation processes . 91
10.c Kunita-Watanabe inequality 94
11. Semimartingales 98
11.a Definition and basic properties 98
11.b Quadratic variation and covariation 99
Chapter II Brownian Motion
1. Gaussian Processes 103
1.a Gaussian random variables in R
k
103
1.b Gaussian processes 109
1.c Isonormal processes 111
2. One Dimensional Brownian Motion 112
2.a One dimensional Brownian motion starting at zero 112
2.b Pathspace and Wiener measure 116
2.c The measures P
x
118
2.d Brownian motion in higher dimensions 118
2.e Markov property . 120

4.c Girsanov’s theorem 175
4.d The Novikov condition 180
5. Representation of Continuous Local Martingales 183
5.a Time change for continuous local martingales 183
5.b Brownian functionals as stochastic integrals 187
5.c Integral representation of square integrable Brownian martingales . 192
5.d Integral representation of Brownian local martingales 195
5.e Representation of positive Brownian martingales 196
5.f Kunita-Watanabe decomposition 196
6. Miscellaneous 200
6.a Ito processes . . 200
6.b Volatilities . . . 203
6.c Call option lemmas 205
6.d Log-Gaussian processes 208
6.e Processes with finite time horizon 209
Chapter IV Application to Finance
1. The Simple Black Scholes Market 211
1.aThemodel 211
1.b Equivalent martingale measure 212
1.c Trading strategies and absence of arbitrage 213
2. Pricing of Contingent Claims 218
2.a Replication of contingent claims 218
2.b Derivatives of the form h = f(S
T
) 221
2.c Derivatives of securities paying dividends 225
3. The General Market Model 228
3.a Preliminaries . . 228
3.b Markets and trading strategies 229
xii Table of Contents

Notation xiii
SUMMARY OF NOTATION
Sets and numbers. N denotes the set of natural numbers (N={1, 2, 3, }), R the
set of real numbers, R
+
=[0, +∞), R =[−∞, +∞] the extended real line and R
n
Euclidean n-space. B(R), B(R) and B(R
n
) denote the Borel σ-field on R, R and R
n
respectively. B denotes the Borel σ-field on R
+
.Fora, b ∈ R set a ∨b = max{a, b},
a ∧ b = min{a, b}, a
+
= a ∨ 0 and a
−
= −a ∧ 0.
Π=[0, +∞) × Ω domain of a stochastic process
P
g
theprogressive σ-field on Π (III.1.a).
P thepredictable σ-field on Π (III.1.a).
[[ S, T ]] = {(t, ω) | S(ω) ≤ t ≤ T (ω) } . . . stochastic interval.
Random variables. (Ω, F,P) the underlying probability space, G⊆Fa sub-σ-
field. For a random variable X set X
+
= X ∨ 0=1
[X>0]

)
t≥0
be a stochastic process and T :Ω→ [0, ∞] an optional
time. Then X
T
denotes the random variable (X
T
)(ω)=X
T (ω)
(ω) (sample of X
along T , I.3.b, I.7.a). X
T
denotes the process X
T
t
= X
t∧T
(process X stopped at
time T). S, S
+
and S
n
denote the space of continuous semimartingales, continuous
positive semimartingales and continuous R
n
-valued semimartingales respectively.
Let X,Y ∈S, t ≥ 0, ∆ = {0=t
0
<t
1

(∆
j
X)
2
. . . . I.9.b, I.10.a, I.11.b.
Q
∆
(X, Y )=

∆
j
X∆
j
Y . I.10.a.
X, Y  covariation process of X, Y (I.10.a, I.11.b).
X, Y 
t
= lim
∆→0
Q
∆
(X, Y ) (limit in probability).
X = X, X quadratic variation process of X (I.9.b).
u
X
(additive) compensator of X (I.11.a).
U
X
multiplicative compensator of X ∈S
+

t
Augmented filtration generated by W (II.2.f).
N(m, C) Normal distribution with mean m ∈ R
k
and
covariance matrix C (II.1.a).
N(d)=P(X ≤ d) X a standard normal variable in R
1
.
n
k
(x)=(2π)
−k/2
exp

−x
2

2

. . Standard normal density in R
k
(II.1.a).
xiv Notation
Stochastic integrals, spaces of integrands. H
•
X denotes the integral process
(H
•
X)

