Lectures on One-dimensional Stochastic
Differential Equations (Part I)
Hans-Jürgen Engelbert
Friedrich Schiller University, Jena, Germany
Research School
CIMPA-UNESCO-MESR-MINECO-MOROCCO
Statistical Methods and Applications
in Actuarial Science and Finance
Marrakesh (8–13 April 2013) and El Kelaa (15–20 April 2013)
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Overview
1. Introduction and Basic Definitions
Semimartingales
Local Time and Tanaka’s Formula
The Occupation Times Formula
The Generalized Itô Formula
SDEs with Generalized Drift
SDEs with Ordinary Drift
Some Examples
SDEs without Drift
2. Integral Functionals of Brownian Motion
Basic Results
The Proof
A Second Theorem
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Overview
3. SDEs without Drift
Formulation of the Problem
First Properties
Existence of Solutions
Some Discussion
Example
Uniqueness in Law
4. SDEs with Generalized Drift
Space Transformation
Existence of Solutions
Uniqueness of Solutions
5. Pathwise Uniqueness and Strong Solutions
Generalized Hölder Condition
The Proof
Generalized Nakao Condition
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
1. Introduction and Basic Definitions
Our aim is to study solutions of one-dimensional stochastic
differential equations (SDEs ). In general, they admit explosions.
Therefore the suitable state space is the extended real line:
R = R ∪ {−∞, +∞} = [−∞, +∞]
equipped with the σ-algebra B R of Borel subsets.
Let (Ω, F, P; F) be a filtered probability space satisfying the
usual conditions. We consider a stochastic process
(X , F)
defined on (Ω, F, P) taking values in R, B R . It is assumed
that X0 is real-valued.
The adequate class of processes (X , F) is that of
semimartingales up to an F-stopping time S which will be
introduced now.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Preliminaries: Semimartingales
Definition (Semimartingales up to S)
Let S be an F-stopping time. (X , F) is called a semimartingale
(resp., local martingale) up to S if there exists an increasing
sequence (Sn ) of F-stopping times such that S = limn→∞ Sn
and the stopped processes X Sn , F are (real-valued)
semimartingales (resp., local martingales).
If (X , F) is a semimartingale up to S then there exists a
decomposition
(1)
Xt = X0 + Mt + Vt
for
t <S
P-a.s.
where (M, F) is a local martingale up to S, M0 = 0 and (V , F)
is right-continuous process with bounded variation on [0, t]
whenever t < S, V0 .
If X is continuous on [0, S] then there is a decomposition (1)
such that M and V are continuous on [0, S] and this
decomposition is unique on [0, S].
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Semimartingales: The Tanaka Formula
We recall some important facts for continuous semimartingales
(see Revuz–Yor (1992)), their extension to continuous
semimartingales up to time S is straightforward.
Local Time and Tanaka’s Formula
The (right) local time LX up to time S is a function of [0, S) × R
into [0, +∞] that the TANAKA F ORMULA holds:
Z t
(2) Xt − y = X0 − y +
sgn (Xu − y ) dXu +LX (t, y ) ∀t < S
|0
{z
}
Itô-Integral
where for every x ∈ R
sgn (x) =


