singular stochastic differential equations - cherny a, englebert h

Lecture Notes in Mathematics 1858
Editors:
J M. Morel, Cachan
F. Takens, Groningen
B. Teissier, Paris

Alexander S. Cherny
Hans-J
¨
urgen Engelbert
Singular Stochastic
Different ial E quat ions
123
Authors
Alexander S. Cherny
Department of Probabilit y Theory
Faculty of Mechanics and Mathematics
Moscow State University
Leninskie Gory
119992, Moscow
Russia
e-mail:
Hans-J
¨
urgen Engelbert
Institut f
¨
ur Stochastik
Fakult
¨
at f

¨
ur Mathematik und Informatik
Friedrich-Schiller-Universit
¨
at Jena
Ernst-Abbe-Platz 1-4
07743 Jena
Germany
e-mail:
LibraryofCongressControlNumber:2004115716
Mathematics Subject Classification (2000): 60-02, 60G17, 60H10, 60J25, 60J60
ISSN 0075-8434
ISBN 3-540-24007-1 Springer Berlin Heidelberg New York
DOI: 10.1007/b104187
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Preface
We consider one-dimensional homogeneous stochastic differential equations
of the form
dX
t
= b(X
t
)dt + σ(X
t
)dB
t
,X
0
= x
0
, (∗)
where b and σ are supposed to be measurable functions and σ =0.
There is a rich theory studying the existence and the uniqueness of solu-
tions of these (and more general) stochastic differential equations. For equa-
tions of the form (∗), one of the best sufficient conditions is that the function
(1 + |b|)/σ
2
should be locally integrable on the real line. However, both in
theory and in practice one often comes across equations that do not satisfy
this condition. The use of such equations is necessary, in particular, if we want
a solution to be positive. In this monograph, these equations are called sin-
gular stochastic differential equations. A typical example of such an equation
is the stochastic differential equation for a geometric Brownian motion.

Apointd ∈ R, at which the function (1 + |b|)/σ
2
is not locally integrable,
is called in this monograph a singular point. We explain why these points
are indeed “singular”. For the isolated singular points, we perform a complete
qualitative classification. According to this classification, an isolated singular
point can have one of 48 possible types. The type of a point is easily computed
through the coefficients b and σ. The classification allows one to find out
whether a solution can leave an isolated singular point, whether it can reach
this point, whether it can be extended after having reached this point, and
so on.
It turns out that the isolated singular points of 44 types do not disturb
the uniqueness of a solution and only the isolated singular points of the
remaining 4 types disturb uniqueness. These points are called here the branch
points. There exists a large amount of “bad” solutions (for instance, non-
Markov solutions) in the neighbourhood of a branch point. Discovering the
branch points is one of the most interesting consequences of the constructed
classification.
The monograph also includes an overview of the basic definitions and facts
related to the stochastic differential equations (different types of existence and
uniqueness, martingale problems, solutions up to a random time, etc.) as well
as a number of important examples.
We gratefully acknowledge financial support by the DAAD and by the
European Community’s Human Potential Programme under contract HPRN-
CT-2002-00281.
Moscow, Jena, Alexander Cherny
October 2004 Hans-J¨urgen Engelbert

Table of Contents
Introduction 1

1 Stochastic Differential Equations 5
1.1 GeneralDefinitions 5
1.2 Sufficient Conditions for Existence and Uniqueness . . . . . . . . . 9
1.3 TenImportantExamples 12
1.4 MartingaleProblems 19
1.5 Solutions upto a Random Time 23
2 One-Sided Classification of Isolated Singular Points 27
2.1 Isolated SingularPoints:The Definition 27
2.2 Isolated SingularPoints:Examples 32
2.3 One-Sided Classification:TheResults 34
2.4 One-SidedClassification:InformalDescription 38
2.5 One-Sided Classification:TheProofs 42
3 Two-Sided Classification of Isolated Singular Points 65
3.1 Two-SidedClassification:The Results 65
3.2 Two-SidedClassification:InformalDescription 66
3.3 Two-SidedClassification:The Proofs 69
3.4 The BranchPoints:Non-MarkovSolutions 73
3.5 The BranchPoints:StrongMarkovSolutions 75
4 Classification at Infinity and Global Solutions 81
4.1 Classification at Infinity: The Results . . . . . . . . . . . . . . . . . . . . . 81
4.2 Classification at Infinity: Informal Description . . . . . . . . . . . . . . 82
4.3 Classification at Infinity: The Proofs . . . . . . . . . . . . . . . . . . . . . . 85
4.4 GlobalSolutions: The Results 86
4.5 GlobalSolutions: The Proofs 88
5 Several Special Cases 93
5.1 PowerEquations: Types ofZero 93
5.2 Power Equations: Types of Infinity . . . . . . . . . . . . . . . . . . . . . . . 97
5.3 Equations with a Constant-Sign Drift: Types of Zero . . . . . . . . 99
5.4 Equations with a Constant-Sign Drift: Types of Infinity . . . . . 102
VIII Table of Contents