M
Doleans measure on (Π, B×F) associated with M (III.2.a)
µ
M
(∆) = E
P


∞
0
1
∆
(s, ω)dM
s
(ω)

,∆∈B×F.
L
2
(M) space L
2
(Π, P
g
,µ
M
) of all progressively measurable processes H
satisfying H
2
L
2

2
loc
(M)ofM-integrable processes H are then defined as follows:
Λ
2
(M) space of all progressively measurable processes H satisfying
1
[0,t]
H ∈ L
2
(M), for all 0 <t<∞.
L(M)=L
2
loc
(M) . . space of all progressively measurable processes H satisfying
1
[[ 0 ,T
n
]]
H ∈ L
2
(M), for some sequence (T
n
) of optional times
increasing to infinity, equivalently

t
0
H
2


∞
0
|H
s
(ω)||dA
s
|(ω) < ∞, for P -ae. ω ∈ Ω.
L
1
loc
(A) thespace of all progressively measurable processes H such that
1
[0,t]
H ∈ L
1
(A), for all 0 <t<∞.
For H ∈ L
1
loc
(A) the integral process I
t
=(H
•
A)
t
=

t
0

For H ∈ L(X) set H
•
X = H
•
A+H
•
M. Then H
•
X is the unique continuous semi-
martingale satisfying (H
•
X)
0
=0,u
H
•
X
= H
•
u
X
and H
•
X, Y  = H
•
X, Y ,
for all Y ∈S(III.4.a.2). In particular H
•
X = H
•

∆→0
S
∆
(H, X)
(limit in probability), where S
∆
(H, X)=

H
t
j−1
(X
t
j
− X
t
j−1
)for∆asabove
(III.2.e.0). The (deterministic) process t defined by t(t)=t, t ≥ 0, is a continuous
semimartingale, in fact a bounded variation process. Thus the spaces L(t) and
L
1
loc
(t) are defined and in fact L(t)=L
1
loc
(t).
Vector valued integrators. Let X ∈S
d
and write X =(X

j
H
j
•
X
j
, (H
•
X)
t
=

t
0
H
s
· dX
s
=

j

t
0
H
j
s
dX
j
s

(X) are defined analogously. If H ∈ Λ
2
(X), then H
•
X is a square
integrable martingale; if H ∈ L
2
(X), then H
•
X ∈ H
2
(III.2.c.3, III.2.f.3).
In particular, if W is an R
d
-valued Brownian motion, then
L
2
(W ) space of all progressively measurable processes H such that
H
2
L
2
(W )
= E
P

∞
0
H
s

•
W 
H
2
= H
L
2
(W )
.If
H ∈ Λ
2
(W ), then H
•
W is a square integrable martingale (III.2.f.3, III.2.f.5).
Stochastic differentials. If X ∈S
n
, Z ∈Swrite dZ = H · dX if H ∈ L(X) and
Z = Z
0
+H
•
X, that is, Z
t
= Z
0
+

t
0
H

X = E(M) is the unique solution to the exponential equation dX
t
= X
t
dM
t
,
X
0
=1. Ifγ ∈ L(M), then all solutions X to the equation dX
t
= γ
t
X
t
dM
t
are
xvi Notation
given by X
t
= X
0
E
t
(γ
•
M). If W is an R
d
-valued Brownian motion and γ ∈ L(W),

2
ds +

t
0
γ
s
· dW
s

(III.4.b).
Finance. Let B be a market (IV.3.b), Z ∈Sand A ∈S
+
.
Z
A
t
= Z
t
/A
t
Z expressed in A-numeraire units.
B(t, T ) Price at time t of the zero coupon bond maturing at time T .
B
0
(t) Riskless bond.
P
A
A-numeraire measure (IV.3.d).
P

extended real line and B(R) and B(R
n
) the Borel σ-fields on R and R
n
respectively.
A random object on (Ω, F,P) is a measurable map X :(Ω, F,P) → (Ω
1
, F
1
)
with values in some measurable space (Ω
1
, F
1
). P
X
denotes the distribution of X
(appendix B.5). If Q is any probability on (Ω
1
, F
1
) we write X ∼ Q to indicate that
P
X
= Q.If(Ω
1
, F
1
)=(R
n