1 if x > 0 ,
 −1 if x ≤ 0 .
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Semimartingales: Local Time
We recall that the right local time LX (t, y ) can always be
chosen such that it is F-adapted, continuous in t < S and
càdlàg in y ∈ R. Unless otherwise stated, LX (t, y ) always
denotes the right local time. The jump of the local time can be
calculated as follows (see Revuz–Yor (1992)):
Z t
X
X
(3)
L (t, y ) − L (t, y −) = 2
1{y } (Xu ) dVu .
0
In particular, if X is a continuous local martingale, then V = 0
and consequently LX (t, y ) is continuous in y , too.
The Occupation Times Formula
If (X , F) is a continuous semimartingale up to S then by
hX , X i := hM, Mi we denote denote the associated continuous
increasing process defined on [0, S), i.e., the unique
continuous increasing process such that M 2 − hM, Mi, F is a
continuous local martingale up to S.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Semimartingales: Local Time
We then have the following so-called occupation times formula:
For every bounded or nonnegative measurable function g on R,
Z
t
Z
g(Xu ) dhX , X iu =
(4)
0
g (y ))LX (t, y ) dy , ∀t < S, P-a.s.
R
Note that the exceptional set does not depend on t and g.
The following important representation formula is an immediate
consequence of the occupation times formula:
(5)
1
L (t, y ) = lim
ε↓0 ε
X
Z
t
1[y ,y +ε] (Xu ) dhX , X iu
P-a.s.
0
The Generalized Itô Formula
For every convex real function F , the generalized Itô formula
asserts:
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Semimartingales: The Generalized Itô Formula
Z
(6) F (Xt ) = F (X0 ) +
0
t
1
D− F (Xu ) dXu +
2
Z
LX (t, y ) n (dy )
R
for t < S P-a.s. where D− F denotes the left derivative of F
and the measure n on (R, B (R)) is the second derivative of F
(in the sense of distributions).
If F is twice continuously differentiable with bounded second
derivative g then the measure n in (6) has the form
dn(y ) = g(y )dy . Comparing (6) with the classical Itô formula
Z t
Z
1 t
(7) F (Xt ) = F (X0 ) +
D− F (Xu ) dXu +
g (Xu ) dhX , X iu
2 0
0
implies the occupation times formula (4) for bounded and
continuous g. An application of the monotone class theorem
extends this formula to general g.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
SDEs with Generalized Drift
In the following we
consider continuous processe (X , F) with
values in R, B R . We introduce
(8)
S∞ := inf {u ≥ 0 : Xu ∈
/ R} .
The F-stopping time S∞ is called the explosion time of X .
Let b : R 7→SR be a measurable function and ν be a set
function on ∞
N=1 B ([−N, N]). We assume that ν is a finite
signed measure on B ([−N, N]), for every N ≥ 1. We shall
consider the SDE with generalized drift
Z
(9)
Xt = X0 +
X
Z
L (t, y ) ν (dy ) +
R
t
b (Xu ) dBu
0
where B is a Brownian motion and LX denotes the local time of
the unknown process X .
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
SDEs with Generalized Drift
Definition (Solution of SDEs with GD)
(1) A continuous R, B R -valued process X defined on OFP
with filtration F is called a solution of Eq. (9) if:
X0 is real-valued.
Xt = Xt∧S∞ for all t ≥ 0.
(X , F) is a semimartingale up to S∞ .
There exists a Brownian motion (B, F) with B0 = X0 such
that Eq. (9) holds for all t < S P-a.s..
(2) (X , F) is called a strong solution if X , FB is a solution of
Eq. (9)
Here FB denotes the filtration generated by B.
The definition includes that both integrals appearing in Eq. (9)
exist and are finite on {t < S}.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
SDEs with Generalized Drift
SDEs with generalized drift have been considered by many
authors, here are a few:
Harrison & Shepp (1981)
Portenko (1982)
Stroock & Yor (1981)
Le Gall (1982, 1983, 1984)
Groh (1984)
Perkins (1982)
E. & Schmidt (1985, 1991)
Bass & Chen (2005)
Blei (2010, 2012)
Blei & Engelbert (2012a, 2012b)
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
SDEs with Ordinary Drift
SDEs of type (9) generalize stochastic equations with “ordinary"
a
drift: Let the measure ν be given by the density 2 where a
b
measurable real function and b the diffusion coefficient as
1
before. Here 2 (y ) = +∞ if b(y ) = 0, and 0 · (+∞) = 0.
b
Suppose that (X , F) is a solution of Eq. (9) such that
{b(X ) = 0} ⊆ {a(X ) = 0}
λ+ ⊗ P-a.e.
where λ+ denotes
measure on R+ . In particular,
the Lebesgue
this is satisfied if b2 (X ) > 0 λ+ ⊗ P-a.e. or if only the
inclusion {b = 0} ⊆ {a = 0} holds. Using
Z
hX , X it =
t
b2 (Xu ) du
for t < S∞ ,
0
we can calculate
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
SDEs with Ordinary Drift
Z
Xt
= X0 +
t
Z
X
L (t, y ) ν (dy ) +
b (Xu ) dBu
Z t
Z
a
X
b (Xu ) dBu
= X0 +
L (t, y ) 2 (y ) dy +
b
0
R
0
R
Z
= X0 +
0
t
b2 (Xu )du
z }| { Z t
a
(Xu ) dhX , X iu +
b (Xu ) dBu
b2
0
and therefore
Z
(10)
Xt = X0 +
t
Z
a (Xu ) du +
0
t
b (Xu ) dBu
0
a stochastic equation with “ordinary" drift a. Now, under the
above condition, it is clear that the converse also holds: Any
solution of Eq. (10) is a solution of Eq. (9) with generalized drift
for the measure dν (y ) = ba2 (y )dy .
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Some Examples
Example (Reflecting Brownian Motion)
We set b = 1, ν = 21 δ0 and assume X0 = 0. Then Eq. (9) looks
like
(11)
Xt =
1 X
L (t, 0) + Bt
2
For constructing a solution, consider a Brownian motion (W , F)
and write the Tanaka formula for it:
Bt
Xt
1 X
L (t,0)
zZ
}|
{
2
t
z W }| {
:= Wt = L (t, 0) +
sgn (Wu ) dWu
0
=
1 X
L (t, 0) +Bt ,
|2 {z }
sup0≥u≥t (−Bu )
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Some Examples
hence (X , F) is a solution which is a reflecting Brownian motion.
The process X is a solution of Skorohod’s reflecting problem
which is pathwise unique and has the following representation:
Xt = sup (−Bu ) + Bt
∀t ≥ 0 .
0≥u≥t
Hence X is the pathwise unique strong solution of SDE (11).
Definition (Pathwise Uniqueness)
The solution of Eq.
(9) is 2called
pathwise unique if for any two
1
solutions X , F and X , F with the same initial value defined
on the same probability space and with the same Brownian
motion (B, F) it follows X 1 = X 2 P-a.s..
For other SDEs , pathwise uniqueness of the solution is defined
in the same way.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Some Examples
Example (Skew Brownian Motion)
Now we consider the case b = 1, ν = αδ0 with α <
simplicity) X0 = 0. Then Eq. (9) looks like
(12)
1
2
and (for
Xt = αLX (t, 0) + Bt
There also exists a pathwise unique strong solution of this SDE
called skew Brownian motion with skewness parameter α.
Harrison and Shepp used symmetric local time:
Xt = β L̂X (t, 0) + Bt
with β ∈ (−1, 1) and there is a one-to-one correspondence
between α and β. Informally, the skew Brownian motion
behaves like a Brownian motion outside of zero but whenever X
1
is in zero it moves up with probability p = 2(1−α)
= 1β
2 and
down with 1 − p. We later return to this example.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
SDEs without Drift
The general solution of Eq. (9) with generalized drift will be
constructed by space transformation from a solution (Y , F) of
an SDE without drift:
Z t
(13)
Yt = Y0 +
σ (Yu ) dBu
0
where σ is a measurable real function and (B, F) a Wiener
process. This is a special case of Eq. (9) with generalized drift,
namely, for ν = 0 and b = σ. The notion of a (strong) solution
and of pathwise uniqueness of a solution is therefore
introduced in the same way as before.
One of our first goals is to study in detail Eq. (13) without drift.
Before, as an important tool, we will investigate the behaviour
of integral functionals of Brownian motion.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
2. Integral Functionals of Brownian Motion
In this section we consider a Brownian motion (W , F) and a
nonegative measurable real function f with values in [0, +∞].
