de Gruyter Expositions in Mathematics 37
Editors
O. H. Kegel, Albert-Ludwigs-Universität, Freiburg
V. P. Maslov, Academy of Sciences, Moscow
W. D. Neumann, Columbia University, New York
R. O.Wells, Jr., Rice University, Houston
de Gruyter Expositions in Mathematics
1The Analytical and Topological Theory of Semigroups, K. H.Hofmann, J. D.Lawson,
J. S.Pym (Eds.)
2 Combinatorial Homotopy and 4-Dimensional Complexes, H. J.Baues
3The Stefan Problem, A. M.Meirmanov
4Finite Soluble Groups, K. Doerk, T. O. Hawkes
5The Riemann Zeta-Function, A. A.Karatsuba, S. M. Voronin
6 Contact Geometry and Linear Differential Equations, V. E. Nazaikinskii, V. E. Shatalov,
B. Yu. Sternin
7 Infinite Dimensional Lie Superalgebras, Yu.A.Bahturin, A. A. Mikhalev, V. M. Petrogradsky,
M. V. Zaicev
8Nilpotent Groups and their Automorphisms, E. I. Khukhro
9 Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug
10 The Link Invariants of the Chern-Simons Field Theory, E. Guadagnini
11 Global Affine Differential Geometry of Hypersurfaces, A M. Li, U. Simon, G. Zhao
12 Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions,
K. Hulek, C. Kahn, S. H.Weintraub
13 Elliptic Problems in Domains with Piecewise Smooth Boundaries, S. A. Nazarov,
B. A. Plamenevsky
14 Subgroup Lattices of Groups, R.Schmidt
15 Orthogonal Decompositions and Integral Lattices, A. I. Kostrikin, P. H. Tiep
16 The Adjunction Theory of Complex Projective Varieties, M. C. Beltrametti, A. J. Sommese
17 The Restricted 3-Body Problem: Plane Periodic Orbits, A.D. Bruno
18 Unitary Representation Theory of Exponential Lie Groups, H. Leptin, J. Ludwig
19 Blow-up in Quasilinear Parabolic Equations, A.A. Samarskii, V.A. Galaktionov,

by
Nicolas Bouleau
≥
Walter de Gruyter · Berlin · New York
Author
Nicolas Bouleau
E
´
cole Nationale des Ponts et Chausse
´
es
6 avenue Blaise Pascal
77455 Marne-La-Valle
´
e cedex 2
France
e-mail:
Mathematics Subject Classification 2000:
65-02; 65Cxx, 91B28, 65Z05, 31C25, 60H07, 49Q12, 60J65, 31-02, 65G99, 60U20,
60H35, 47D07, 82B31, 37M25
Key words:
error, sensitivity, Dirichlet form, Malliavin calculus, bias, Monte Carlo, Wiener space,
Poisson space, finance, pricing, portfolio, hedging, oscillator.
Ț
ȍ Printed on acid-free paper which falls within the guidelines
of the ANSI to ensure permanence and durability.
Library of Congress Ϫ Cataloging-in-Publication Data
Bouleau, Nicolas.
Error calculus for finance and physics : the language of Dirichlet
forms / by Nicolas Bouleau.

of works. Without retracing the whole history of these investigations, we can draw
the main lines of the present inquiry.
The first approach is to represent the errors as random variables. This simple idea
offers the great advantage of using only the language of probability theory, whose
power has now been proved in many fields. This approach allows considering error
biases and correlations and applying statistical tools to guess the laws followed by
errors. Yet this approach also presents some drawbacks. First, the description is
too rich, for the error on a scalar quantity needs to be described by knowledge of
a probability law, i.e. in the case of a density, knowledge of an arbitrary function
(and joint laws with the other random quantities of the model). By definition however,
errors are poorly known and the probability measure of an error is very seldom known.
Moreover, in practical cases when using this method, engineers represent errors by
means of Gaussian random variables, which means describing them by only their bias
and variance. This way has the unavoidable disadvantage of being incompatible with
nonlinear calculations. Secondly, this approach makes the study of error transmission
extremely complex in practice since determining images of probability measures is
theoretically obvious, but practically difficult.
The second approach is to represent errors as infinitely small quantities. This of
course does not prevent errors from being more or less significant and from being
compared in size. The errors are actually small but not infinitely small; this approach
therefore is an approximate representation, yet does present the very significant ad-
vantage of enabling errors to be calculated thanks to differential calculus which is a
very efficient tool in both the finite dimension and infinite dimension with derivatives
in the sense of Fréchet or Gâteaux.
If we apply classical differential calculus, i.e. formulae of the type
dF(x,y) = F

