singular stochastic differential equations - cherny a, englebert h

It turns out that the isolated singular points of 44 types do not disturbthe uniqueness of a solution and only the isolated singular points of the remaining 4 types disturb uniqueness..

Lecture Notes in Mathematics 1858Editors:

J. M Morel, Cachan

F Takens, Groningen

B Teissier, Paris

Alexander S Cherny

Department of Probability Theory

Faculty of Mechanics and Mathematics

Moscow State University

Institut f¨ur Stochastik

Fakult¨at f¨ur Mathematik und Informatik

Library of Congress Control Number:2004115716

Mathematics Subject Classification (2000):60-02, 60G17, 60H10, 60J25, 60J60ISSN0075-8434

ISBN3-540-24007-1 Springer Berlin Heidelberg New York

DOI:10.1007/b104187

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We consider one-dimensional homogeneous stochastic differential equations

of the form

dX t = b(X t )dt + σ(X t )dB t , X0= x0, (∗)

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 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

solu-(1 +|b|)/σ2 should be locally integrable on the real line However, both in

theory and in practice one often comes across equations that do not satisfythis 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

A point d ∈ R, at which the function (1 + |b|)/σ2is 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 singularpoint 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 reachthis 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 disturbthe 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 thebranch points is one of the most interesting consequences of the constructedclassification

The monograph also includes an overview of the basic definitions and factsrelated to the stochastic differential equations (different types of existence anduniqueness, 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 theEuropean Community’s Human Potential Programme under contract HPRN-CT-2002-00281

Table of Contents

1.1 General Definitions 5

1.2 Sufficient Conditions for Existence and Uniqueness 9

1.3 Ten Important Examples 12

1.4 Martingale Problems 19

1.5 Solutions up to a Random Time 23

2 One-Sided Classification of Isolated Singular Points 27 2.1 Isolated Singular Points: The Definition 27

2.2 Isolated Singular Points: Examples 32

2.3 One-Sided Classification: The Results 34

2.4 One-Sided Classification: Informal Description 38

2.5 One-Sided Classification: The Proofs 42

3 Two-Sided Classification of Isolated Singular Points 65 3.1 Two-Sided Classification: The Results 65

3.2 Two-Sided Classification: Informal Description 66

3.3 Two-Sided Classification: The Proofs 69

3.4 The Branch Points: Non-Markov Solutions 73

3.5 The Branch Points: Strong Markov Solutions 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 Global Solutions: The Results 86

4.5 Global Solutions: The Proofs 88

5 Several Special Cases 93 5.1 Power Equations: Types of Zero 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

A.1 Local Times 105

A.2 Random Time-Changes 107

A.3 Bessel Processes 108

A.4 Strong Markov Families 110

A.5 Other Facts 111

Appendix B: Some Auxiliary Lemmas 113 B.1 Stopping Times 113

B.2 Measures and Solutions 114

B.3 Other Lemmas 116

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 byFeller (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 anapproach 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 byGikhman [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 existenceand uniqueness were obtained by Girsanov [21], Itˆo [25], Krylov [31], [32],Skorokhod [42], Stroock and Varadhan [44], Zvonkin [49], and others Theevolution of the theory has shown that it is reasonable to introduce dif-ferent types of solutions (weak and strong solutions) and different types ofuniqueness (uniqueness in law and pathwise uniqueness); see Liptser andShiryaev [33], Yamada and Watanabe [48], Zvonkin and Krylov [50] Moreinformation 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 , X0= x0, (1)one of the weakest sufficient conditions for weak existence and uniqueness inlaw 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) = 0 for any x ∈ R and

2 Introduction

Condition (2) is rather weak Nevertheless, SDEs that do not satisfy thiscondition often arise in theory and in practice Such are, for instance, theSDE for a geometric Brownian motion

dX t = µX t dt + σX t dB t , X0= x0(the Black-Scholes model !) and the SDE for a δ-dimensional Bessel process (δ > 1):

dX t=δ − 1

2X t dt + dB t , X0= x0.

In practice, SDEs that do not satisfy (2) arise, for example, in the followingsituation Suppose that we model some process as a solution of (1) Assumethat this process is positive by its nature (for instance, this is the price of astock 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|

σ2 ∈ L / 1

loc(d).