Appendix A: Some Known Facts 105
A.1 LocalTimes 105
A.2 RandomTime-Changes 107
A.3 BesselProcesses 108
A.4 Strong Markov Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A.5 OtherFacts 111
Appendix B: Some Auxiliary Lemmas 113
B.1 StoppingTimes 113
B.2 MeasuresandSolutions 114
B.3 OtherLemmas 116
References 119
Index of Notation 123
Index of Terms 127
Introduction
The basis of the theory of diffusion processes was formed by Kolmogorov [30]
(the Chapman–Kolmogorov equation, forward and backward partial differ-
ential equations). This theory was further developed in a series of papers by
Feller (see, for example, [16], [17]).
Both Kolmogorov and Feller considered diffusion processes from the point
of view of their finite-dimensional distributions. Itˆo [24], [25] proposed an
approach to the “pathwise” construction of diffusion processes. He introduced
the notion of a stochastic differential equation (abbreviated below as SDE).
At about the same time and independently of Itˆo, SDEs were considered by
Gikhman [18], [19]. Stroock and Varadhan [44], [45] introduced the notion of
a martingale problem that is closely connected with the notion of a SDE.
Many investigations were devoted to the problems of existence, unique-
ness, and properties of solutions of SDEs. Sufficient conditions for existence
and uniqueness were obtained by Girsanov [21], Itˆo [25], Krylov [31], [32],
Skorokhod [42], Stroock and Varadhan [44], Zvonkin [49], and others. The
evolution of the theory has shown that it is reasonable to introduce dif-

ferent types of solutions (weak and strong solutions) and different types of
uniqueness (uniqueness in law and pathwise uniqueness); see Liptser and
Shiryaev [33], Yamada and Watanabe [48], Zvonkin and Krylov [50]. More
information on SDEs and their applications can be found in the books [20],
[23], [28, Ch. 18], [29, Ch. 5], [33, Ch. IV], [36], [38, Ch. IX], [39, Ch. V], [45].
For one-dimensional homogeneous SDEs, i.e., the SDEs of the form
dX
t
= b(X
t
)dt + σ(X
t
)dB
t
,X
0
= x
0
, (1)
one of the weakest sufficient conditions for weak existence and uniqueness in
law was obtained by Engelbert and Schmidt [12]–[15]. (In the case, where
b = 0, there exist even necessary and sufficient conditions; see the paper [12]
by Engelbert and Schmidt and the paper [1] by Assing and Senf.) Engelbert
and Schmidt proved that if σ(x) =0foranyx ∈ R and
1+|b|
σ
2
∈ L
1
loc

(R), (2)
then there exists a unique solution of (1). (More precisely, there exists a
unique solution defined up to the time of explosion.)
A.S. Cherny and H J. Engelbert: LNM 1858, pp. 1–4, 2005.
c
 Springer-Verlag Berlin Heidelberg 2005
2 Introduction
Condition (2) is rather weak. Nevertheless, SDEs that do not satisfy this
condition often arise in theory and in practice. Such are, for instance, the
SDE for a geometric Brownian motion
dX
t
= µX
t
dt + σX
t
dB
t
,X
0
= x
0
(the Black-Scholes model !) and the SDE for a δ-dimensional Bessel process
(δ>1):
dX
t
=
δ − 1
2X
t

dt + dB
t
,X
0
= x
0
.
In practice, SDEs that do not satisfy (2) arise, for example, in the following
situation. Suppose that we model some process as a solution of (1). Assume
that this process is positive by its nature (for instance, this is the price of a
stock or the size of a population). Then a SDE used to model such a process
should not satisfy condition (2). The reason is as follows. If condition (2) is
satisfied, then, for any a ∈ R, the solution reaches the level a with strictly
positive probability. (This follows from the results of Engelbert and Schmidt.)
The SDEs that do not satisfy condition (2) are called in this monograph
singular SDEs. The study of these equations is the subject of the monograph.
We investigate three main problems:
(i) Does there exist a solution of (1)?
(ii) Is it unique?
(iii) What is the qualitative behaviour of a solution?
In order to investigate singular SDEs, we introduce the following defini-
tion. A point d ∈ R is called a singular point for SDE (1) if
1+|b|
σ
2
/∈ L
1
loc
(d).
We always assume that σ(x) =0foranyx ∈ R. This is motivated by the

desire to exclude solutions which have sojourn time in any single point. (In-
deed, it is easy to verify that if σ = 0 at a point z ∈ R,thenanysolution
of (1) spends no time at z. This, in turn, implies that any solution of (1) also
solves the SDE with the same drift and the diffusion coefficient σ −σ(z)I
{z}
.
“Conversely”, if σ = 0 at a point z ∈ R and a solution of (1) spends no time
at z, then, for any η ∈ R, it also solves the SDE with the same drift and the
diffusion coefficient σ + ηI
{z}
.)
The first question that arises in connection with this definition is: Why are
these points indeed “singular”? The answer is given in Section 2.1, where we
explain the qualitative difference between the singular points and the regular
points in terms of the behaviour of solutions.
Using the above terminology, we can say that a SDE is singular if and only
if the set of its singular points is nonempty. It is worth noting that in practice
one often comes across SDEs that have only one singular point (usually, it
is zero). Thus, the most important subclass of singular points is formed by
the isolated singular points.(Wecalld ∈ R an isolated singular point if d is
Introduction 3
singular and there exists a deleted neighbourhood of d that consists of regular
points.)
In this monograph, we perform a complete qualitative classification of
the isolated singular points. The classification shows whether a solution can
leave an isolated singular point, whether it can reach this point, whether
it can be extended after having reached this point, and so on. According
to this classification, an isolated singular point can have one of 48 possible
types. The type of a point is easily computed through the coefficients b and
σ. The constructed classification may be viewed as a counterpart (for SDEs)