1
(x, y)dx + F


1
+ F

2
ψ

1

ds +

F

1
ϕ

2
+ F

2
ψ

2

dt.
In the case of Brownian motion however and, more generally, of continuous semi-
martingales, Itô calculus displays a second-order differential calculus. Similarly, it
is indeed simple to see that error biases (see Chapter I, Section 1) involve second
derivatives in their transmission by nonlinear functions.
The objective of this book is to display that errors may be thought of as germs
of Itô processes.Wepropose, for this purpose, introducing the language of Dirichlet

versities and supposes as prerequisite a preliminary training in probability theory.
Textbook references are given in the bibliography at the end of each chapter.
Acknowledgements.Iexpress my gratitude to mathematicians, physicists and fi-
nance practitioners who have reacted to versions of the manuscript or to lectures
on error calculus by fruitful comments and discussions. Namely Francis Hirsch,
Paul Malliavin, Gabriel Mokobodzki, Süleyman Üstünel, Dominique Lépingle, Jean-
Michel Lasry, Arnaud Pecker, Guillaume Bernis, Monique Jeanblanc-Picqué, Denis
Talay, Monique Pontier, Nicole El Karoui, Jean-François Delmas, Christophe Chorro,
François Chevoir and Michel Bauer. My students have also to be thanked for their sur-
prise reactions and questions. I must confess that during the last years of elaboration
of the text, the most useful discussions occurred from people, colleagues and students,
who had difficulties understanding the new language. This apparent paradox is due
to the fact that the matter of the book is emerging and did not yet reach a definitive
form. For the same reason is the reader asked to forgive the remaining obscurities.
Paris, October 2003 Nicolas Bouleau

Contents
Preface v
I Intuitive introduction to error structures 1
1 Error magnitude 1
2 Description of small errors by their biases and variances 2
3 Intuitive notion of error structure 8
4How to proceed with an error calculation 10
5 Application: Partial integration for a Markov chain 12
Appendix. Historical comment: The benefit of randomizing physical
or natural quantities 14
Bibliography for Chapter I 16
II Strongly-continuous semigroups and Dirichlet forms 17
1 Strongly-continuous contraction semigroups on a Banach space 17
2 The Ornstein–Uhlenbeck semigroup on R and the associated

2 From an instantaneous error structure to a pricing model 143
3 Error calculations on the Black–Scholes model 155
4 Error calculations for a diffusion model 165
Bibliography for Chapter VII 185
VIII Applications in the field of physics 187
1 Drawing an ellipse (exercise) 187
2 Repeated samples: Discussion 190
3 Calculation of lengths using the Cauchy–Favard method (exercise) 195
4Temperature equilibrium of a homogeneous solid (exercise) 197
5 Nonlinear oscillator subject to thermal interaction:
The Grüneisen parameter 201
6 Natural error structures on dynamic systems 219
Bibliography for Chapter VIII 229
Index 231
Chapter I
Intuitive introduction to error structures
Learning a theory is made easier thanks to previous practical training, e.g. probability
theory is usually taught by familiarizing the student with the intuitive meaning of ran-
dom variables, independence and expectation without emphasizing the mathematical
difficulties. We will pursue the same course in this chapter: managing errors without
strict adherence to symbolic rigor (which will be provided subsequently).
1 Error magnitude
Let us consider a quantity x with a small centered error εY ,onwhich a nonlinear
regular function f acts. Initially we thus have a random variable x +εY with no bias
(centered at the true value x) and a variance of ε
2
σ
2
Y
: bias

= E[f(x+ εY ) − f(x)]=
ε
2
σ
2
Y
2
f

(x) +ε
3
O(1)
variance
1
= E

(f (x + εY ) −f(x))
2

= ε
2
σ
2
Y
f

2
(x) +ε
3
O(1).