We always assume that σ(x) = 0 for any x ∈ 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, then any solution

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 arethese points indeed “singular”? The answer is given in Section 2.1, where weexplain the qualitative difference between the singular points and the regularpoints 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 practiceone 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 (We call d ∈ R an isolated singular point if d is

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 Markovprocesses

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 ofexistence and uniqueness (see Figure 1.1 on p 8) and cite some classical con-ditions that guarantee existence and uniqueness This chapter also includesseveral 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 thearguments why these points are indeed “singular” Then we study the ex-istence, the uniqueness, and the qualitative behaviour of a solution in theright-hand neighbourhood of an isolated singular point This leads to theone-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 itative behaviour of a solution in the two-sided neighbourhood of an isolatedsingular point We consider the effects brought by the combination of rightand left types Since there exist 7 possible right types and 7 possible lefttypes, there are 49 feasible combinations One of these combinations corre-sponds to a regular point, and therefore, an isolated singular point can haveone of 48 possible types It turns out that the isolated singular points of only

qual-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 bourhood of +∞” This leads to the classification at infinity According to

neigh-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 explodeinto +∞ 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

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 andTable 5.2 on p 103)

The known results from the stochastic calculus used in the proofs are tained in Appendix A, while the auxiliary lemmas are given in Appendix B.The monograph includes 7 figures with simulated paths of solutions ofsingular SDEs

con-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 theexistence and the uniqueness of solutions as well as some general theoremsthat 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 ious combinations of the existence and the uniqueness of solutions Most ofthese examples (but not all) are well known We also find all the possiblecombinations of existence and uniqueness

var-Section 1.4 includes the definition of a martingale problem We also recallthe relationship between the martingale problems and the SDEs

In Section 1.5, we define a solution up to a random time

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 ,Qsuch 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;

(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) A solution (Z, B) is called a strong solution if Z is

F B t

-adapted, whereF B

t is the σ-field generated by σ(B s ; s ≤ t) and by the subsets

of theQ-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 Here we call them simply solutions 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 hasQ{∀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 erties

prop-Proposition 1.5 Let (Z, B) be a strong solution of (1.1).

(i) There exists a measurable map

(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+,Rm ), B−→C(R+,Rn ), Bsuch 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 tions It was proved with no additional assumptions by Cherny [7]

assump-The crucial fact needed to prove Proposition 1.7 is the following result Itshows 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 ability space (see Examples 1.16, 1.17)

prob-It may also happen that on some filtered probability space there exists asolution (or there are even several solutions with the same Brownian motion),while on some other filtered probability space with a Brownian motion thereexists no solution (see Examples 1.18, 1.19, 1.20, and 1.24)

8 1 Stochastic Differential Equations

weak

existence

strongexistence

uniqueness

in law

pathwiseuniqueness

weakexistence

strongexistence

uniqueness

in law

pathwiseuniqueness

strongexistence

uniqueness

in law

pathwiseuniqueness

uniqueness

in law

the bestpossiblesituation



TT





T

TT

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 spacewith a Brownian motion (see Proposition 1.5) However, it may happen inthis case that there are several solutions with the same Brownian motion (seeExamples 1.21–1.23)

If pathwise uniqueness holds for (1.1) and there exists a solution on somefiltered probability space, then on any other filtered probability space with aBrownian 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 uniquenesstogether with weak existence guarantee that the situation is the best possible.Proposition 1.7 shows that uniqueness in law together with strong existenceguarantee 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 ) and σ t (X) = σ(t, X t ), where b :R+× R n → R n and σ :R+× R n →

Rn×m are measurable functions

We begin with sufficient conditions for strong existence and pathwiseuniqueness The first result of this type was obtained by Itˆo

Proposition 1.9 (Itˆo) Suppose that, for a SDE

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 ,

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 , X0= x0, 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 dimensional SDE

Then strong existence and pathwise uniqueness hold.