of Feller’s classification of boundary behaviour of continuous strong Markov
processes.
The monograph is arranged as follows.
Chapter 1 is an overview of basic definitions and facts related to SDEs,
more precisely, to the problems of the existence and the uniqueness of solu-
tions. In particular, we describe the relationship between different types of
existence and uniqueness (see Figure 1.1 on p. 8) and cite some classical con-
ditions that guarantee existence and uniqueness. This chapter also includes
several important examples of SDEs. Moreover, we characterize all the pos-
sible combinations of existence and uniqueness (see Table 1.1 on p. 18).
In Chapter 2, we introduce the notion of a singular point and give the
arguments why these points are indeed “singular”. Then we study the ex-
istence, the uniqueness, and the qualitative behaviour of a solution in the
right-hand neighbourhood of an isolated singular point. This leads to the
one-sided classification of isolated singular points. According to this classifi-
cation, an isolated singular point can have one of 7 possible right types (see
Figure 2.2 on p. 39).
In Chapter 3, we investigate the existence, the uniqueness, and the qual-
itative behaviour of a solution in the two-sided neighbourhood of an isolated
singular point. We consider the effects brought by the combination of right
and left types. Since there exist 7 possible right types and 7 possible left
types, there are 49 feasible combinations. One of these combinations corre-
sponds to a regular point, and therefore, an isolated singular point can have
one of 48 possible types. It turns out that the isolated singular points of only
4 types can disturb the uniqueness of a solution. We call them the branch
points and characterize all the strong Markov solutions in the neighbourhood
of such a point.
In Chapter 4, we investigate the behaviour of a solution “in the neigh-
bourhood of +∞”. This leads to the classification at infinity. According to
this classification, +∞ can have one of 3 possible types (see Figure 4.1 on

p. 83). The classification shows, in particular, whether a solution can explode
into +∞. Thus, the well known Feller’s test for explosions is a consequence
of this classification.
All the results of Chapters 2 and 3 apply to local solutions, i.e., solutions
up to a random time (this concept is introduced in Chapter 1). In the second
4 Introduction
part of Chapter 4, we use the obtained results to study the existence, the
uniqueness, and the qualitative behaviour of global solutions, i.e., solutions
in the classical sense. This is done for the SDEs that have no more than one
singular point (see Tables 4.1–4.3 on pp. 88, 89).
In Chapter 5, we consider the power equations, i.e., the equations of the
form
dX
t
= µ|X
t
|
α
dt + ν|X
t
|
β
dB
t
and propose a simple procedure to determine the type of zero and the type
of infinity for these SDEs (see Figure 5.1 on p. 94 and Figure 5.2 on p. 98).
Moreover, we study which types of zero and which types of infinity are pos-
sible for the SDEs with a constant-sign drift (see Table 5.1 on p. 101 and
Table 5.2 on p. 103).
The known results from the stochastic calculus used in the proofs are con-

tained in Appendix A, while the auxiliary lemmas are given in Appendix B.
The monograph includes 7 figures with simulated paths of solutions of
singular SDEs.
1 Stochastic Differential Equations
In this chapter, we consider general multidimensional SDEs of the form (1.1)
given below.
In Section 1.1, we give the standard definitions of various types of the
existence and the uniqueness of solutions as well as some general theorems
that show the relationship between various properties.
Section 1.2 contains some classical sufficient conditions for various types
of existence and uniqueness.
In Section 1.3, we present several important examples that illustrate var-
ious combinations of the existence and the uniqueness of solutions. Most of
these examples (but not all) are well known. We also find all the possible
combinations of existence and uniqueness.
Section 1.4 includes the definition of a martingale problem. We also recall
the relationship between the martingale problems and the SDEs.
In Section 1.5, we define a solution up to a random time.
1.1 General Definitions
Here we will consider a general type of SDEs, i.e., multidimensional SDEs
with coefficients that depend on the past. These are the equations of the form
dX
i
t
= b
i
t
(X)dt +
m


j=1
σ
ij
t
(X)dB
j
t
,X
0
= x
0
(i =1, ,n), (1.1)
where n ∈ N, m ∈ N, x
0
∈ R
n
,and
b : C(R
+
, R
n
) × R
+
→ R
n
,
σ : C(R
+
, R
n