= bias
1
g

(f (x)) +
1
2
variance
1
g

(f (x)) +ε
3
O(1)
variance
2
= variance
1
· g

2
(f (x)) +ε
3
O(1).
A similar relation could easily be obtained for applications from R
p
into R
q
.
Formula (∗) deserves additional comment. If our interest is limited to the main


= E[X
2
| Y ]−(E[X | Y ])
2
.
There exists ψ such that var[X | Y ]=ψ(Y) and var[X | Y = y] means ψ(y), which
is defined for P
Y
-almost-every y.
I.2 Description of small errors by their biases and variances 3
2.1. Suppose that the assessment of pollution in a river involves the concentration
C of some pollutant, with the quantity C being random and able to be measured by
an experimental device whose result exhibits an error C. The random variable C
is generally correlated with C (for higher river pollution levels, the device becomes
dirtier and fuzzier). The classical probabilistic approach requires the joint law of the
pair (C, C) in order to model the experiment, or equivalently the law of C and the
conditional law of C given C.
For pragmatic purposes, we now adopt the three following assumptions:
A1. We consider that the conditional law of C given C provides excessive infor-
mation and is practically unattainable. We suppose that only the conditional variance
var[C | C] is known and (if possible) the bias E[C | C].
A2. We suppose that the errors are small. In other words, the simplifications typically
performed by physicists and engineers when quantities are small are allowed herein.
A3. We assume the biases E[C | C] and variances var[C | C] of the errors to be
of the same order of magnitude.
With these hypotheses, is it possible to compute the variance and bias of the error
on a function of C, say f(C)?
Let us remark that by applying A3 and A2, (E[C | C])
2

f

◦ C ·var[C | C].
Let us introduce the two functions γ and a, defined by
var[C | C]=γ(C)ε
2
E[C | C]=a(C)ε
2
,
where ε is a size parameter denoting the smallness of errors; (1) can then be written
(2)



var[(f ◦C) | C]=f

2
◦ C ·γ(C)ε
2
E[(f ◦C) | C]=f

◦ C ·a(C)ε
2
+
1
2
f

◦ C ·γ(C)ε
2

(x)γ (x)ε
2
for P
C
-a.e. x.
The object (R, B(R), P
C
,
C
) with, in this case 
C
[f ]=f

2
· γ , suitably ax-
iomatized will be called an error structure and 
C
will be called the quadratic error
operator of this error structure.
2.2. What happens when C is a two-dimensional random variable? Let us take an
example.
Suppose a duration T
1
follows an exponential law of parameter 1 and is measured
in such a manner that T
1
and its error can be modeled by the error structure




2
T
2
1
ε
2
.
Similarly, suppose a duration T
2
following the same law is measured by another
device such that T
2
and its error can be modeled by the following error structure:



S
2
=

R
+
, B(R
+
), e
−y
1
[0,∞[
(y) dy, 
2

the image probability space of (T
1
,T
2
) is

R
2
+
, B(R
2
+
), 1
[0,∞[
(x)1
[0,∞[
(y)e
−x−y
dx dy

.
I.2 Description of small errors by their biases and variances 5
The error on a regular function F of T
1
and T
2
is


F(T

1
,T
2
)T
2
1
+ F

12
(T
1
,T
2
)T
1
T
2
+
1
2
F

22
(T
1
,T
2
)T
2
2

1
)
2
| T
1
,T
2
]
+ 2F

1
(T
1
,T
2
)F

2
(T
1
,T
2
)E[T
1
T
2
| T
1
,T
2

,U
2
| V
1
,V
2
]=E[U
1
| V
1
]·E[U
2
| V
2
].
Once again we obtain with A1 to A3:
var[(F (T
1
,T
2
)) | T
1
,T
2
]=F

1
2
(T
1

, B(R
2
+
), 1
[0,∞[
(x)1
[0,∞[
(y)e
−x−y
dx dy,

satisfies
[F ](x, y) = 
1
[F(·,y)](x) +
2
[F(x,·)](y).
If we consider that the conditional laws of errors are very concentrated Gaussian
laws with dispersion matrix
M = ε
2

α
2
x
2
0
0 β
2
y

= 1.
T
2
y
O
x
T
1
b) Let us now weaken the independence hypothesis by supposing T
1
and T
2
to be
independent but their errors not. This assumption means that the quantity
E[T
1
T
2
| T
1
,T
2
]−E[T
1
| T
1
,T
2
]E[T
2

no longer vanishes, but remains a function of T
1
and T
2
. This quantity is called the
conditional covariance of T
1
and T
2
given T
1
, T
2
and denoted by cov[(T
1
,T
2
) |
T
1
,T
2
].
As an example, we can take
cov[(T
1
,T
2
) | T
1

]
cov[(T
1
,T
2
) | T
1
,T
2
] var[T
2
| T
1
,T
2
]

=

α
2
T
2
1
ρT
1
T
2
ρT
1

(T
1
,T
2
)α
2
T
2
1
ε
2
+ 2F

1
(T
1
,T
2
)F

2
(T
1
,T
2
)ρT
1
T
2
ε


2
(x, y)ρxy +F

2
2
(x, y)β
2
y
2
.
If, as in the preceding case, we consider that the conditional laws of errors are very
concentrated Gaussian laws with dispersion matrix
M = ε
2