For the proof, see [15, Th 5.53]

The following proposition guarantees only pathwise uniqueness Its maindifference 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 dimensional SDE

one-dX t = b(t, X t )dt + σ(t, X t )dB t , X0= x0, there exist a constant C > 0 and a strictly increasing function h :R+→ R+with 0+

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

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

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

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 thediagram corresponds to weak existence; the second – to strong existence; thethird – to uniqueness in law; the fourth – to pathwise uniqueness Thus, the

statement “for the SDE , we have + − + − ” 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

Proof Suppose that there exists a solution (Z, B) Then almost all paths

of Z satisfy the integral equation

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

1.3 Ten Important Examples 13

Proof Suppose that (Z, B) is a solution of (1.5) Then

The process Z is a continuous semimartingale with t = t Hence, by the

occupation times formula,

(for the precise definition of sgn, see (1.3)), we have + − + −

Proof Let W be a Brownian motion on (Ω, G, Q) We set

then Z is a continuous ( G t , t = t It follows from

P L´evy’s characterization theorem that Z is a Brownian motion This implies

This implies thatF B

t =F t |Z| (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

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 , X0= x0,

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 , X0= x0,

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 Thenthe SDEs of Examples 1.18–1.20 have no solution on

Ω, G, (G t ),Qwith the

Brownian motion B Indeed, if (Z, B) is a solution, then Z 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 inlaw

Example 1.21 (no uniqueness in law) For the SDE

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 tis notF 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

The occupation times formula and Proposition A.6 (ii) ensure that A tis a.s

continuous and finite It follows from Proposition A.9 that A t −−−→a.s.

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,

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

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 , X0= x0 (1.10)

(here the starting point x0 is arbitrary) with 1 < δ < 2, we have + + − − ;

for SDE (1.10) with δ ≥ 2, x0= 0, we have + + + + 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 :Rn → 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

For the proof, see [35] or [40]

We finally present one more example Its characteristic diagram is differentfrom 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 , X0= 0 (1.11)

with

σ(t, x) = sgn x if t ≤ 1,

I(x = 1) sgn x 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 = Z t∧τ Then ( Z, B) is another solution Thus, there is no uniqueness in

law and no pathwise uniqueness

18 1 Stochastic Differential Equations

Table 1.1 Possible and impossible combinations of existence and uniqueness As

an example, the combination “+− +−” in line 11 corresponds to a SDE, for which

there exists a solution, there exists no strong solution, there is uniqueness in law,and there is no pathwise uniqueness The table shows that such a SDE is provided

by each of Examples 1.18–1.20

Exam-Let us mention one of the applications of the results given above ForSDE (1.1), each of the following properties:

weak existence,

strong existence,

uniqueness in law,

pathwise uniqueness

1.4 Martingale Problems 19

may hold or may not hold Thus, there are 16 (= 24) feasible combinations.Some of these combinations are impossible (for instance, if there is pathwiseuniqueness, then there must be uniqueness in law) For each of these com-binations, Propositions 1.6, 1.7 and Examples 1.16–1.25 allow one either toprovide an example of a corresponding SDE or to prove that this combination

is impossible It turns out that there are only 5 possible combinations (seeTable 1.1)

1.4 Martingale Problems

Let n ∈ N, x0∈ R n and

b : C(R+,Rn)× R+→ R n ,

a : C(R+,Rn)× R+→ R n×n

be predictable functionals Suppose moreover that, for any t ≥ 0 and

ω ∈ C(R+,Rn ), the matrix a t (ω) is positively definite.

Throughout this section, X = (X t ; t ≥ 0) will denote the coordinate process on C(R+,Rn), i.e., the process defined by

X t : C(R+,Rn) ω −→ ω(t) ∈ R n

By (F t ) we will denote the canonical filtration on C(R+), i.e., F t =

σ(X s ; s ≤ t), and F will stand for the σ-fieldt≥0 F t = σ(X s ; s ≥ 0) Note

that F coincides with the Borel σ-field B(C(R+,Rn))

Definition 1.26 A solution of the martingale problem (x0, b, a) is a measure

20 1 Stochastic Differential Equations

Let us now consider SDE (1.1) and set

a t (ω) = σ t (ω)σ ∗ t (ω), t ≥ 0, ω ∈ C(R+,Rn ), where σ ∗ denotes the transpose of the matrix σ Then the martingale problem (x0, b, a) is called a martingale problem corresponding to SDE (1.1) The

relationship between (1.1) and this martingale problem becomes clear fromthe following statement

Theorem 1.27 (i) Let (Z, B) be a solution of (1.1) Then the measure

P = Law(Z t ; t ≥ 0) is a solution of the martingale problem (x0, b, a).

(ii) Let P be a solution of the martingale problem (x0, b, a) Then there exist a filtered probability space

Ω, G, (G t ),Qand a pair of processes (Z, B)

on this space such that (Z, B) is a solution of (1.1) and Law(Z t ; t ≥ 0) = P.