We introduce the integral functional
Z
(14)
t+
Tt :=
f (Bu ) du ,
t ∈ [0, +∞] ,
0
and look for conditions for convergence and divergence of Tt .
Define
E :=
Z
x ∈R:
f (y ) dy = +∞, ∀ open G 3 x
G
which is a closed subset of R and let
DE = DE (W ) := inf {t ≥ 0 : Wt ∈ E}
the first entry time if W into E.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Basic Results
Theorem
Suppose that λ ({f > 0}) > 0 (λ Lebesgue measure on R).
Rt
P-a.s.
0 f (Wu ) du < +∞ ∀ t < DE
R DE +
f (Wu ) du = +∞ P-a.s.
0
Corollary (0–1 Law)
The following conditions are equivalent:
nR
o
t
(i) P
f
(W
)
du
<
+∞,
∀t
≥
0
= 1.
u
0
nR
o
t
(i) P
> 0.
0 f (Wu ) du < +∞, ∀t ≥ 0
(i) f is locally integrable, i.e., E = ∅.
Corollary
If λ ({f > 0}) > 0 then
R∞
0
f (Wu ) du = +∞
Hans-Jürgen Engelbert, FSU Jena, Germany
P-a.s.
Lectures on One-dimensional SDEs
The Proof
Proof. We recall the occupation times formula
Z
Z t
f (Wu ) du =
f (y ) LW (t, y ) dy .
0
R
and collect some properties of the local time LW (t, y ).
LW (t, y ) is continuous in t and y .
For any strictly positive random time τ , LW (τ, 0) > 0 P-a.s.
LW (∞, 0) := limt→∞ LW (t, 0) = +∞ P-a.s.
The first property holds because W is a continuous martingale.
For the second and third properties we recall the Tanaka
formula together with the the uniqueness of the solution of
Skorohod’s problem:
Bt
sup0≥u≥t (−Bu )
Wt =
z
}|
{
Z
z }| {
t
LW (t, 0) +
sgn (Wu ) dWu .
0
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
The Proof
The fourth property we need is
LW (t, y ) = 0 for y ∈
/ [mt , Mt ] P-a.s.
with the notation mt = inf0≤u≤t Wu and Mt = sup0≤u≤t Wu .
Proof of (i): For t < DE (ω) we have [mt (ω) , Mt (ω)] ⊆ R \ E
and is compact, hence f is integrable over [mt (ω) , Mt (ω)] and
consequently, using the fourth property above,
Z
t
Z
f (Wu ) du =
0
f (y ) LW (t, y ) dy
R
Z
=
≤
f (y ) LW (t, y ) dy
[mt (ω),Mt (ω)]
Z
W
max
L (t, x)
x∈[mt (ω),Mt (ω)]
Hans-Jürgen Engelbert, FSU Jena, Germany
f (y ) dy < +∞ .
[mt (ω),Mt (ω)]
Lectures on One-dimensional SDEs
Proof
Proof of (ii): 1) First we have to consider the case E = ∅. Then
DE = +∞ and we observe
Z ∞
Z t
f (Wu ) du = lim
f (Wu ) du
t→∞ 0
0
Z
= lim
f (y ) LW (t, y ) dy
t→∞ R
Z
=
f (y ) LW (∞, y ) dy = +∞
R
because of the third property.
2) We consider the case E 6= ∅. Then DE < +∞ P-a.s. and we
f = WD +t − WD
see that W
is a Brownian motion. WDE
E
E t≥0
takes at most two values x1 ≤ 0 ≤ x2 from E. (If 0 ∈ E then
f = W .) On the set WD = xi we can
DE = 0 P-a.s. and W
E
estimate
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Proof
DE +t
Z
f (Wu ) du
0
Z
DE +t
≥
f WDE +u du
f (Wu ) du =
DE
t
Z
0
fu
f xi + W
=
0
≥
t
Z
Z
du =
Z +ε
f (xi + y ) LW (t, y ) dy
f
R
min LW (t, x)
f (xi + y ) dy = +∞ .
x∈[−ε,ε]
{z
}
|
{z
} | −ε
f
=+∞
>0
Here we have chosen ε such that minx∈[−ε,ε] LW (t, x) > 0
f
which is possible since LW (t, x) is continuous in x and
f
LW (t, y ) t0 > 0 P-a.s. by the second property.
f
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
A Second Theorem
We conclude this section with another important result on the
behaviour of integral functionals of Brownian motion. Let (W , F)
be a Brownian motion and a > 0. We introduce
Da (W ) = inf {t ≥ 0 : Wt = a} .
We are interested in the behaviour of the integral functional
Z Da (W )
TDa (W ) − =
f (Wu ) du .
0
Theorem
Suppose that f is locally integrable, i.e., E = ∅. Then the
following conditions are equivalent:
(i) P TDa (W ) − < +∞ = 1.
(ii) P TDa (W ) − < +∞ > 0.
Ra
(iii) 0 (a − y ) f (y ) dy < +∞.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
3. SDEs without Drift
Now we start with the investigation of SDEs without drift
Z
(15)
Yt = y0 +
t
σ (Yu ) dBu
0
where y0 ∈ R, σ is a measurable real function and (B, F) a
Wiener process. Recall that a solution (Y , F) is a continuous
local martingale up to S∞ = S∞ (Y ) such that there exists a
Brownian motion (B, F) satisfying (15) for all t < S∞ (Y ). This
Rt
includes that 0 σ 2 (Yu ) du < +∞ , t < S∞ (Y ) , P-a.s. which
ensures the existence of the stochastic integral in (15).
By the definition of a solution, limt↑S∞ Yt ∈ {−∞, +∞} but this
is not the behaviour of a continuous local martingale up to
S∞ (Y ) which is oscillating. The formal proof is skipped.
Therefore, S∞ (Y ) = +∞ P-a.s. and in the following we are
always looking for nonexploding solutions (15).
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
SDEs without Drift
We continue with a useful lemma.
Lemma
Let (M, F) be a continuous local martingale with M0 = 0 and
LM (t, 0) = 0 for all t ≥ 0 P-a.s. Then M ≡ 0 P-a.s.
Proof. Using the Tanaka formula
Z
|Mt | = |M0 | +
|{z}
=0
0
t
sgn (Mu ) dMu + LM (t, 0)
| {z }
=0
it follows that (|M|, F) is a nonnegative continuous local
martingale and hence a nonnegative supermartingale starting
at 0. Therefore E [|Mt |] = 0 which implies Mt = 0 P-a.s.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
First Properties
We now introduce the following sets:
N = Nσ := {x ∈ R : σ (x) = 0}
and
E = Eb :=
Z
x ∈R:
σ
−2
(y ) dy = ∞ ∀ open G 3 x
.
G
They will play a basic role in the following. Next we collect
some important properties of solutions (Y , F).
Proposition
Let (Y , F) be an arbitrary solution of Eq. (15). We then have:
LY (t, x) = 0 ∀ (t, x) ∈ [0, +∞] × E P-a.s.
hY , Y it = hY , Y it∧DE (Y ) ∀ t P-a.s.
Yt = Yt∧DE (Y ) ∀ t P-a.s.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Proof of Proposition
Proof. (i) Suppose that x ∈ E and define
1
1
1
Y
.
An := L (t, y ) > , ∀ y ∈ Gn , Gn = x − , x +
n
n
n
On An , we then obtain
Z
Z t
Z
1
−2
−2
Y
σ (y ) dy ≤ σ (y ) L (t, y ) dy =
σ −2 (Yu ) dhY , Y iu
n Gn
R
0
{z
}
|
+∞
dhY ,Y i
u
Z t
z }| { Z t
1N c (Yu ) ≤ t
= σ −2 (Yu ) σ 2 (Yu ) du =
0
0
which
({An }) = 0. From this follows
Y implies P P L (t, x) > 0 = 0 and (i) is verified.
(ii) immediately follows from (iii).
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Proof of Proposition
(iii) We define the continuous local martingale (Z , F) by
Z = Y − Y DE , Y DE is the stopped process in DE = DE (Y ). The
claim is that Z ≡ 0 P-a.s. Obviously, Z0 = 0 and in view of the
lemma above it is sufficient to verfify that LZ (t, 0) = 0 P-a.s.
Note that hZ , Z i = hY , Y i − hY , Y iDE . We can calculate P-a.s.
Z
1 t
L (t, 0) = lim
1[0,ε] (Zu ) dhZ , Z iu
ε↓0 ε 0
Z
1 t
1[0,ε] Yu − YDE dhY , Y iu
= lim
ε↓0 ε t∧DE
Z
1 t
= lim
1
(Yu ) dhY , Y iu
ε↓0 ε t∧DE [YDE ,YDE +ε]
= LY t, YDE − LY t ∧ DE , YDE = 0
because LY t, YDE = LY t ∧ DE , YDE = 0 on DE ≤ t in view
of YDE ∈ E on DE < +∞ and statement (i) of the Proposition. Z
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Existence of Solutions
Before we pass on to the existence of solutions we introduce
distinguished solutions (Y , F) which will be called fundamental
solutions. Recall that N = {x ∈ R : σ (x) = 0}.
Definition
A solution (Y , F) of Eq. (15) is called fundamental solution if
(16)
σ 2 (Yt (ω)) > 0,
0 ≤ t < DE (Y ) (ω) ,
λ × P-a.e.
Remark
If σ (x) 6= 0 identically then every solution (Y , F) of SDE (15) is
a fundamental solution.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Existence of Solutions
Theorem (Existence of Solutions)
The following conditions are equivalent:
(i) For any initial value x0 ∈ R, there exists a fundamental
solution (Y , F) of SDE (15) without drift.
(ii) For any initial value x0 ∈ R, there exists a solution (Y , F) of
SDE (15) without drift.
(iii) Eb ⊆ N.
Proof. (i)⇒(ii) is trivial.
(ii)⇒(iii) Suppose that y0 ∈ E. Let (Y , F) be a solution with
initial value y0 . Then DE = 0 and by part (iii) of the Proposition
we obtain Yt = Yt∧DE = Y0 = y0 . This yields
Z t
Z t
2
0 = hY , Y it =
σ (Yu ) du =
σ 2 (y0 ) du ,
0
0
hence σ 2 (y0 ) = 0 and therefore y0 ∈ N, proving E ⊆ N.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Proof of Theorem
(iii)⇒(i) 1) Our aim is to construct a solution (Y , F) with given