) × R
+
→ R
n×m
are predictable functionals. (The definition of a predictable process can be
found, for example, in [27, Ch. I, §2a]or[38,Ch.IV,§ 5].)
Remark. We fix a starting point x
0
together with b and σ. In our terminology,
SDEs with the same b and σ and with different starting points are different
SDEs.
A.S. Cherny and H J. Engelbert: LNM 1858, pp. 5–25, 2005.
c
 Springer-Verlag Berlin Heidelberg 2005
6 1 Stochastic Differential Equations
Definition 1.1. (i) A solution of (1.1) is a pair (Z, B) of adapted processes
on a filtered probability space

Ω, G, (G
t
)
t≥0
, Q

such that
(a) B is a m-dimensional (G
t
)-Brownian motion, i.e., B is a m-dimensional
Brownian motion started at zero and is a (G
t

, Q)-martingale;
(b) for any t ≥ 0,

t
0

n

i=1
|b
i
s
(Z)| +
n

i=1
m

j=1
(σ
ij
s
(Z))
2

ds < ∞ Q-a.s.;
(c) for any t ≥ 0, i =1, ,n,
Z
i
t

= x
i
0
+

t
0
b
i
s
(Z)ds +
m

j=1

t
0
σ
ij
s
(Z)dB
j
s
Q-a.s.
(ii) There is weak existence for (1.1) if there exists a solution of (1.1) on
some filtered probability space.
Definition 1.2. (i) Asolution(Z, B) is called a strong solution if Z is

F
B

t

-
adapted, where
F
B
t
is the σ-field generated by σ(B
s
; s ≤ t) and by the subsets
of the Q-null sets from σ(B
s
; s ≥ 0).
(ii) There is strong existence for (1.1) if there exists a strong solution
of (1.1) on some filtered probability space.
Remark. Solutions in the sense of Definition 1.1 are sometimes called weak
solutions.Herewecallthemsimplysolutions. However, the existence of a
solution is denoted by the term weak existence in order to stress the difference
between weak existence and strong existence (i.e., the existence of a strong
solution).
Definition 1.3. There is uniqueness in law for (1.1) if for any solutions
(Z, B)and(

Z,

B) (that may be defined on different filtered probability
spaces), one has Law(Z
t
; t ≥ 0) = Law(


Z
t
; t ≥ 0).
Definition 1.4. There is pathwise uniqueness for (1.1) if for any solutions
(Z, B)and(

Z,B) (that are defined on the same filtered probability space),
one has Q{∀t ≥ 0,Z
t
=

Z
t
} =1.
Remark. If there exists no solution of (1.1), then there are both uniqueness
in law and pathwise uniqueness.
The following 4 statements clarify the relationship between various prop-
erties.
Proposition 1.5. Let (Z, B) be a strong solution of (1.1).
(i) There exists a measurable map
1.1 General Definitions 7
Ψ:

C(R
+
, R
m
), B

−→


C(R
+
, R
n
), B

(here B denotes the Borel σ-field) such that the process Ψ(B) is

F
B
t

-adapted
and Z =Ψ(B) Q-a.s.
(ii) If

B is a m-dimensional (

F
t
)-Brownian motion on a filtered proba-
bility space


Ω,

G, (

G

t
),

Q

and

Z := Ψ(

B),then(

Z,

B) is a strong solution
of (1.1).
For the proof, see, for example, [5].
Now we state a well known result of Yamada and Watanabe.
Proposition 1.6 (Yamada, Watanabe). Suppose that pathwise unique-
ness holds for (1.1).
(i) Uniqueness in law holds for (1.1);
(ii) There exists a measurable map
Ψ:

C(R
+
, R
m
), B

−→


C(R
+
, R
n
), B

such that the process Ψ(B) is

F
B
t

-adapted and, for any solution (Z, B)
of (1.1), we have Z =Ψ(B) Q-a.s.
For the proof, see [48] or [38, Ch. IX, Th. 1.7].
The following result complements the theorem of Yamada and Watanabe.
Proposition 1.7. Suppose that uniqueness in law holds for (1.1) and there
exists a strong solution. Then pathwise uniqueness holds for (1.1).
This theorem was proved by Engelbert [10] under some additional assump-
tions. It was proved with no additional assumptions by Cherny [7].
The crucial fact needed to prove Proposition 1.7 is the following result. It
shows that uniqueness in law implies a seemingly stronger property.
Proposition 1.8. Suppose that uniqueness in law holds for (1.1).Then,for
any solutions (Z, B) and (