Proof (i) Conditions (a), (b) of Definition 1.26 are obviously satisfied Let

us check condition (c) Set

N t = Z t −

 t

0 b s (Z)ds, t ≥ 0.

(We use here the vector form of notation.) For m ∈ N, we consider the

stopping time S m (N ) = inf {t ≥ 0 : N t ≥ m} Since N is a (G t ,Q)-local

martingale, the stopped process N S m (N) is a (G t ,Q)-martingale Hence, forany 0≤ s < t and C ∈ F s, we have

We extend the processes b, σ, a from Ω1 to Ω and the process W from Ω2

to Ω in the obvious way

For any t ≥ 0, ω ∈ Ω, the matrix σ t (ω) corresponds to a linear operator

Rm → R n Let ϕ t (ω) be the m × m-matrix of the operator of orthogonal

projection onto (ker σ t (ω)) ⊥ , where ker σ t (ω) denotes the kernel of σ t (ω); let

ψ t (ω) be the m × m-matrix of the operator of orthogonal projection onto

ker σ t (ω) Then ϕ = (ϕ t ; t ≥ 0) and ψ = (ψ t ; t ≥ 0) are predictable R m×m

-valued processes For any t ≥ 0, ω ∈ Ω, the restriction of the operator σ t (ω)

1.4 Martingale Problems 21

to (ker σ t (ω)) ⊥ is a bijection from (ker σ t (ω)) ⊥ ⊆ R m onto Im σ t (ω) ⊆ R n,

where Im σ t (ω) denotes the image of σ t (ω) Let us define the operator χ t (ω) :

Rn → R m as follows: χ t (ω) maps Im σ t (ω) onto (ker σ t (ω)) ⊥ as the inverse

of σ t (ω); χ t (ω) vanishes on (Im σ t (ω)) ⊥ Obviously, χ = (χ t ; t ≥ 0) is a

predictableRm×n -valued process We have χ t (ω)σ t (ω) = ϕ t (ω).

Define the process Z as Z t (ω1, ω2) = ω1(t) and the process M as

By the multidimensional version of P L´evy’s characterization theorem

(see [38, Ch IV, Th 3.6]), we deduce that B is a m-dimensional ( G tBrownian motion

)-Set ρ t (ω) = σ t (ω)χ t (ω) Let us consider the process

22 1 Stochastic Differential Equations

Comparing (1.14) with (1.15), we deduce that i − M i  = 0 Hence,

M = x0+ N As a result, the pair (Z, B) is a solution of (1.1)

In this monograph, we will investigate only weak solutions and uniqueness

in law for SDE (1) It will be more convenient for us to consider a solution

of (1) as a solution of the corresponding martingale problem rather than totreat it in the sense of Definition 1.1 The reason is that in this case a solution

is a single object and not a pair of processes as in Definition 1.1 This approach

is justified by Theorem 1.27 Thus, from here on, we will always deal withthe following definition, which is a reformulation of Definition 1.26 for thecase of the SDEs having the form (1)

Definition 1.28 A solution of SDE (1) is a measure P on B(C(R+)) such

Remark If one accepts Definition 1.28, then the existence and uniqueness

of a solution are defined in an obvious way It follows from Theorem 1.27that the existence of a solution in the sense of Definition 1.28 is equivalent

to weak existence (Definition 1.1); the uniqueness of a solution in the sense

of Definition 1.28 is equivalent to uniqueness in law (Definition 1.3)

Definition 1.29 (i) A solutionP of (1) is positive if P{∀t ≥ 0, X t ≥ 0} = 1.

(ii) A solutionP of (1) is strictly positive if P{∀t ≥ 0, X t > 0} = 1.

The negative and strictly negative solutions are defined in a similar way.