α
2
x
2
ρxy
ρxy β
2
y
2

,
the elliptic level curves of these Gaussian densities with equation
(u v)M
−1

= ε
2
[T
1
cos θ +T
2
sin θ](x, y),
hence
√
u
2
+ v
2
is the standard deviation of the error in the direction θ .
T
2
y
O
x
T
1
c) We can also abandon the hypothesis of independence of T
1
and T
2
. The most
general error structure on (R
2
+
, B(R

where the matrix

a(x, y) b(x,y)
b(x, y) c(x, y)

is positive semi-definite. Nevertheless, we will see further below that in order to
achieve completely satisfactory error calculus, a link between the measure µ and the
operator  will be necessary.
Exercise. Consider the error structure of Section 2.2.a):




R
2
+
, B(R
2
+
), 1
[0,∞[
(x)1
[0,∞[
(y)e
−x−y
dx dy,

[F ](x, y) = F

1

2
2

.
What is the conditional variance of the error on H ?
Being bivariate, the random variable H possesses a bivariate error and we are thus
seeking a 2 ×2-matrix.
Setting F(x,y) = x ∧ y, G(x, y) =
x+y
2
,wehave
[F ](x, y) = 1
{x≤y}
α
2
x
2
+ 1
{y≤x}
β
2
y
2
[G](x, y) =
1
4
α
2
x
2

1
,T
2
] cov[(H
1
,H
2
) | T
1
,T
2
]
cov[(H
1
,H
2
) | T
1
,T
2
] var[H
2
| T
1
,T
2
]

=


α
2
T
2
1
+
1
2
1
{T
2
≤T
1
}
β
2
T
2
2
1
2
1
{T
1
≤T
2
}
α
2
T

2

.
3 Intuitive notion of error structure
The preceding example shows that the quadratic error operator  naturally polarizes
into a bilinear operator (as the covariance operator in probability theory), which is a
first-order differential operator.
I.3 Intuitive notion of error structure 9
3.1. We thus adopt the following temporary definition of an error structure.
An error structure is a probability space equipped with an operator  acting upon
random variables
(, X, P,)
and satisfying the following properties:
a) Symmetry
[F,G]=[G, F ];
b) Bilinearity



i
λ
i
F
i
,

j
µ
j
G


i
(F
1
, ,F
p
)

j
(G
1
, ,G
q
)[F
i
,G
j
].
3.2. In order to take in account the biases, we also have to introduce a bias operator
A,alinear operator acting on regular functions through a second order functional
calculus involving :
A[(F
1
, ,F
p
)]=

i



of errors to a modelisation by an error structure. We have to consider that
(, X, P)
represents what can be obtained by experiment and that the errors are small and only
known by their two first conditional moments with respect to the σ -field X. Then, up
to a size renormalization, we must think  and A as
[X]=E[(X)
2
|X]
A[X]=E[X|X]
where X is the error on X. These two quantities have the same order of magnitude.
4How to proceed with an error calculation
4.1. Suppose we are drawing a triangle with a graduated rule and a protractor: we take
the polar angle of OA, say θ
1
, and set OA = 
1
;nextwetake the angle (OA, AB),
say θ
2
, and set AB = 
2
.
y
O
θ
1
A
θ
2
B

dθ
1
π
dθ
2
π
, D,

where
D =

f ∈ L
2

d
1
L
d
2
L
dθ
1
π
dθ
2
π

:
∂f
∂

[f ]=
2
1

∂f
∂
1

2
+
1

2
∂f
∂
1
∂f
∂
2
+
2
2

∂f
∂
2

2
+


(i.e. no term in
∂f
∂
i
∂f
∂θ
j
). Such a hypothesis proves natural
when measurements are conducted using different instruments. The bilinear operator
associated with  is
[f, g]=
2
1
∂f
∂
1
∂g
∂
1
+
1
2

1

2

∂f
∂
1


∂f
∂θ
1
∂g
∂θ
2
+
∂f
∂θ
2
∂g
∂θ
1

+
∂f
∂θ
2
∂g
∂θ
2
.
2) Compute the errors on significant quantities using functional calculus on 
(Property 3d)). Take point B for instance:
X
B
= 
1
cos θ

2
+ 2 sin θ
1
sin(θ
1
+ θ
2
))
+ 
2
2
(1 + 2 sin
2
(θ
1
+ θ
2
))
[Y
B
]=
2
1
+ 
1

2
(cos θ
2
+ 2 cos θ

2
sin(2θ
1
+ 2θ
2
).
For the area of the triangle, the formula area(OAB) =
1
2

1

2
sin θ
2
yields
[area(OAB)]=
1
4

2
1

2
2
(1 + 2 sin
2
θ
2
).