Z,

B)(that may be defined on different filtered
probability spaces), one has Law(Z

t
,B
t
; t ≥ 0) = Law(

Z
t
,

B
t
; t ≥ 0).
For the proof, see [7].
The situation with solutions of SDEs can now be described as follows.
It may happen that there exists no solution of (1.1) on any filtered prob-
ability space (see Examples 1.16, 1.17).
It may also happen that on some filtered probability space there exists a
solution (or there are even several solutions with the same Brownian motion),
while on some other filtered probability space with a Brownian motion there
exists no solution (see Examples 1.18, 1.19, 1.20, and 1.24).
8 1 Stochastic Differential Equations
weak
existence
strong
existence
uniqueness
in law
pathwise
uniqueness
weak

existence
strong
existence
uniqueness
in law
pathwise
uniqueness
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅









weak
existence
strong
existence
uniqueness
in law
pathwise
uniqueness
















❅
❅
❅
❅
❅

❅
❅
❅
weak
existence
pathwise
uniqueness
strong
existence
uniqueness
in law
the best
possible
situation
✔
✔
✔
❚
❚
❚
✔
✔
✔
❚
❚
❚
Fig. 1.1. The relationship between various types of existence and uniqueness. The
top diagrams show obvious implications and the implications given by the Yamada–
Watanabe theorem. The centre diagram shows an obvious implication and the im-
plication given by Proposition 1.7. The bottom diagram illustrates the Yamada–

Watanabe theorem and Proposition 1.7 in terms of the “best possible situation”.
1.2 Sufficient Conditions for Existence and Uniqueness 9
If there exists a strong solution of (1.1) on some filtered probability space,
then there exists a strong solution on any other filtered probability space
with a Brownian motion (see Proposition 1.5). However, it may happen in
this case that there are several solutions with the same Brownian motion (see
Examples 1.21–1.23).
If pathwise uniqueness holds for (1.1) and there exists a solution on some
filtered probability space, then on any other filtered probability space with a
Brownian motion there exists exactly one solution, and this solution is strong
(see the Yamada–Watanabe theorem). This is the best possible situation.
Thus, the Yamada–Watanabe theorem shows that pathwise uniqueness
together with weak existence guarantee that the situation is the best possible.
Proposition 1.7 shows that uniqueness in law together with strong existence
guarantee that the situation is the best possible.
1.2 Sufficient Conditions for Existence and Uniqueness
The statements given in this section are related to SDEs, for which b
t
(X)=
b(t, X
t
)andσ
t
(X)=σ(t, X
t
), where b : R
+
× R
n
→ R

n
and σ : R
+
× R
n
→
R
n×m
are measurable functions.
We begin with sufficient conditions for strong existence and pathwise
uniqueness. The first result of this type was obtained by Itˆo.
Proposition 1.9 (Itˆo). Suppose that, for a SDE
dX
i
t
= b
i
(t, X
t
)dt +
m

j=1
σ
ij
(t, X
t
)dB
j
t

,X
0
= x
0
(i =1, ,n),
there exists a constant C>0 such that
b(t, x) −b(t, y) + σ(t, x) − σ(t, y)≤Cx − y,t≥ 0,x,y∈ R
n
,
b(t, x)+ σ(t, x)≤C(1 + x),t≥ 0,x∈ R
n
,
where
b(t, x) :=

n

i=1
(b
i
(t, x))
2

1/2
,
σ(t, x) :=

n

i=1

m

j=1
(σ
ij
(t, x))
2

1/2
.
Then strong existence and pathwise uniqueness hold.
For the proof, see [25], [29, Ch. 5, Th. 2.9], or [36, Th. 5.2.1].
10 1 Stochastic Differential Equations
Proposition 1.10 (Zvonkin). Suppose that, for a one-dimensional SDE
dX
t
= b(t, X
t
)dt + σ(t, X
t
)dB
t
,X
0
= x
0
,
the coefficient b is measurable and bounded, the coefficient σ is continuous
and bounded, and there exist constants C>0, ε>0 such that
|σ(t, x) −σ(t, y)|≤C


|x − y|,t≥ 0,x,y∈ R,
|σ(t, x)|≥ε, t ≥ 0,x∈ R.
Then strong existence and pathwise uniqueness hold.
For the proof, see [49].
For homogeneous SDEs, there exists a stronger result.
Proposition 1.11 (Engelbert, Schmidt). Suppose that, for a one-
dimensional SDE
dX
t
= b(X
t
)dt + σ(X
t
)dB
t
,X
0
= x
0
,
σ =0at each point, b/σ
2
∈ L
1
loc
(R), and there exists a constant C>0 such
that
|σ(x) − σ(y)|≤C


|x − y|,x,y∈ R,
|b(x)| + |σ(x)|≤C(1 + |x|),x∈ R.
Then strong existence and pathwise uniqueness hold.
For the proof, see [15, Th. 5.53].
The following proposition guarantees only pathwise uniqueness. Its main
difference from Proposition 1.10 is that the diffusion coefficient here need not
be bounded away from zero.
Proposition 1.12 (Yamada, Watanabe). Suppose that, for a one-
dimensional SDE
dX
t
= b(t, X
t
)dt + σ(t, X
t
)dB
t
,X
0
= x
0
,
there exist a constant C>0 and a strictly increasing function h : R
+
→ R
+
with