− X
−
.
For nonnegative X let E(X)=

Ω
XdP and let E(P ) denote the family of all
random variables X such that at least one of E(X
+
), E(X
−
) is finite. For X ∈E(P )
set E(X)=E(X
+
) − E(X
−
)(expected value of X). This quantity will also be
denoted E
P
(X) if dependence on the probability measure P is to be made explicit.
If X ∈E(P ) and A ∈Fthen 1
A
X ∈E(P) and we write E(X; A)=E(1
A
X).
The expression “P -almost surely” will be abbreviated “P -as.”. Since random vari-
ables X, Y are extended real valued, the sum X + Y is not defined in general.
However it is defined (P -as.) if both E(X
+
) and E(Y

i.o.)=0.
(b) If the events A
n
are independent and

n
P (A
n
)=∞ then P (A
n
i.o.)=1.
(c) If P (A
n
) ≥ δ, for all n ≥ 1, then P (A
n
i.o.) ≥ δ.
Proof. (a) Let m ≥ 1. Then 0 ≤ P(A
n
i.o.) ≤

n≥m
P (A
n
) → 0, as m ↑∞.
(b) Set A =[A
n
i.o.]. Then P (A
c
) = lim
m

A
n

.
2 1.a Forms of convergence.
1. CONVERGENCE OF RANDOM VARIABLES
1.a Forms of convergence. Let X
n
, X, n ≥ 1, be random variables on the prob-
ability space (Ω, F,P) and 1 ≤ p<∞. We need several notions of convergence
X
n
→ X:
(i) X
n
→ XinL
p
,ifX
n
− X
p
p
= E

|X
n
− X|
p

→ 0, as n ↑∞.

(−∞) − (−∞)=0andZ
p
is allowed to assume the value +∞. Recall that the
finiteness of the probability measure P implies that Z
p
increases with p ≥ 1.
Thus X
n
→ XinL
p
implies that X
n
→ XinL
r
, for all 1 ≤ r ≤ p.
Convergence in L
1
will simply be called convergence in norm.ThusX
n
→ X
in norm if and only if X
n
− X
1
= E

|X
n
− X|


k
n
(ω) → X(ω) and so
X
n
(ω) → X(ω). (b) Note that P

|X
n
− X|≥

≤ 
−1


X
n
− X


1
.
1.a.1. Convergence in probability implies almost sure convergence of a subsequence.
Proof. Assume that X
n
→ X in probability and choose inductively a sequence
of integers 0 <n
1
<n
2

→ X, P -as.
Remark. Thus convergence in norm implies almost sure convergence of a subse-
quence. It follows that convergence in L
p
implies almost sure convergence of a
subsequence. Let L
0
(P ) denote the space of all (real valued) random variables on
(Ω, F,P). As usual we identify random variables which are equal P -as. Conse-
quently L
0
(P ) is a space of equivalence classes of random variables.
It is interesting to note that convergence in probability is metrizable, that
is, there is a metric d on L
0
(P ) such that X
n
→ X in probability if and only if
Chapter I: Martingale Theory 3
d(X
n
,X) → 0, as n ↑∞, for all X
n
,X ∈ L
0
(P ). To see this let ρ(t)=1∧ t,
t ≥ 0, and note that ρ is nondecreasing and satisfies ρ(a + b) ≤ ρ(a)+ρ(b), a, b ≥ 0.
From this it follows that d(X, Y )=E

ρ(|X − Y |)

k
∈F, k ≥ 1, and A =

k
A
k
.IfX
n
→ X in probability on each set
A
k
, then X
n
→ X in probability on A.
Proof. Replacing the A
k
with suitable subsets if necessary, we may assume that the
A
k
are disjoint. Let , δ > 0 be arbitrary, set E
m
=

k>m
A
k
and choose m such
that P

E

n
− X| >

∩ A

≤ P (E
m
) <δ. Since
here δ>0 was arbitrary, this lim sup is zero, that is, P

|X
n
− X| >

∩ A

→ 0,
as n ↑∞.
1.b Norm convergence and uniform integrability. Let X be a random variable
and recall the notation E(X; A)=E(1
A
X)=

A
XdP. The notion of uniform
integrability is motivated by the following observation:
1.b.0. X is integrable if and only if lim
c↑∞
E



|X|;[|X|≥c]

<.IfA ∈F
with P (A) < /c is any set, we have
E

|X|1
A

= E

|X|; A ∩ [|X| <c]