1.5 Solutions up to a Random Time 23

1.5 Solutions up to a Random Time

There are several reasons why we consider solutions up to a random time.First, a solution may explode Second, a solution may not be extended after

it reaches some level Third, we can guarantee in some cases that a solutionexists up to the first time it leaves some interval, but we cannot guaranteethe existence of a solution after that time (see Chapter 2)

In order to define a solution up to a random time, we replace the space

C(R+) of continuous functions by the space C(R+) defined below We need

this space to consider exploding solutions Let π be an isolated point added

to the real line

Definition 1.30 The space C(R+) consists of the functions f :R+→ R ∪ {π} with the following property: there exists a time ξ(f) ∈ [0, ∞] such that

f is continuous on [0, ξ(f )) and f = π on [ξ(f ), ∞) The time ξ(f) is called

the killing time of f

Throughout this section, X = (X t ; t ≥ 0) will denote the coordinate process on C(R+), i.e.,

X t : C(R+) ω −→ ω(t) ∈ R ∪ {π},

(F t ) will denote the canonical filtration on C(R+), i.e., F t = σ(X s ; s ≤ t),

andF will stand for the σ-fieldt≥0 F t = σ(X s ; s ≥ 0).

Remark There exists a metric on C(R+) with the following properties

(a) It turns C(R+) into a Polish space

(b) The convergence f n → f in this metric is equivalent to:

(In particular, C(R+) is a closed subspace in this metric.)

(c) The Borel σ-field on C(R+) with respect to this metric coincides with

σ(X t ; t ≥ 0).

In what follows, we will need two different notions: a solution up to S and

a solution up to S −.

Definition 1.31 Let S be a stopping time on C(R+) A solution of (1) up

to S (or a solution defined up to S) is a measure P on F S such that

24 1 Stochastic Differential Equations

In the following, we will often say that (P, S) is a solution of (1)

Remarks (i) The measure P is defined on F S and not onF since otherwise

it would not be unique

(ii) In the usual definition of a local martingale, the probability measure

is defined onF Here P is defined on a smaller σ-field F S However, in view

of the equality M S = M , the knowledge of P only on F S is sufficient to

verify the inclusion M ∈ M c

loc(F t ,P) that arises in (d) In other words, if

P and P are probability measures onF such that P|F S = P |F S =P, then

M ∈ M c

loc(F t ,  P) if and only if M ∈ M c

loc(F t , P) (so we can write simply

Similarly, in order to verify conditions (a), (b), (c), and (e), it is sufficient

to know the values ofP only on F S

Definition 1.32 (i) A solution (P, S) is positive if P{∀t ≤ S, X t ≥ 0} = 1.

(ii) A solution (P, S) is strictly positive if P{∀t ≤ S, X t > 0 } = 1.

The negative and strictly negative solutions are defined in a similar way Recall that a function S : C(R+)→ [0, ∞] is called a predictable stopping time if there exists a sequence (S n) of (F t)-stopping times such that

1.5 Solutions up to a Random Time 25

Definition 1.33 Let S be a predictable stopping time on C(R+) with a

predicting sequence (S n ) A solution of (1) up to S − (or a solution defined

up to S−) is a measure P on F S− such that, for any n ∈ N, the restriction

ofP to F S n is a solution up to S n

In the following, we will often say that (P, S−) is a solution of (1)

Remarks (i) Obviously, this definition does not depend on the choice of a

predicting sequence for S.

(ii) Definition 1.33 implies thatP{∀t < S, X t = π} = 1.

(iii) When dealing with solutions up to S, one may use the space C(R+).

The space C(R+) is essential only for solutions up to S −.

In this monograph, we will use the following terminology: a solution in

the sense of Definition 1.28 will be called a global solution, while a solution in the sense of Definition 1.31 or Definition 1.33 will be called a local solution.

The next statement clarifies the relationship between these two notions

Theorem 1.34 (i) Suppose that ( P, S) is a solution of (1) in the sense

of Definition 1.31 and S = ∞ P-a.s Then P admits a unique extension

P to F Let Q be the measure on C(R+) defined as the restriction of  P to

{ξ = ∞} = C(R+) Then Q is a solution of (1) in the sense of Definition 1.28.

(ii) Let Q be a solution of (1) in the sense of Definition 1.28 Let P be

the measure on C(R+) defined as P(A) = Q(A ∩ {ξ = ∞}) Then (P, ∞) is

a solution of (1) in the sense of Definition 1.31.

Proof (i) The existence and the uniqueness of P follow from Lemma B.5

The latter part of (i) as well as statement (ii) are obvious. 