initial value y0 by time change. Replacing σ by σ (y0 + ·)
without loss of generality we can assume y0 = 0. We start with
a Brownian motion (W , G) and define
Z t+
(17)
Tt =
σ −2 (Wu ) du , t ≥ 0 ,
0
where
(18)
1
0
= +∞. Let
At = inf {s ≥ 0 : Ts > t} ,
t ≥ 0,
be the right inverse of (Tt ). Then we have:
(At ) is a G-time change.
T∞ = +∞ P-a.s.
At is finite P-a.s. and continuous.
We now introduce the time-changed process (Y , F) by
Y = W ◦ A,
F=G◦A
which is a continuous local martingale with hY , Y i = A.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Proof of Theorem
2) We compute A = hY , Y i. Note that we have λ (E c ∩ N) = 0.
Using Fubini’s theorem we get
W ∈ E ∪ N c λ ⊗ P-a.e.
Together with A∞ = DE (W ) this yields
σ 2 (Wu ) > 0, u < A∞ , λ ⊗ P-a.e.
The equality A∞ = DE (W ) also yields WA∞ ∈ E on
{A∞ < +∞} and because of the assumption E ⊆ N
σ 2 (WA∞ ) = 0 on {A∞ < +∞} .
Using these properties and the definition of (Tt ) it follows
Z At
Z TA
t
2
At =
σ (Wu ) dTu =
σ 2 (WAu ) du
0
Z
=
0
TAt
σ 2 (WAu ) du
0
 R
 t σ 2 (Yu ) du if At < A∞
0
=
 R ∞ σ 2 (Y ) du if A = A
0
u
Hans-Jürgen Engelbert, FSU Jena, Germany
t
∞
Lectures on One-dimensional SDEs
Proof of Theorem
Using σ 2 (Yu ) = σ 2 (WA∞ ) = 0 if t ≤ u and At = A∞ yields
Z t
σ 2 (Yu ) du.
At =
0
3) For proving that (Y , F) is a solution, we take a Brownian
motion (B 0 , F) and define the continuous local martingale
Z t
Z t
−1
σ (Yu ) 1{σ6=0} (Yu ) dYu +
1{σ=0} (Yu ) dBu0
Bt =
0
0
with hB, Bit = t for all t, hence (B, F) is a Brownian motion and
Z t
σ (Yu ) dBu
0
Z t
Z t
=
σ (Yu ) 1{σ6=0} dBu =
σ (Yu ) 1{σ6=0} σ −1 (Yu ) dYu
0
0
Z t
Z t
Z t
1{σ6=0} dYu =
1{σ6=0} dYu +
1{σ=0} dYu
=
0
0
0
= Yt − Y0 = Yt ,
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Discussion
so (Y , F) is a solution. By construction we have
σ 2 (Yu ) = σ 2 (WAu ) > 0
if
Au < A∞ = DE (W )
or, equivalently, σ 2 (Yu ) > 0 if u < DE (Y ). Hence (Y , F) is a
fundamental solution.
Corollary
If σ −2 is locally integrable then there is a (fundamental) solution
(Y , F) for every initial condition Y0 = y0 .
Proof. σ −2 is locally integrable iff E = Eb = ∅.
Definition
A solution (Y , F) is called trivial if Y = Y0 P-a.s.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Discussion
Corollary
If (Y , F) is a solution with Y0 = y0 ∈ E then (Y , F) is trivial.
Proof. This follows from the Proposition (Part (iii)) since
DE (Y ) = 0 in this case.
Corollary
For any initial value y0 , there exists a non-trivial solution (Y , F)
if and only if σ −2 is locally integrable.
Proof. Because every fundamental solution starting from
y0 ∈ E c is nontrivial the statement immediately follows from the
first two corollaries.
Corollary
In the following cases, the existence condition E ⊆ N is
satisfied: (i) σ is continuous; (ii) σ has right and left limits and
if σ (y ) 6= 0 then σ (y +) 6= 0 and σ (y −) 6= 0.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Example
Trivial solutions cannot be excluded, they naturally arise, e.g.,
for Lipschitz diffusion coefficients σ.
Example
Suppose that σ is locally Lipschitz, for instance, σ (y ) = y or
σ (y ) = sin y . Obviously, N c ⊆ E c (as for every continuous
function) and hence E ⊆ N. But the converse is also true: If
σ (y0 ) = 0 then |σ (y ) − σ (y0 ) | ≤ K |y − y0 | for some constant
K > 0 and for y ∈ G, with some open G 3 y0 . Hence
σ −2 (y ) ≥ K |y − y0 |2 and we get y0 ∈ E.
Thus E = N and for all local Lipschitz functions σ, every
solution (Y , F) starting from an initial value y0 with σ (y0 ) = 0
must be trivial.
This observation extends to functions σ which are locally
Hölder continuous with exponent α ≥ 12 .
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Uniqueness in Law
Definition
We say that the solution to Eq. (15) is unique
(in law) if
1
1
whenever there are two solutions Y , F and Y 2 , F2 with
the same initial value, possibly, on different probability spaces
then the distributions of Y 1 and Y 2 coincide.
Theorem (Uniqueness of Fundamental Solution)
The fundamental solution (Y , F) of Eq. (15) without drift is
unique in law.
Theorem (Existence and Uniqueness)
For every initial value y0 ∈ R there exists a solution (Y , F) to
Eq. (15) without drift and uniqueness in law holds if and only if
the condition E = N is satisfied:
σ (y ) 6= 0 iff σ −2 is integrable in a neighbourhood of y .
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Proof of Uniqueness in Law
Proof of Theorem (on Existence and Uniqueness) 1) If E = N
then, for any initial condition y0 ∈ R, there exists a solution
(Y , F) and it must be a fundamental solution (there are no zeros
of σ outside of E). By the first theorem, the solution is unique.
2) If for any initial condition there exists a (fundamental)
solution, we have E ⊆ N by the existence theorem. If E 6= N we
can choose y0 ∈ N ∩ E c and a fundamental solution (Y , F)
starting from y0 . On the other side, there is the trivial solution
Y t ≡ y0 . This contradicts the uniqueness assumption and
hence E = N.
Before we are going to proof the uniqueness of fundamental
solutions, we first state a lemma. Let (Y , F) be an arbitrary
fundamental solution of (13), A0 = hY , Y i and T 0 its right
inverse:
Tt0 = inf s ≥ 0 : A0s > t , t ≥ 0 .
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Proof of Uniqueness in Law
Let W be a Dambis–Dubins–Schwarz Brownian motion for Y
which is defined, in general, on an enlarged probability space:
Wt∧A0∞ = YTt0 ,
Yt = WA0t ,
t ≥ 0.
Lemma
Tt0
Z
=
t+
σ −2 (Wu ) du
∀t ≥0
P-a.s.
0
Proof. Without loss of generality we assume that y0 = 0. By
time change in the Lebesgue–Stieltjes integral we obtain
Z t∧A0∞
Z A0 0
T
t
−2
σ (Wu ) du =
σ −2 (Wu ) du
0
0
Z
Tt0
=
0
Z
(19)
=
0
Hans-Jürgen Engelbert, FSU Jena, Germany
Tt0
σ −2 (Yu ) dA0u
σ −2 (Yu ) σ 2 (Yu ) du = Tt0 .
Lectures on One-dimensional SDEs
Proof of Uniqueness in Law
Since (Y , F) is a fundamental solution, we observe that
Z t∧A0∞
Z A0 0
T
t
1{σ2 (Wu )=0} du =
1{σ2 (Wu )=0} du
0
0
Z
Tt0
=
0
Z
Tt0
=
0
Z
=
0
Tt0
1{σ2 (Yu )=0} dA0u
1{σ2 (Yu )=0} dA0u
1{σ2 (Yu )=0} σ 2 (Yu ) du = 0
{Tt0
on
< DE (Y )}. This yields that the right hand side of (19) is
equal to Tt0 and hence
Z t∧A0∞
Tt0 =
σ −2 (Wu ) du
0
o n
on Tt0 < DE (Y ) = t < A0DE (Y ) = t < A0∞
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Proof of Uniqueness in Law
where the last equality follows from the above Proposition, (iii).
This yields the statement of the lemma on the set {t < A0∞ }.
But on {t ≥ A0∞ } we have Tt0 = +∞ as well as
Z t+
σ −2 (Wu ) du = +∞
0
because of the theorem on integral functionals of Brownian
motion and the identity A0∞ = A0DE (Y ) = DE (W ).
Proof of Theorem (on Uniqueness of Fundamental Solutions)
Let (Y , F) be a fundamental solution of Eq. (15) without drift.
We have seen that Yt = WA0t , t ≥ 0, for a
Dambis–Dubins–Schwarz Brownian motion and by the Lemma
T 0 and hence A0 is a well-determined measurable functional of
W . This yields that Y is a well-determined measurable
functional of W . Consequently, the distribution of Y is
well-determined.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Uniqueness of Solutions
Corollary
Suppose that σ −2 is locally integrable. Then the solution (Y , F)
is unique in law if and only if Nσ = ∅.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
4. SDEs with Generalized Drift
In this section we will construct solutions of the SDE
Z
Z t
X
(20)
Xt = X0 +
L (t, y ) ν (dy ) +
b (Xu ) dBu
0
R
with generalized drift by space transformation. We shall
assume that the signed measure ν satisfies the “atom
condition" ν ({x}) < 12 for each x.
We postpone the “reflecting case" ν ({x}) =
1
2
for some x.
In the case ν ({x}) > 12 for some x it can be shown that there
does not exist a solution (at least for some initial values).
Let g denote the unique solution of the integral equation