+ E

|X|; A ∩ [|X|≥c]

≤ cP (A)+E

|X|;[|X|≥c]

<+  =2.
Thus lim
P (A)→0
E

|X|1
A


4 1.b Norm convergence and uniform integrability.
that is, lim
c↑∞
E

|X
i
|;[|X
i
|≥c]

= 0, uniformly in i ∈ I. The family F is called
uniformly P -continuous if it satisfies
lim
P (A)→0
sup
i∈I
E

1
A
|X
i
|

=0,
that is, lim
P (A)→0
E


i
| i ∈ I }
is uniformly integrable it suffices to show that for each >0 there exists a c ≥ 0
such that sup
i∈I
E

|X
i
|;[|X
i
|≥c]

≤ .
(b) To show that the family F = {X
i
| i ∈ I } is uniformly P-continuous we must
show that for each >0 there exists a δ>0 such that sup
i∈I
E

1
A
|X
i
|

<, for
all sets A ∈Fwith P(A) <δ. This means that the family {µ
i

1
-bounded.
Proof. Let F be uniformly integrable and choose ρ such that E

|X
i
|;[|X
i
|≥ρ]

< 1,
for all i ∈ I. Then X
i

1
= E(

|X
i
|;[|X
i
|≥ρ]

+ E(

|X
i
|;[|X
i
| <ρ]


+ E

|X
i
|; A ∩ [|X
i
|≥c]

≤ cP (A)+E(

|X
i
|;[|X
i
|≥c]

<+  =2, for every i ∈ I.
Thus the family F is uniformly P -continuous. Conversely, let F be uniformly P-
continuous and L
1
-bounded. We must show that lim
c↑∞
E(

|X
i
|;[|X
i
|≥c]

|≥c]) ≤ r/c<δand so E(

|X
i
|;[|X
i
|≥c]

<, for
all i ∈ I.
Chapter I: Martingale Theory 5
1.b.3 Norm convergence. Let X
n
,X ∈ L
1
(P ). Then the following are equivalent:
(i) X
n
→ X in norm, that is, X
n
− X
1
→ 0,asn ↑∞.
(ii) X
n
→ X in probability and the sequence (X
n
) is uniformly integrable.
(iii) X
n

A
|X|

</3, for all sets
A ∈F. Now choose c ≥ 1 such that
E

|X|;[|X|≥c − 1]

</3 (0)
and finally N such that n ≥ N implies X
n
−X
1
<δ</3 and let n ≥ N. Then
|X
n
|≤|X
n
− X| + |X| and so
E

|X
n
|;[|X
n
|≥c]

≤ E



.
Let A =[|X
n
|≥c] ∩ [|X| <c− 1] and B =[|X
n
|≥c] ∩ [|X|≥c − 1]. Then
|X
n
−X|≥1 on the set A and so P(A) ≤ E

1
A
|X
n
−X|

≤X
n
−X
1
<δwhich
implies E

1
A
|X|

</3. Using (0) it follows that
E

consequently for all n ≥ 1. Then sup
n≥1
E

|X
n
|;[|X
n
|≥c]

≤  as desired.
(b) ⇒ (c): Uniform integrability implies uniform P -continuity.
(c) ⇒ (a): Assume now that the sequence (X
n
) is uniformly P -continuous and
converges to X ∈ L
1
(P ) in probability. Let >0 and set A
n
=[|X
n
− X|≥].
Then P(A
n
) → 0, as n ↑∞. Since the sequence (X
n
) is uniformly P -continuous
and X ∈ L
1
(P ) is integrable, we can choose δ>0 such that A ∈Fand P (A) <δ


|X
n
− X|; A
n

+ E

|X
n
− X|; A
c
n

≤ E

|X
n
|; A
n

+ E

|X|; A
n

+ P (A
c
n
) ≤  +  +  =3.