2
cos θ
2
+ 
2
2
we obtain
[OB
2
]=4

(
2
1
+ 
2
2
)
2
+ 3(
2
1
+ 
2
2
)
1

2

we have
[OB]
OB
2
= 1 −

1

2
cos θ
2
OB
2
,
thereby providing the result that theproportional error onOB is minimal when 
1
= 
2
and θ
2
= 0. In this case
([OB])
1/2
OB
=
√
3
2
.
12 I Intuitive introduction to error structures

t
)dt

.
Suppose that the Markov chain (X
x
n
) is a discrete approximation of (X
t
) and
simulated by
(3) X
x
n+1
= (X
x
n
,U
n+1
), X
x
0
= x,
where U
1
,U
2
, ,U
n
, is a sequence of i.i.d. random variables uniformly dis-

(1 − x)
2
.
Then, for regular functions h, k,

1
0
[h, k](x) dx =

1
0
h

(x)k

(x)x
2
(1 − x
2
)dx
yields by partial integration
(5)

1
0
[h, k](x) dx =−

1
0
h(x)

[F(U
1
, ,U
n
, ),G(U
1
, ,U
n
, )]
= F

1
(U
1
, ,U
n
, )G

1
(U
1
, ,U
n
, )U
2
1
(1 − U
1
)
2

, )U
2
1
(1 − U
1
)
2

.
(7)
The derivative of interest to us then becomes
dP
dx
= E

∞

n=0
e
−nt
∂(f(X
x
n
))
∂x
t

and by the representation in (3)
(8)
∂f (X

= f

(X
x
n
)
n−1

i=1


1
(X
x
i
,U
i+1
)

2
(x, U
1
),
and comparing (8) with (9) yields
dP
dx
= E

∞


1
, ,U
n
, ) =
∞

n=0
e
−nt
t
∂f (X
x
n
)
∂U
1
G

1
(U
1
, ,U
n
, )U
2
1
(1 − U
1
)
2

∂
∂U
1



1
(x, U
1
)


2
(x, U
1
)


.
Formula (10) is atypical integration by partsformula, useful in Monte Carlo simulation
when simultaneously dealing with several functions f .
One aim of error calculus theory is to generalize such integration by parts formulae
to more complex contexts.
14 I Intuitive introduction to error structures
We must now focus on making such error calculations more rigorous. This process
will be carried out in the following chapters using a powerful mathematical toolbox,
the theory of Dirichlet forms. The benefit consists of the possibility of performing
error calculations in infinite dimensional models, as is typical in stochastic analysis
and in mathematical finance in particular. Other advantages will be provided thanks
to the strength of rigorous arguments.

Gauss tackled this question in the following way. He first assumed – we will return
to this idea later on – that the quantity to be measured is random and can vary within
the domain of the measurement device according to an a priori law. In more modern
language, let X be this random variable and µ its law. The results of the measurement
operations are other random variables X
1
, ,X
n
and Gauss assumes that:
a) the conditional law of X
i
given X is of the form
P{X
i
∈ E | X = x}=

E
ϕ(x
1
− x) dx
1
,
b) the variables X
1
, ,X
n
are conditionally independent given X.
He then easily computed the conditional law of X given the measurement results: it
displays a density with respect to µ. This density being maximized at the arithmetic
Appendix 15

while the errors need not be, when performed with the same instrument. He did not
develop any new mathematical formalism for this idea, but emphasized the advantage
of assuming small errors: This allows Gauss’ argument for the normal law to become
compatible with nonlinear changes of variables and to be carried out by differential
calculus. This focus is central to the field of error calculus.
Twelve years after his demonstration that led to the normal law, Gauss became
interested in the propagation of errors and hence must be considered as the founder
of error calculus. In Theoria Combinationis (1821) he states the following problem.
Given a quantity U = F(V
1
,V
2
, )function of the erroneous quantities V
1
,V
2
, ,
compute the potential quadratic error to expect on U , with the quadratic errors σ
2
1
,
σ
2
2
, on V
1
,V
2
, being known and assumed to be small and independent. His
response consisted of the following formula:

it has a coherence property. With a formula such as
(12) σ
U
=




∂F
∂V
1




σ
1
+




∂F
∂V
2




σ