 1 − 2R
[0,x] g (y −) ν (dy ) for x ≥ 0 ,
g (x) =
R
 1+2
g (y −) ν (dy ) for x < 0 ,
(x,0)
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
SDEs with Generalized Drift
The solution can be written explicitly by

 exp (−2ν ([0, x])) Q
0≤y ≤x (1 − 2ν ({y })) exp (2ν ({y }))
g (x) =
Q
 exp (2ν ((x, 0)))
(1 − 2ν ({y }))−1 exp (−2ν ({y }))
x<y <0
This concrete form is less important but we need the relation
(21)
dg −1 (x) = 2g −1 (x) ν (dx) .
Now we put R
x
G (x) = 0 g (y ) dy , x ∈ R which is strictly increasing and
continuous.
The inverse of G defined on (G (−∞) , G (∞)) and denoted
by H is strictly increasing and continuous, too.
We extend H, g, b by setting

−∞ for x ∈ [−∞, G (−∞)] ,
g (±∞) = b (±∞) = 0 , H (x) =
+∞ for x ∈ [G (+∞) , +∞] .
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
SDEs with Generalized Drift
Note that we have the relation (recall a−1 = +∞ if a = 0)
Z x
(22)
H (x) =
(g ◦ H)−1 (y ) dy , x ∈ R .
0
Proposition (Space Transformation)
(i) Let (Y , F) be a (resp., fundamental) solution of of Eq.
(15)(σ) with σ = (g ◦ H) (b ◦ H) and Y0 = G (x0 ). Put
X = H (Y ). Then (X , F) is a (resp., fundamental) solution of Eq.
(20)(b, ν) with X0 = x0 .
(ii) Let (X , F) be a (resp., fundamental) solution of Eq.
(20)(b, ν) with generalized drift and X0 = x0 . Put Y = G (X ).
Then (Y , F) is a (resp., fundamental) solution of Eq. (15)(σ)
with σ = (g ◦ H) (b ◦ H) and Y0 = G (x0 ).
(iii) In both cases we have the relation
(23)
LX (t, y ) = g −1 (y ) LY (t, G (y )) for t < S∞ (X ) P-a.s.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Space Transformation: The Proof
Proof. (iii) Suppose that (Y , F) is a solution of Eq. (15). Note
that H is the difference of two convex functions by (22).
Applying the generalized Itô formula to X = H (Y ) we obtain
that X is a continuous semimartingale up to S∞ (X ) with
martingale part
Z t
Mt =
(g ◦ H)−1 (Yu ) dYu , t < S∞ (X ) ,
0
hence
Z
hX , X it =
t
(g ◦ H)−2 (Yu ) dhY , Y iu , t < S∞ (X ) .
0
Using the occupation times formula, for each Borel f ≥ 0 we get
Z
f (y ) g (y ) LX (t, y ) dy
R
Z t
=
f (Xu ) g (Xu ) dhX , X iu
0
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Space Transformation: The Proof
=dhX ,X iu
Z
t
=
z
}|
{
(f ◦ H) (Yu ) (g ◦ H) (Yu ) (g ◦ H)−2 (Yu ) dhY , Y iu
0
=0 on E
Z
=
(∗)
=
=dH(z)
b
}|
{
z }| { z
(f ◦ H) (z) LY (t, z) (g ◦ H)−1 (z) dz
ZR
(f ◦ H) (z) LY (t, z) dH (z)
(G(−∞),G(+∞))
Z
=
f (y ) LY (t, G (y )) dy
R
(*) Note that Ebc ⊆ (G (−∞) , G (+∞)). By comparing
g (y ) LX (t, y ) = LY (t, G (y )) dy λ-a.e. P-a.s.
and, by the right continuity of both members in y and t, we get
statement (iii).
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Space Transformation: The Proof
(i) If (Y , F) is a solution, using the generalized Itô formula,
Xt = H (Yt )
Z
=
t
=dYu
−1
z
}|
{
(Yu ) (g ◦ H) (Yu ) (b ◦ H) (Yu ) dBu
(g ◦ H)
H (y0 ) +
0
Z
1
+
LY (t, z) d (g ◦ H)−1 (z)
2 R
=2g −1 (y )ν(dy )
Z
=
=
(iii)
=
t
Z
z }| {
1
(b ◦ H) (Yu ) dBu +
LY (t, G (y )) dg −1 (y )
2 R
0
Z t
Z
x0 +
b (Xu ) dBu +
g −1 (y ) LY (t, G (y )) ν (dy )
0
R
Z t
Z
x0 +
b (Xu ) dBu +
LX (t, y ) ν (dy ) ,
x0 +
0
R
hence (X , F) is a solution.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Space Transformation: The Proof
(ii) If (X , F) is a solution with explosion time S∞ (X ), then by
Itô’s formula for t < S∞ (X )
Z t
Yt = G (Xt ) = G (x0 ) +
g (Xu −) b (Xu ) dBu
0
=g(y −) Z
=2g(y −)ν(dy )
Z t z }| {
Z
z }| {
1
+
g (Xu −) LX (du, y ) ν (dy ) −
LX (t, y ) dg (y )
2 R
0
R
Z t
Z
= y0 +
g (Xu ) b (Xu ) dBu +
g (y −) LX (t, y ) ν (dy )
R
Z 0
1
−
LX (t, y ) 2g (y −) ν (dy )
2 R
Z t
= y0 +
(g ◦ H) (Yu ) (b ◦ H) (Yu ) dBu
0
Z t
= y0 +
σ (Yu ) dBu .
0
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Space Transformation: The Proof
Let Sn = inf {t ≥ 0 : |Xt | ≥ n}. Then Sn ↑ S∞ (X ) and passing
to the limit in the Eq. just obtained we get
Z S∞ (X )
YS∞ (X ) = y0 +
σ (Yu ) dBu on {S∞ (X ) < +∞} .
0
By the definition of Y we have Yt = Yt∧S∞ (X ) for each t. By the
definition of σ we have
Z t
Z t∧S∞ (X )
σ (Yu ) dBu =
σ (Yu ) dBu .
0
0
Summarizing we obtain, for each t,
Z t
Yt = y0 +
σ (Yu ) dBu ,
0
so that (Y , F) is a solution to Eq. (15). The statement
concerning fundamental solutions immediately follows from the
lemma below.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Existence of Solutions
Lemma
Let σ = (g ◦ H) (b ◦ H). Then
Nσc = G (Nbc ) ,
Ebc = G (Ebc ) .
Proof. omitted
Theorem (Existence of Solutions)
The following conditions are equivalent:
(i) For every x0 ∈ R, there exists a fundamental solution (X , F)
to Eq. (20) with generalized drift such that X0 = x0 .
(ii) For every x0 ∈ R, there exists a solution (X , F) to Eq. (20)
with generalized drift such that X0 = x0 .
(iii) Eb ⊆ Nb .
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Existence of Solutions: The Proof
Proof. 1) Suppose (iii): Eb ⊆ Nb . Let x0 ∈ R. From the Lemma
we get Eb ⊆ Nσ . Hence there is a fundamental solution (Y , F)
with Y0 = G (x0 ). The above Proposition yields that (X , F) with
X = H (Y ) is a fundamental solution of Eq. (20) with X0 = x0 .
2) The implication (i)⇒(ii) is trivial.
3) (ii)⇒(iii) Let y0 ∈ (G (−∞) , G (+∞)) , y0 = G (x0 ). By
assumption (ii) there exists a solution (X , F) of Eq. (20) with
X0 = x0 . The Proposition, (ii), then implies that (Y , F) defined
by Y = G (X ) is a solution of Eq. (15) such that Y0 = y0 . In the
/ (G (−∞) , G (+∞)), there is the trivial solution (Y , F)
case y0 ∈
with Y0 = y0 , because σ(y0 ) = 0. Hence, for any y0 ∈ R, there
exists a solution (Y , F) of Eq. (15) with Y0 = y0 . As a necessary
condition, we obtain from the existence theorem for equations
without drift that Eb ⊆ Nσ . The Lemma implies Eb ⊆ Nb .
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Existence of Solutions
Corollary
Suppose that b−2 is locally integrable. Then, for each x0 ∈ R,
there exists a (fundamental) solution to Eq. (20) with X0 = x0 .
Corollary
Suppose that b has right hand left limits b (x+) and b (x−) at
every x ∈ R and that b (x) = 0 whenever
|b (x+) | ∧ |b (x−) | = 0. Then, for each x0 ∈ R, there exists a
(fundamental) solution to Eq. (20) with X0 = x0 . In particular,
the condition is satisfied if b is continuous.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Existence of Solutions
Remark
1) For every solution (X , F), we have
(24)
Xt = Xt∧DE
b
(X )
∀ t ≥ 0 P-a.s.
Using space transformation and the Lemma, this property is
obtained from the corresponding property Yt = Yt∧DEσ (Y ) for
the solition (Y , F).
2) From (24) immediately follows that
(25)
LX (t, y ) = 0 ∀ (t, y ) ∈ [0, S∞ ) × Eb . P-a.s.
Therefore, without loss of generality we can assume that ν
vanishes on Eb .
3) The results can easily be extended to the case that ν is a
locally finite signed measure on the open set R \ Eb . However,
the condition (24) need not hold any longer.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Uniqueness of Solutions
Theorem (Uniqueness of Solutions)
(i) The fundamental solution (X , F) of Eq. (20) with generalized
drift is unique in law.
(ii) For every initial value x0 ∈ R there exists a solution (X , F) to
Eq. (20) with generalized drift and uniqueness in law holds if
and only if the condition Eb = Nb is satisfied:
b (x) 6= 0 iff b−2 is integrable in a neighbourhood of x.
Proof. By space transformation (see the above Proposition)
the statements are reduced to the corresponding statements (i)
on the uniqueness of the fundamental solution and (ii) on the
existence and uniqueness for SDEs without drift.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
5. Pathwise Uniqueness and Strong Solutions
We consider the SDE with generalized drift
Z
(26)
Xt = x0 +
X
Z
L (t, y ) ν (dy ) +
R
t
b (Xu ) dBu .
0
Recall that a solution (X , F) is called strong if it is FB -adapted.
(We consider only deterministic initial values X0 = x0 .)
We say that the (resp., fundamental) solution to Eq. (26) is
pathwise
unique if for any two (resp., fundamental) solutions
X 1 , F and X 2 , F on the same probability space and with the
same Brownian motion (B, F) starting from the same initial
value it follows X 1 = X 2 .
The next theorem gives general conditions for pathwise
uniqueness and existence of strong solutions.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Pathwise Uniqueness and Strong Solutions
Theorem (Pathwise Uniqueness I)
Suppose
 that f : R \ Eb 7→ [0, +∞] and h : R 7→ [0, +∞] such

 (i) f b−2 is locally integrable in R \ Eb .