∈ L
1
(P ), n ≥ 1, and assume that X
n
→ X almost surely.
Then the following are equivalent:
(i) X ∈ L
1
(P ) and X
n
→ X in norm.
(ii) The sequence (X
n
) is uniformly integrable.
Proof. (i) ⇒ (ii) follows readily from 1.b.3. Conversely, if the sequence (X
n
)
is uniformly integrable, especially L
1
-bounded, then the almost sure convergence
X
n
→ X and Fatou’s lemma imply that X
1
= E(|X|)=E

lim inf
n
|X
n

x↑∞
φ(x)/x =+∞ and sup
i∈I
E(φ(|X
i
|)) < ∞. (1)
The function φ can be chosen to be convex and nondecreasing.
Proof. (⇐): Let φ be such a function and C = sup
i∈I
E(φ(|X
i
|)) < +∞. Set
ρ(a)=Inf
x≥a
φ(x)/x. Then ρ(a) →∞,asa ↑∞, and φ(x) ≥ ρ(a)x, for all x ≥ a.
Thus
E

|X
i
|;[|X
i
|≥a]

= ρ(a)
−1
E

ρ(a)|X
i

-bounded and so δ(0) = sup
i∈I
X
i

1
< ∞.
We seek a piecewise linear convex function φ as in (1) with φ(0) = 0. Such a
function has the form φ(x)=φ(a
k
)+α
k
(x − a
k
), x ∈ [a
k
,a
k+1
], with 0 = a
0
<
a
1
< <a
k
<a
k+1
→∞and increasing slopes α
k
↑∞.

φ(|X
i
|); [a
k
≤|X
i
| <a
k+1
]

=

∞
k=0
E

φ(a
k
)+α
k
(|X
i
|−a
k
); [a
k
≤|X
i
| <a
k+1

]) ≤ a
−1
k
E

|X
i
|;[|X
i
|≥a
k
]

and observing that
φ(a
k
)/a
k
≤ α
k
by the increasing nature of the slopes (Figure 1.1), we obtain
E(φ(|X
i
|)) ≤

∞
k=0
2α
k
E

treated separately. Recall that δ(a
0
)=δ(0) < ∞ and choose 0 <α
0
< 2 so that
α
0
δ(a
0
) < 1=(2/3)
0
.Fork ≥ 1 set α
k
=2
k
. It follows that
E(φ(|X
i
|)) ≤

∞
k=0
2(2/3)
k
=6, for all i ∈ I.
1.b.6 Example. If p>1 then the function φ(x)=x
p
satisfies the assumptions
of Theorem 1.b.5 and E(φ(|X
i

uniformly integrable.
Proof. Let i ∈ I, c>0 and q be the exponent conjugate to p (1/p+1/q = 1). Using
the inequalities of Hoelder and Chebycheff we can write
E

|X
i
|1
[|X
i
|≥c]

≤1
[|X
i
|≥c]

q
X
i

p
= P

|X
i
|≥c

1
q

8 2.a Sigma fields, information and conditional expectation.
2. CONDITIONING
2.a Sigma Þelds, information and conditional expectation. Let E(P ) denote the
family of all extended real valued random variables X on (Ω, F,P) such that
E(X
+
) < ∞ or E(X
−
) < ∞ (i.e., E(X) exists). Note that E(P )isnotavec-
tor space since sums of elements in E(X) are not defined in general.
2.a.0. (a) If X ∈E(P ), then 1
A
X ∈E(P ), for all sets A ∈F.
(b) If X ∈E(P ) and α ∈ R, then αX ∈E(P ).
(c) If X
1
,X
2
∈E(P ) and E(X
1
)+E(X
2
) is defined, then X
1
+ X
2
∈E(P ).
Proof. We show only (c). We may assume that E(X
1
) ≤ E(X

) < ∞ and, since
(X
1
+ X
2
)
−
≤ X
−
1
+ X
−
2
, also E

(X
1
+ X
2
)
−

< ∞.ThusX
1
+ X
2
∈E(P ).
2.a.1. Let G⊆Fbe a sub-σ-field, D ∈Gand X
1
,X

1
A
) ≤ E(X
2
1
A
), for all G-measurable subsets A ⊆ D.
If P

[X
1
>X
2
] ∩ D

> 0 then there exist real numbers α<βsuch that the
event A =[X
1
>β>α>X
2
] ∩ D ∈Ghas positive probability. But then
E(X
1
1
A
) ≥ βP(A) >αP(A) ≥ E(X
2
1
A
), contrary to assumption. Thus we must

satisfies
E
Q
A
(X)=P (A)
−1
E(X1
A
), for all random variables X ∈E(P ).
At any given time the family of all events A, for which it is known whether they
occur or not, is a sub-σ-field of F. For example it is known that ∅ does not occur,