0+
0

h
−2
(x)dx =+∞ such that
|b(t, x) −b(t, y)|≤C|x − y|,t≥ 0,x,y∈ R,
|σ(t, x) −σ(t, y)|≤h(|x − y|),t≥ 0,x,y∈ R.
Then pathwise uniqueness holds.
1.2 Sufficient Conditions for Existence and Uniqueness 11
For the proof, see [29, Ch. 5, Prop. 2.13], [38, Ch. IX, Th. 3.5], or [39, Ch. V,
Th. 40.1].
We now turn to results related to weak existence and uniqueness in law.
The first of these results guarantees only weak existence; it is almost covered
by further results, but not completely. Namely, here the diffusion matrix σ
need not be elliptic (it might even be not a square matrix).
Proposition 1.13 (Skorokhod). Suppose that, for a SDE
dX
i
t
= b
i
(t, X
t
)dt +
m

j=1
σ
ij
(t, X
t
)dB

j
t
,X
i
0
= x
i
0
(i =1, ,n),
the coefficients b and σ are continuous and bounded. Then weak existence
holds.
For the proof, see [42] or [39, Ch. V, Th. 23.5].
Remark. The conditions of Proposition 1.13 guarantee neither strong exis-
tence (see Example 1.19) nor uniqueness in law (see Example 1.22).
In the next result, the conditions on b and σ are essentially relaxed as
compared with the previous proposition.
Proposition 1.14 (Stroock, Varadhan). Suppose that, for a SDE
dX
i
t
= b
i
(t, X
t
)dt +
n

j=1
σ
ij

(t, X
t
)dB
j
t
,X
0
= x
0
(i =1, ,n),
the coefficient b is measurable and bounded, the coefficient σ is continuous
and bounded, and, for any t ≥ 0, x ∈ R
n
, there exists a constant ε(t, x) > 0
such that
σ(t, x)λ≥ε(t, x)λ,λ∈ R
n
.
Then weak existence and uniqueness in law hold.
For the proof, see [44, Th. 4.2, 5.6].
In the next result, the diffusion coefficient σ need not be continuous.
However, the statement deals with homogeneous SDEs only.
Proposition 1.15 (Krylov). Suppose that, for a SDE
dX
i
t
= b
i
(X
t

)dt +
n

j=1
σ
ij
(X
t
)dB
j
t
,X
0
= x
0
(i =1, ,n),
the coefficient b is measurable and bounded, the coefficient σ is measurable
and bounded, and there exist a constant ε>0 such that
σ(x)λ≥ελ,x∈ R
n
,λ∈ R
n
.
Then weak existence holds. If moreover n ≤ 2, then uniqueness in law holds.
12 1 Stochastic Differential Equations
For the proof, see [32].
Remark. In the case n>2, the conditions of Proposition 1.15 do not guar-
antee uniqueness in law (see Example 1.24).
1.3 Ten Important Examples
In the examples given below, we will use the characteristic diagrams

to illustrate the statement of each example. The first square in the
diagram corresponds to weak existence; the second – to strong existence; the
third – to uniqueness in law; the fourth – to pathwise uniqueness. Thus, the
statement “for the SDE ,wehave
+ −+ − ” should be read as follows:
“for the SDE , there exists a solution, there exists no strong solution,
uniqueness in law holds, and pathwise uniqueness does not hold”.
We begin with examples of SDEs with no solution.
Example 1.16 (no solution). For the SDE
dX
t
= −sgn X
t
dt, X
0
=0, (1.2)
where
sgn x =

1ifx>0,
−1ifx ≤ 0,
(1.3)
we have
−−++ .
Proof. Suppose that there exists a solution (Z, B). Then almost all paths
of Z satisfy the integral equation
f(t)=−

t
0

sgn f(s)ds, t ≥ 0. (1.4)
Let f be a solution of this equation. Assume that there exist a>0, t>0
such that f(t)=a.Setv =inf{t ≥ 0:f (t)=a}, u =sup{t ≤ v : f (t)=0}.
Using (1.4), we get a = f(v) − f (u)=−(v − u). The obtained contradiction
shows that f ≤ 0. In a similar way we prove that f ≥ 0. Thus, f ≡ 0, but
then it is not a solution of (1.4). As a result, (1.4), and hence, (1.2), has no
solution. 
The next example is a SDE with the same characteristic diagram and
with σ ≡ 1.
Example 1.17 (no solution). For the SDE
dX
t
= −
1
2X
t
I(X
t
=0)dt + dB
t
,X
0
=0, (1.5)
we have
−−++ .
1.3 Ten Important Examples 13
Proof. Suppose that (Z, B) is a solution of (1.5). Then
Z
t
= −


t
0
1
2Z
s
I(Z
s
=0)ds + B
t
,t≥ 0.
By Itˆo’s formula,
Z
2
t
= −

t
0
2Z
s
1
2Z
s
I(Z
s
=0)ds +

t
0

2Z
s
dB
s
+ t
=

t
0
I(Z
s
=0)ds +

t
0
2Z
s
dB
s
,t≥ 0.
The process Z is a continuous semimartingale with Z
t
= t. Hence, by the
occupation times formula,