R



(ii) For every open U 3 0, U h−1 (y ) dy = +∞ .


that:
(iii) There exists a number c > 0 such that





(b (x + y ) − b (x))2 ≤ f (x) h (y ) ,





x, x + y ∈ R \ Eb , y ∈ (−c, c) .
Then the fundamental solution to Eq. (26) is pathwise unique.
Corollary (Strong Solutions)
If additionally Eb ⊆ Nb , then every fundamental solution to Eq.
(26) is strong. Furthermore, for every Brownian motion (B, F) ,
there exists a pathwise unique strong solution (X , F) .
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
The Proof
Proof (of the Corollary). The additional condition guarantees
weak existence (for every initial condition). Pathwise
uniqueness holds in view of the theorem. Applying the
Yamada–Watanabe theorem (restricted to the class of
fundamental solutions) yields the result.
Sketch of Proof (of the Theorem).
1) Consider two fundamental solutions X 1 , F and X 1 , F ,
X01 = X02 = x0 , on the same probability space and with the
same Brownian motion (B, F). We have to show: X 1 = X 2
P-a.s.
2) By localizing we can achieve that R \ Eb has compact
closure and hence f b−2 is integrable over R \ Eb . Note that
then explosion cannot occur.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
The Proof
3) The crucial point is the following lemma:
Lemma
We have LX
1 −X 2
(t, 0) = 0 P-a.s.
Proof. Let
X = X 1 − X 2,
Di := DEb X i ,
D = D1 ∧ D2 .
X is constant on [D1 ∨ D2 , ∞), therefore
LX (t, 0) = LX (t ∧ (D1 ∨ D2 ), 0) P-a.s.
Since X 6= 0 on [D1 ∧ D2 , D1 ∨ D2 ] and the local time is carried
on X = 0 we get
LX (t, 0) = LX (t ∧ D, 0) P-a.s.
and it is sufficient to verify that LX (t ∧ D, 0) = 0 P-a.s..
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
The Proof
Consider the sets An := LX (D, y ) > n1 ∀ y ∈ 0, n1 . Using
the occupation times formula, on An for n sufficiently large we
obtain
Z 1
n
+∞ =
h−1 (y ) dy
0
Z c
≤
h−1 (y ) LX (D, y ) dy
0
Z
D
=
1[0,c) (Xu ) h−1 (Xu ) dhX , X iu
0
Z
D
=
0
Z
D
≤
0
Z
≤
0
D
2
1[0,c) (Xu ) h−1 (Xu ) b Xu1 − b Xu2
du
1[0,c) (Xu ) h−1 (Xu ) f Xu2 h (Xu ) du
f Xu2 du
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
The Proof
Z
=
Z0
=
=
b2 (Xu2 ) du
z
}| {
D 2
−2
2
f Xu b
Xu dhX 2 , X 2 i
2
f (y ) b−2 (y ) LX (D, y ) dy
R
Z
X2
max L (D, y ) f (y ) b−2 (y ) dy < +∞ .
y ∈R\Eb
R
Here we have used:
R \ Eb has compact closure.
2
LX (D, y ) is bounded in y on compact sets.
2
LX (D, y ) vanishes outside R \ Eb .
f b−2 is integrable over R \ Eb .
This yields that An must have probability zero for all n
sufficiently large which implies the claim.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
The Proof
If h−1 is integrable in a right neighbourhood but not integrable in
any left neighbourhood of zero one has to invert the order and
to prove that X 2 − X 1 has left local time zero which yields that
X 1 − X 2 has right local time zero.
4) Now, in the final step, we put X = X 1 ∨ X 2 . Based on the
Lemma and using the generalized Itô formula we can show that
X is again a fundamental solution of Eq. (26). For this it is
needed:
Z t
2
X
L (t, y ) =
1{X 1 ≤X 2 } (u) LX (du, y )
0
Z t
1
+
1{X 2 <X 1 } (u) LX (du, y ) P-a.s.
0
Because of the uniqueness in law of the fundamental solution,
this yields that Xt and Xti have the same distribution and
because Xt ≥ Xti we get Xt = Xt1 = Xt2 P-a.s.The continuity of
the processes implies X 1 = X 2 P-a.s.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Another Result on Pathwise Uniqueness
The following result is a generalization of the theorem of Nakao.
Theorem (Pathwise Uniqueness II)
Suppose that g : R \ Eb 7→ [0, +∞] and h : R 7→ [0, +∞] such
that:


(i) g is increasing .




R


(ii) For every open U 3 0 of zero, U h−1 (y ) dy = +∞ .







(iii) There exists a number c > 0 such that



|g (x + y ) − g (x) |
(b (x + y ) − b (x))2 ≤ h (y )
,

|y |





x, x + y ∈ R \ Eb , y ∈ (−c, c) .






(iv) For every compact set K ⊂ R \ Eb





infx∈K b (x) > 0 .
Then the fundamental solution to Eq. (26) is pathwise unique.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Another Result on Pathwise Uniqueness
Corollary (Strong Solutions)
If, additionally to the generalized Nakao condition, Eb ⊆ Nb ,
then every fundamental solution to Eq. (26) is strong.
Furthermore, for every Brownian motion (B, F) , there exists a
pathwise unique strong solution (X , F) .
Proof. The proof of the Corollary is the same as in the first
case above.
The proof of the theorem is similar to the case of a generalized
Hölder condition (see above) and will be omitted.
Remark
1) There us a certain overlapping between the generalized
Hölder condition and the generalized Nakao condition. Indeed,
if we put f ≡ 1 in the first one and g (x) = x (the identity), the
conditions coincide.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Another Result on Pathwise Uniqueness
Remark
2) If we suppose h (y ) = |y | then the generalized Nakao
condition means that g is of locally bounded square variation.
In the case of SDEs with “ordinary" drift, under additional
assumptions, Nakao has proven pathwise uniqueness if b is of
bounded variation. Le Gall extended this result to functions b of
bounded square variation.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Lectures on One-dimensional Stochastic
Differential Equations (Part II)
Hans-Jürgen Engelbert
Friedrich Schiller University, Jena, Germany
Research School
CIMPA-UNESCO-MESR-MINECO-MOROCCO
Statistical Methods and Applications
in Actuarial Science and Finance
Marrakesh (8–13 April 2013) and El Kelaa (15–20 April 2013)
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Overview
6. SDEs with Ordinary Drift
Existence and Uniqueness
Pathwise Uniqueness I
Pathwise Uniqueness II
Some Corollaries
7. Non-Explosion of Solutions
Why Explosion Does Occur?
The Behaviour of Integral Functionals Revisited
Characterization of Non-Explosion
Jeulin’s Lemma
Proof of the Theorem
8. Fundamental Solutions and Time Delay
Fundamental Solutions
Time Delay
Theorem on Time Delay
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Overview
Some Examples
Structure of the General Solution
Time Delay and Local Time
9. Reflecting SDEs
Reflecting SDEs with generalized Drift
Reflecting SDEs with ordinary Drift
Reduction to the Non-Reflected Case
10. The Predictable Representation Property
The Definition
A General Theorem on Integral Representation
The Proof
The PRP for Solutions of SDEs
11. Financial Markets: An Example
Linear or Geometric Stocks
No-Arbitrage
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Overview
Equivalent Martingale Measures
Complete Markets
Uniqueness of the Equivalent Martingale Measure
12. On the Yamada-Watanabe Theorem
The Yamada-Watanabe Theorem
Joint Uniqueness in Law
Extension of the Yamada-Watanabe Theorem
The Proof
Equivalence of Conditions
No Strong Solutions
Tanaka’s Example
On Skew Brownian Motion
Space Transformation
Barlow’s Example
Extension of Barlow’s Example
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
6. SDEs with Ordinary Drift
In the present section, we summarize the results concerning
SDEs
Z t
Z t
(1)
Xt = X0 +
a (Xu ) du +
b (Xu ) dBu
0
0
with diffusion coefficient b and “ordinary" drift a. As we already
know from the Introduction, the link to SDEs with generalized
a (x)
dx.
drift is given by the drift measure ν (dx) = 2
b (x)
We assume:

 (i) E ⊆ Na .
b
(2)
 (ii) a b−2 is locally integrable in R .
(ii) is motivated by the desire to introduce the locally finite
signed measure ν. (i) is needed because the solution X of the
SDE with generalized drift is stopped after reaching Eb . The
same should be true for the solution of SDE (1)(a,b).
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
SDEs with Ordinary Drift
Theorem (Existence and Uniqueness)
(i) For every x0 ∈ R there exists a fundamental solution (X , F)
to Eq. (1)(a,b) with ordinary drift such that X0 = x0 if and only if
Eb ⊆ Nb .
(ii) Suppose that Nb ⊆ Na . Then, for every x0 ∈ R there exists
a solution (X , F) to Eq. (1)(a,b) with ordinary drift such that
X0 = x0 if and only if Eb ⊆ Nb .
(iii) Suppose that Nb ⊆ Na . Then for any x0 ∈ R there exists a
unique solution to Eq. (1)(a,b) with ordinary drift such that
X0 = x0 if and only if Eb = Nb .
(iv) For every x0 ∈ R, the fundamental solution (X , F) to
Eq. (1)(a,b) with ordinary drift such that X0 = x0 is unique.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
SDEs with Ordinary Drift
Theorem (Pathwise Uniqueness I)
(A) Let


(i)







(ii)


(iii)










f : R \ Eb 7→ [0, +∞] and h : R 7→ [0, +∞] satisfy:
f b−2 is locally integrable in R \ Eb .
R
For every open U 3 0, U h−1 (y ) dy = +∞ .
There exists a number c > 0 such that
(b (x + y ) − b (x))2 ≤ f (x) h (y ) ,
x, x + y ∈ R \ Eb , y ∈ (−c, c) .
Then the fundamental solution (X , F) to Eq. (1)(a,b) is pathwise
unique.
(B) If additionally Eb ⊆ Nb , then every fundamental solution to
Eq. (1)(a,b) is strong. Furthermore, for every Brownian motion
(B, F), there exists a pathwise unique strong solution (X , F).
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
SDEs with Ordinary Drift
Theorem (Pathwise Uniqueness II)
(A) Let


(i)







(ii)





 (iii)














g : R \ Eb 7→ [0, +∞] and h : R 7→ [0, +∞] satisfy:
g is increasing.
For every open U 3 0,
R
U
h−1 (y ) dy = +∞ .
There exists a number c > 0 such that
|g (x + y ) − g (x) |
(b (x + y ) − b (x))2 ≤ h (y )
,
|y |
x, x + y ∈ R \ Eb , y ∈ (−c, c) .
(iv) For every compact K ⊂ R \ Eb , infx∈K b (x) > 0 .
Then the solution (X , F) to Eq. (1)(a,b) is pathwise unique.
Every solution (X , F) to Eq. (1)(a,b) is a strong solution. If
additionally Eb ⊆ Nb holds then for any x0 ∈ R and every
Wiener process (B, F), there exists a pathwise unique strong
solution (X , F) with X0 = x0 .
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
SDEs with Ordinary Drift
Theorem (Continued)
(B) Suppose that Nb ⊆ Eb and let f : R \ Eb 7→ [0, +∞] and
h
: R 7→ [0, +∞] satisfy:

(i) f b−2 is locally integrable in R \ Eb .




R



(ii) For every open U 3 0, U h−1 (y ) dy = +∞ .