2 One-Sided Classification

of Isolated Singular Points

In this chapter, we consider SDEs of the form (1)

Section 2.1 deals with the following question: Which points should be called

singular for SDE (1)? This section contains the definition of a singular point

as well as the reasoning that these points are indeed “singular”

Several natural examples of SDEs with isolated singular points are given

in Section 2.2 These examples illustrate how a solution may behave in theneighbourhood of such a point

In Section 2.3 we investigate the behaviour of a solution of (1) in the hand neighbourhood of an isolated singular point We present a completequalitative classification of different types of behaviour

right-Section 2.4 contains an informal description of the constructed tion

classifica-The statements that are formulated in Section 2.3 are proved in tion 2.5

Sec-Throughout this chapter, we assume that σ(x) = 0 for all x ∈ R.

2.1 Isolated Singular Points: The Definition

In this section, except for Proposition 2.2 and Theorem 2.8, we will deal withglobal solutions, i.e., solutions in the sense of Definition 1.28

Throughout the section, except for Proposition 2.2 and Theorem 2.8, X denotes the coordinate process on C(R+) and (F t) stands for the canonical

filtration on C(R+)

Definition 2.1 (i) A measurable function f : R → R is locally integrable at

a point d ∈ R if there exists δ > 0 such that

(ii) A measurable function f is locally integrable on a set D ⊆ R if f is

locally integrable at each point d ∈ D We will use the notation: f ∈ L1

28 2 One-Sided Classification of Isolated Singular Points

Proposition 2.2 (Engelbert, Schmidt) Suppose that, for SDE (1),

1 +|b|

σ2 ∈ L1

Then there exists a unique solution of (1) defined up to S −, where S =

supninf{t ≥ 0 : |X t | = n} and X denotes the coordinate process on C(R+).

For the proof, see [15]

Remark Under the conditions of Proposition 2.2, there need not exist a

global solution because the solution may explode within a finite time rem 4.5 shows, which conditions should be added to (2.1) in order to guar-antee the existence of a global solution

Theo-In Chapter 2, we prove the following local analog of Proposition 2.2 (see

Theorem 2.11) If the function (1 + |b|)/σ2 is locally integrable at a point d,

then there exists a unique solution of (1) “in the neighbourhood of d”

There-fore, it is reasonable to call such a point “regular” for SDE (1)

Definition 2.3 (i) A point d ∈ R is called a singular point for SDE (1) if

1 +|b|

σ2 ∈ L / 1

loc(d).

A point that is not singular will be called regular.

(ii) A point d ∈ R is called an isolated singular point for (1) if d is singular

and there exists a deleted neighbourhood of d that consists of regular points.

The next 5 statements are intended to show that the singular points inthe sense of Definition 2.3 are indeed “singular”

Proposition 2.4 Suppose that |b|/σ2 ∈ L1

loc(R) and 1/σ2∈ L / 1

loc(d) Then

there exists no solution of (1) with X0= d.

For the proof, see [15, Th 4.35]

Theorem 2.5 Let I ⊆ R be an open interval Suppose that |b|/σ2∈ L / 1

loc(x)

for any x ∈ I Then, for any x0∈ I, there exists no solution of (1).

Proof (Cf also [11].) Suppose thatP is a solution By the occupation timesformula and by the definition of a solution, we have

2.1 Isolated Singular Points: The Definition 29

Thus, for the stopping time S = 1 ∧ inf{t ≥ 0 : X t ∈ I}, one has /

(Here we used the fact that L x

S (X) = 0 for x / ∈ I; see Proposition A.5.) This

Remark The above statement shows that a solution cannot enter an open

interval that consists of singular points

Theorem 2.6 Suppose that d is a singular point for (1) and P is a solution

of (1) Then, for any t ≥ 0, we have

If (2.3) is satisfied, then (2.2), together with the right-continuity of L y t (X)

in y, ensures that, for any t ≥ 0, L d

t (X) = 0P-a.s If (2.4) is satisfied, then,

30 2 One-Sided Classification of Isolated Singular Points

Theorem 2.7 Let d be a regular point for (1) and P be a solution of (1).

Suppose moreover that P{T d < ∞} > 0, where T d = inf{t ≥ 0 : X t = d } Then, for any t ≥ 0, on the set {t > T d } we have

L d t (X) > 0, L d− t (X) > 0 P-a.s

This statement is proved in Section 2.5

Theorems 2.6 and 2.7 may be informally described as follows Singularpoints for (1) are those and only those points, at which the local time of asolution vanishes

Consider now SDE (1) with x0 = 0 If the conditions of Proposition 2.2are satisfied, then the behaviour of a solution is regular in the following sense:

– there exists a solution up to S −;

de-Theorem 2.8 Suppose that

1 There exists no solution up to S −.

2 There exists a unique solution up to S −, and it is positive.

3 There exists a unique solution up to S −, and it is negative.

4 There exist a positive solution as well as a negative solution up to S − (In this case alternating solutions may also exist.)