t
0
I(Z
s
=0)ds =


R
I(x =0)L
x
t
(Z)dx =0,t≥ 0,
where L
x
t
(Z) denotes the local time of the process Z (see Definition A.2). As a
result, Z
2
is a positive local martingale, and consequently, a supermartingale.
Since Z
2
≥ 0andZ
2
0
= 0, we conclude that Z
2
=0a.s.Butthen(Z, B)is
not a solution of (1.5). 
Now we turn to the examples of SDEs that possess a solution, but no
strong solution.
Example 1.18 (no strong solution; Tanaka). For the SDE
dX
t
=sgnX
t
dB

t
,X
0
=0 (1.6)
(for the precise definition of sgn,see(1.3)), we have
+ − + −
.
Proof. Let W be a Brownian motion on (Ω, G, Q). We set
Z
t
= W
t
,B
t
=

t
0
sgn W
s
dW
s
,t≥ 0
and take G
t
= F
W
t
. Obviously, (Z, B) is a solution of (1.6) on


Ω, G, (G
t
), Q

.
If (Z, B) is a solution of (1.6) on a filtered probability space

Ω, G, (G
t
), Q

,
then Z is a continuous (G
t
, Q)-local martingale with Z
t
= t. It follows from
P. L´evy’s characterization theorem that Z is a Brownian motion. This implies
uniqueness in law.
If (Z, B) is a solution of (1.6), then
B
t
=

t
0
sgn Z
s
dZ
s

,t≥ 0.
This implies that F
B
t
= F
|Z|
t
(see [38, Ch. VI, Cor. 2.2]). Hence, there exists
no strong solution.
If (Z, B) is a solution of (1.6), then (−Z, B) is also a solution. Thus, there
is no pathwise uniqueness. 
14 1 Stochastic Differential Equations
The next example is a SDE with the same characteristic diagram, b =0,
and a continuous σ.
Example 1.19 (no strong solution; Barlow). There exists a continuous
bounded function σ : R → (0, ∞) such that, for the SDE
dX
t
= σ(X
t
)dB
t
,X
0
= x
0
,
we have
+ − + −
.

For the proof, see [2].
The next example is a SDE with the same characteristic diagram and
with σ ≡ 1. The drift coefficient in this example depends on the past.
Example 1.20 (no strong solution; Tsirelson). There exists a bounded
predictable functional b : C(R
+
) ×R
+
→ R such that, for the SDE
dX
t
= b
t
(X)dt + dB
t
,X
0
= x
0
,
we have
+ − + −
.
For the proof, see [46], [23, Ch. IV, Ex. 4.1], or [38, Ch. IX, Prop. 3.6].
Remark. Let B be a Brownian motion on (Ω, G, Q). Set G
t
= F
B
t
.Then

the SDEs of Examples 1.18–1.20 have no solution on

Ω, G, (G
t
), Q

with the
Brownian motion B. Indeed, if (Z, B)isasolution,thenZ is (G
t
)-adapted,
which means that (Z, B) is a strong solution.
We now turn to examples of SDEs, for which there is no uniqueness in
law.
Example 1.21 (no uniqueness in law). For the SDE
dX
t
= I(X
t
=0)dB
t
,X
0
=0, (1.7)
we have
++−−
.
Proof. It is sufficient to note that (B,B)and(0,B) are solutions of (1.7) on

Ω, G, (G
t

), Q

whenever B is a (G
t
)-Brownian motion. 
Remark. Let B be a Brownian motion on (Ω, G, Q)andη be a random vari-
able that is independent of B with P{η =1} = P{η = −1} =1/2. Consider
Z
t
(ω)=

B
t
(ω)ifη(ω)=1,
0ifη(ω)=−1
and take G
t
= F
Z
t
.Then(Z,B) is a solution of (1.7) on

Ω, G, (G
t
), Q

that
is not strong. Indeed, for each t>0, η is not
F
B

t
-measurable. Since the sets
{η = −1} and {Z
t
=0} are indistinguishable, Z
t
is not F
B
t
-measurable.
1.3 Ten Important Examples 15
The next example is a SDE with the same characteristic diagram, b =0,
and a continuous σ.
Example 1.22 (no uniqueness in law; Girsanov). Let 0 <α<1/2.
Then, for the SDE
dX
t
= |X
t
|
α
dB
t
,X
0
=0, (1.8)
we have
++−−
.
Proof. Let W be a Brownian motion started at zero on (Ω, G, Q)and

A
t
=

t
0
|W
s
|
−2α
ds, t ≥ 0,
τ
t
=inf{s ≥ 0:A
s
>t},t≥ 0,
Z
t
= W
τ
t
,t≥ 0.
The occupation times formula and Proposition A.6 (ii) ensure that A
t
is a.s.
continuous and finite. It follows from Proposition A.9 that A
t
a.s.
−−−→
t→∞

∞.
Hence, τ is a.s. finite, continuous, and strictly increasing. By Proposi-
tion A.16, Z is a continuous (F
W
τ
t
)-local martingale with Z
t
= τ
t
.Using
Proposition A.18, we can write
τ
t
=

τ
t
0
ds =

τ
t
0
|W
s
|
2α
dA
s

=

A
τ
t
0
|W
τ
s
|
2α
ds =

t
0
|Z
s
|
2α
ds, t ≥ 0.
(We have A
τ
t
= t due to the continuity of A and the property A
t
a.s.
−−−→
t→∞
∞.)
Hence, the process

B
t
=

t
0
|Z
s
|
−α
dZ
s
,t≥ 0
is a continuous (F
W
τ
t
)-local martingale with B
t
= t. According to P. L´evy’s
characterization theorem, B is a (F
W
τ
t
)-Brownian motion. Thus, (Z, B)isa
solution of (1.8).
Now, all the desired statements follow from the fact that (0,B)isanother
solution of (1.8). 
The next example is a SDE with the same characteristic diagram and
with σ ≡ 1.