2.1 Isolated Singular Points: The Definition 31

AAA

AAA

AA

Fig 2.1 Qualitative difference between the regular points and the singular points.

The top graph shows the “typical” behaviour of a solution in the neighbourhood

of a regular point The other 4 graphs illustrate 4 possible types of behaviour of asolution in the neighbourhood of a singular point As an example, the sign “ ”

in the bottom left-hand graph indicates that there is no positive solution The sign

“ ? ” in the bottom right-hand graph indicates that an alternating solution mayexist or may not exist

32 2 One-Sided Classification of Isolated Singular Points

2.2 Isolated Singular Points: Examples

A SDE with an isolated singular point is provided by Example 1.17 For thisSDE, there is no solution

Another SDE with an isolated singular point is the SDE for a Besselprocess, which has been considered as Example 1.23 Here we will study it

for all starting points x0

As in the previous section, we deal here with global solutions

Example 2.9 (SDE for a Bessel process) Let us consider the SDE

(ii) If x0= 0 or 1 < δ < 2, then this equation possesses different solutions.

For any solution P, we have P{∃t ≥ 0 : X t= 0} = 1.

Proof (i) Let P be the distribution of a δ-dimensional Bessel process started

at x0 Proposition A.21 (combined with Theorem 1.27) shows that P is asolution of SDE (2.6) Suppose that there exists another solutionP Set

Law(|X t |; t ≥ 0 | P ) = Law(|X t |; t ≥ 0 | P). (2.8)Proposition A.20 (i) guarantees that P{∀t ≥ 0, X t > 0 } = 1 This, together

with (2.8), implies that P {∀t ≥ 0, X t = 0} = 1 Since the paths of X are

continuous and P {X0 = x0 > 0 } = 1, we get P  {∀t ≥ 0, X t > 0 } = 1.

Using (2.8) once again, we obtainP=P

(ii) We first suppose that x0 = 0 Let P be defined as above and P be

the image ofP under the map

C(R+) ω −→ −ω ∈ C(R+).

It is easy to verify thatP is also a solution of (2.6) The solutionsP and P

are different since

P{∀t ≥ 0, X t ≥ 0} = 1, P  {∀t ≥ 0, X t ≤ 0} = 1.

2.2 Isolated Singular Points: Examples 33

(Moreover, for any α ∈ (0, 1), the measure P α = α P + (1 − α)P  is also a

Now, let P be an arbitrary solution of (2.6) Let us prove that P{∃t ≥

0 : X t = 0} = 1 For x0 = 0, this is clear Assume now that x0 > 0, so

that 1 < δ < 2 The measure Q = Law(X2

t ; t ≥ 0 | P) is a solution of (2.7).

As there is weak uniqueness for (2.7), Q is the distribution of the square

of a δ-dimensional Bessel process started at x20 By Proposition A.20 (ii),

Q{∃t > 0 : X t= 0} = 1, which yields P{∃t > 0 : X t= 0} = 1

Example 2.10 (SDE for a geometric Brownian motion) Let us

con-sider the SDE

dX t = µX t dt + (X t + ηI(X t = 0))dB t , X0= x0 (2.9)

with µ ∈ R, η = 0.

(i) If x0> 0, then there exists a unique solution P of this equation It is

strictly positive If µ > 1/2, then P{lim t→∞ X t=∞} = 1; if µ < 1/2, then

P{lim t→∞ X t= 0} = 1.

(ii) If x0= 0, then there exists no solution.

Remark The term ηI(X t = 0) is added in order to guarantee that σ = 0 at

each point The choice of η = 0 does not influence the properties of (2.9) as

seen from the reasoning given below

Proof of Example 2.10 If P is a solution of (2.9), then, for any t ≥ 0,

dX t = µX t dt + X t dB t , X0= x0. (2.10)Propositions 1.6 and 1.9 combined together show that there is uniqueness inlaw for (2.10) Applying Itˆo’s formula and Theorem 1.27, we deduce that thesolution of (2.10) is given by