Example 1.23 (no uniqueness in law; SDE for a Bessel process). For
the SDE
dX
t
=
δ −1
2X
t
I(X
t
=0)dt + dB
t
,X
0
=0 (1.9)
with δ>1, we have
++−−.
16 1 Stochastic Differential Equations
Proof. It follows from Proposition A.21 that there exists a solution (Z, B)
of (1.9) such that Z is positive. By Itˆo’s formula,
Z
2
t
=

t
0
(δ −1)I(Z
s
=0)ds +2


t
0
Z
s
dB
s
+ t
= δt −

t
0
(δ −1)I(Z
s
=0)ds +2

t
0

|Z
2
s
|dB
s
,t≥ 0.
By the occupation times formula,

t
0
I(Z

s
=0)ds =

t
0
I(Z
s
=0)dZ
s
=

R
I(x =0)L
x
t
(Z)dx =0,t≥ 0.
Hence, the pair (Z
2
,B) is a solution of the SDE
dX
t
= δdt +2

|X
t
|dB
t
,X
0
=0.

Propositions 1.6 and 1.12 combined together show that Z
2
is

F
B
t

-adapted.
As Z is positive, Z is also

F
B
t

-adapted, which means that (Z, B)isastrong
solution.
By Proposition 1.5 (i), there exists a measurable map Ψ : C(R
+
) →
C(R
+
) such that the process Ψ(B)is

F
B
t

-adapted and Z =Ψ(B)a.s.For
any t ≥ 0, we have

Ψ
t
(B)=

t
0
δ −1
2Ψ
s
(B)
I

Ψ
s
(B) =0

ds + B
t
a.s.
The process

B = −B is a Brownian motion. Hence, for any t ≥ 0,
−Ψ
t
(−B)=

t
0
δ −1
−2Ψ

s
(−B)
I

−Ψ
s
(−B) =0

ds + B
t
a.s.
Consequently, the pair (

Z,B), where

Z = −Ψ(−B), is a (strong) solution
of (1.9). Obviously, Z is positive, while

Z is negative. Hence, Z and

Z have
a.s. different paths and different laws. This implies that there is no uniqueness
in law and no pathwise uniqueness for (1.9). 
Remark. More information on SDE (1.9) can be found in [5]. In particular,
it is proved in [5] that this equation possesses solutions that are not strong.
Moreover, it is shown that, for the SDE
dX
t
=
δ −1

2X
t
I(X
t
=0)dt + dB
t
,X
0
= x
0
(1.10)
(here the starting point x
0
is arbitrary) with 1 <δ<2, we have ++−−;
for SDE (1.10) with δ ≥ 2, x
0
=0,wehave
++++
. The SDE for a Bessel
process is also considered in Sections 2.2, 3.4.
1.3 Ten Important Examples 17
The following rather surprising example has multidimensional nature.
Example 1.24 (no uniqueness in law; Nadirashvili). Let n ≥ 3.There
exists a function σ : R
n
→ R
n×n
such that
ελ≤σ(x)λ≤Cλ ,x∈ R
n

,λ∈ R
n
with some constants C>0, ε>0 and, for the SDE
dX
i
t
=
n

j=1
σ
ij
(X
t
)dB
j
t
,X
0
= x
0
(i =1, ,n),
we have
+ −−
.
For the proof, see [35] or [40].
We finally present one more example. Its characteristic diagram is different
from all the diagrams that appeared so far.
Example 1.25 (no strong solution and no uniqueness). For the SDE
dX

t
= σ(t, X
t
)dB
t
,X
0
= 0 (1.11)
with
σ(t, x)=

sgn x if t ≤ 1,
I(x =1)sgnx if t>1
(for the precise definition of sgn,see(1.3)), we have
+ −−−
.
Proof. If W is a Brownian motion, then the pair
Z
t
= W
t
,B
t
=

t
0
sgn W
s
dW

s
,t≥ 0 (1.12)
is a solution of (1.11).
Let (Z, B) be the solution given by (1.12). Set τ =inf{t ≥ 1:Z
t
=1},

Z
t
= Z
t∧τ
.Then(

Z,B) is another solution. Thus, there is no uniqueness in
law and no pathwise uniqueness.
If (Z, B) is a solution of (1.12), then
Z
t
=

t
0
sgn Z
s
dB
s
,t≤ 1.
The arguments used in the proof of Example 1.18 show that (Z, B)isnota
strong solution. 