(iii) There exists a number c > 0 such that





(b (x + y ) − b (x))2 ≤ f (x) h (y ) ,





x, x + y ∈ R \ Eb , y ∈ (−c, c) .
Then the same conclusions hold as in part (i).
Proof. The proof of all statements is based on the equivalence
of Eq. (1)(a,b) and the SDE with generalized drift: (i) for
fundamental solutions (ii) for general solutions if the additional
assumptions Nb ⊆ Na or Nb ⊆ Eb come into play.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
SDEs with Ordinary Drift
Corollary (Existence and Uniqueness)
Suppose that
|a| + 1
is locally integrable and (ii) b (x) 6= 0 ∀ x ∈ R .
(i)
b2
Then for all x0 ∈ R, there exists a unique solution (X , F) of
Eq. (1)(a,b) with X0 = x0 .
Corollary (Strong Solutions)
Suppose that a is locally integrable and consider the SDE
Z
(3)
t
Xt = X0 +
a (Xu ) du + Bt .
0
Then for all x0 ∈ R, there exists a pathwise unique strong
solution (X , F) of Eq. (3)(a) with X0 = x0 . Moreover, every
solution of Eq. (3)(a) is a strong solution.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
7. Non-Explosion of Solutions
Solutions (X , F) are, in general, exploding.
The reason is the following: The process Y = G (X ) solves
Z t
(4)
Yt = Y0 +
σ (Yu ) dBu
0
where σ = (b ◦ H) (g ◦ H) and H is the inverse of G on
(c, d) := (G (−∞) , G (+∞)). Note that, σ being zero outside of
(c, d), the process Y is stopped after leaving (c, d). For
simplicity: X0 = 0, hence Y0 = 0. We define
Dx (Y ) = inf {t ≥ 0 : Yt = x} .
Observation
(i) X does not explode P-a.s. iff Dc (Y ) ∧ Dd (Y ) = +∞ P-a.s.
(ii) In particular, if d = +∞ then X does not explode to +∞ and
if c = −∞ then X does not explode to −∞.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Non-Explosion of Solutions
Now recall the construction of Y :
Z
Yt = WAt ,
At := hY , Y it =
t
σ 2 (Yu ) du ,
t ≥ 0.
0
Therefore, Y not reaching c or d is equivalent to
At < Dc (W ) ∧ Dd (W ) ∀ t ≥ 0 but this is equivalent to
TDc (W )− = +∞ and TDd (W )− = +∞ P-a.s.
R t+
Recall: Tt = 0 σ −2 (Wu ) du .
Theorem (Integral Behaviour for General f )
Suppose that a > 0 and f is locally integrable in R, i.e., E = ∅.
Then the following conditions are equivalent:
(i) P TDa (W )− < +∞ = 1.
(ii) P TDa (W )− < +∞ > 0.
Ra
(iii) 0 (a − y ) f (y ) dy < +∞.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Non-Explosion of Solutions
From this we can derive the following non-explosion criterion:
Theorem (Non-Explosion)
X does not explode P-a.s.if and only if the following conditions
are satisfied:
(i) If d < +∞ and σ −2 is locally integrable on [0, d) then
d
Z
(d − y ) σ
−2
+∞
Z
(y ) dy =
0
0
d − G (x)
dx = +∞ .
g (x) b2 (x)
(ii) If −∞ < c and σ −2 is locally integrable on (c, 0] then
Z
0
(y − c) σ
−2
Z
0
(y ) dy =
c
Hans-Jürgen Engelbert, FSU Jena, Germany
−∞
G (x) − c
dx = +∞ .
g (x) b2 (x)
Lectures on One-dimensional SDEs
Non-Explosion of Solutions
Proof. Suppose X does not explode. Then TDd (W )− = +∞. To
prove (i) we assume that d < +∞ and that σ −2 is locally
integrable on [0, d) and thus on (−ε, d) for some ε > 0. Setting
f = σ −2 on [−ε, d) and zero outside, we see that f is locally
integrable and on {Dd (W ) < D−ε (W )} (with positive
probability)
Z Dd (W )
Z Dd (W )
f (Wu ) du =
σ −2 (Wu ) du = +∞ ,
0
0
Rd
by the Theorem we get 0 (d − y ) σ −2 (y ) dy = +∞, which
proves (i). The proof of (ii) is analogous. Conversely, suppose
(i) and show TDd (W )− = +∞ P-a.s. This is clear for d = +∞. If
σ −2 is not locally integrable on [0, d) then [0, d) ∩ E 6= ∅ and we
get
Z DE +
Z Dd (W )
−2
+∞ =
σ (Wu ) du ≤
σ −2 (Wu ) du = TDd (W )− ,
0
0
hence X does not explode.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Non-Explosion of Solutions
Thus we can assume that d < +∞ and that σ −2 is locally
integrable on [0, d). By condition (i) it follows that
Z
d
(d − y ) σ −2 (y ) dy = +∞.
0
Setting f = σ −2 on [0, d) and zero outside, the Theorem yields
Z
Dd (W )
Z
f (Wu ) du ≤
+∞ =
0
0
Dd (W )
σ −2 (Wu ) du = TDd (W )− .
Analogously, from (ii) follows TDc (W )− = +∞ P-a.s. In summary,
X does not explode.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Jeulin’s Lemma
There is an elementary proof of the theorem on the bevaviour
of TDa (W )− based on a lemma of Jeulin.
Lemma (Jeulin’s Lemma)
Let (Ly )y ∈[0,a) be a family of nonnegative random variables
(measurable in (ω, y )) such that for a function
L
ϕ : [0, a) 7→ (0, +∞), the rv ϕ(yy ) is equally distributed to an
L
integrable rv ξ: ϕ(yy ) ∼ ξ, E (ξ) > 0. Let m be a measure on
[0, a) such that ϕ dm is σ-finite. Then the following assertions
are equivalent:
nR
o
(i) P
L
m
(dy
)
<
+∞
= 1.
[0,a) y
nR
o
(ii) P
> 0.
[0,a) Ly m (dy ) < +∞
R
(iii) [0,a) ϕ (y ) m (dy ) < +∞.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Jeulin’s Lemma
Proof. Using Fubini’s theorem, (iii)⇒(i) follows from
Z
Z
E
Ly m (dy ) = E
[0,a)
[0,a)
Ly
ϕ (y ) m (dy )
ϕ (y )
Z
= E (ξ)
ϕ (y ) m (dy ) < +∞ .
[0,a)
(i)⇒(ii) is trivial.
nR
We show (ii)⇒(iii): Set AN =
theorem,
o
L
m
(dy
)
≤
N
. By Fubini’s
y
[0,a)
!
Z
N ≥ E 1 AN
Ly
E 1AN
ϕ (y )
[0,a)
Z
=
Ly m (dy )
[0,a)
Hans-Jürgen Engelbert, FSU Jena, Germany
ϕ (y ) m (dy )
Lectures on One-dimensional SDEs
Jeulin’s Lemma
+∞
Ly
=
>u
du ϕ (y ) m (dy )
P AN ∩
ϕ (y )
[0,a) 0
+
Z
Z +∞ Ly
≥
du
ϕ (y ) m (dy )
P (AN ) − P
≤u
ϕ (y )
[0,a)
0
Z
Z +∞
=
ϕ (y ) m (dy )
(P (AN ) − P ({ξ ≤ u}))+ du .
{z
}
|
[0,a)
0
|
{z
}
→N→∞ P({ξ>u})
{z
}
|
<+∞
Z
Z
→N→∞ E(ξ)>0
Hence for N sufficiently large we conclude that
Z
ϕ (y ) m (dy ) < +∞
[0,a)
and (iii) is satisfied.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Proof of the Theorem
Proof of the Theorem. By the occupation times formula,
Z Da (W )
Z
f (Wu ) du =
LW (Da (W ), y ) f (y ) dy
0
ZR
=
LW (Da (W ), y ) m (dy )
R
with the measure m defined by m (dy ) = f (y ) dy . We set
ϕ (y ) = a − y on [0, a) and Ly = LW (Da (W ), y ). For proving
the theorem it is now sufficient to verify that the distribution of
LW (Da (W ), y )
does not depend on y and is different from δ0 .
a−y
Firstly, for 0 ≤ y < a we have
LW (Da (W ), y ) = LW (Dy (W ), y ) + LW ((Dy (W ) , Da (W )], y ) ,
|
{z
} |
{z
}
=0
∼d LW (Da−y (W ),0)
the second identity holds in view of the strong Markov property.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Proof of the Theorem
Therefore,
LW (Da (W ), y ) ∼d LW (Da−y (W ), 0) .
Replacing a − y by a and introducing the Brownian motion
Wt0 := a1 Wa2 t , we calculate
=a2 D1 (W 0 )
LW (Da (W ), 0) =
=
v :=
ε0 :=
u
a2
=
ε
a
=
0
=a W u
z }| {
a2
Z Da (W )
z }| {
1
lim
1(−ε,+ε) (Wu ) du
ε↓0 2ε 0
Z a2 D1 (W 0 )
1
0
1(−ε,+ε) a W u du
lim
ε↓0 2ε 0
a2
Z
0
a2 D1 (W )
lim
1(−ε,+ε) a Wv0 dv
ε↓0 2ε 0
Z D1 (W 0 )
a
1(− ε ,+ ε ) Wv0 dv
lim ε
a
a
ε↓0 2 a 0
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Proof of the Theorem
ε0 :=
ε
a
Z
D1 (W 0 )
=
1
a lim
0
ε ↓0 2ε0
=
a LW
∼d
a LW (D1 (W ), 0) .
0
1(−ε0 ,+ε0 ) Wv0 dv
0
D1 W 0 , 0
As a result,
1 W
L (Da (W ), 0) ∼d LW (D1 (W ), 0)
a
and hence
1
LW (Da−y (W ), 0) ∼d LW (D1 (W ), 0) ,
a−y
0 ≤ y < a.
This proves the claim.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
8. Fundamental Solutions and Time Delay
Now we are going to study the structure if the general solution
of the SDE with generalized drift
Z
Z t
X
(5)
Xt = X0 +
L (t, y ) ν (dy ) +
b (Xu ) dBu
0
R
where B is a Brownian motion and
the unknown process X .
LX
denotes the local time of
We assume: Eb ⊆ Nb (existence condition).
We already know: Xt = Xt∧DE (X ) P-a.s. We also have
b
introduced fundamental solutions as particular solutions of (5):
b2 (X ) > 0 on 0, DEb (X ) P-a.s.
In this section we assume: Ebc ∩ Nb 6= ∅. As a consequence,
there exist solutions with positive sojourn time in Ebc ∩ Nb . For
example, we can stop the fundamental solution after reaching
some point a ∈ Ebc ∩ Nb .
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Time Delay
Starting from any solution (X , F) , we now construct new
solutions by time delay.
Definition (Time Delay)
An F-adapted right continuous increasing process V with
values in [0, +∞] is called an (X , F) -delay if
Z +∞
1M c (Xu ) dVu = 0 P-a.s.
0
where V0− = 0 and M = Ebc ∩ Nb .
The Time-Delayed Solution: We define
t ≥ 0.
Then T = (Tt ) is a continuous F-time change. Let X , F be the
process (X ◦ T , F ◦ T ) obtained from (X , F) by the delay V .
At = t + Vt ,
Tt = inf {s ≥ 0 : As > t} ,
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Time Delay
Theorem (Time Delay)
The process X , F is again a solution of Eq. (5).
Proof. X , F is a continuous semimartingale up to S∞ X ,
Z
Z Tt
X t = X0 +
LX (Tt , y ) ν (dy ) +
b (Xu ) dBu , t < S∞ X .
R
{z
}
|0
=:Mt
By the Tanaka formula we get
LX (Tt , y ) = LX (t, y ) ∀ (t, y ) ∈ 0, S∞ X × R P-a.s.
For dealing with the martingale part we need the following
Lemma (New Clock)
Z t
Z t
Tt =
1M X u dTu +
1M c X u du .
0
Hans-Jürgen Engelbert, FSU Jena, Germany
0
Lectures on One-dimensional SDEs
Time Delay
Remark
T increases in “normal clock" whenever X moves outside of M.
Now we can calculate
hX , X it
Tt
t
= hM, Mit =
b (Xu ) du =
b2 X u dTu
0
0
Z t
Z t
=
b2 X u 1M X u dTu +
b2 X u 1M c X u du
0
|0
{z
}
Z
Z
2
=0
Z
t
b
=
2
Z
X u 1M c X u du =
t
b2 X u du .
0
0
This yields (on a, possibly, Zenlarged filtered probability space)
t
Mt =
b X u dB u
0
for some BM B, possibly, on an enlargment of (Ω, F, P).
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Time Delay
Example
(i) Let F ⊆ M be closed.
 Define
 0
for t < DF (X ) ,
Vt =
 +∞ otherwise .
Then X is just the process X stopped at DF (X ):
X t = Xt∧DF (X ) , t ≥ 0 .
(ii) Let U be an F-stopping time such that b (XU ) = 0 on
{U < +∞}. For example, U = DF (X ), F ⊆ M closed. Let S be
a nonnegative FU -measurable
random variable and define

 0 for t < U ,
Vt =
 S otherwise .
Then X is constant on [U, U + S] (taking values in M) and
behaves like X outside [U, U + S].
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Time Delay
Remark
It can be shown that the fundamental solution (X , F) of Eq. (5)
is a continuous strong Markov process. Example (i) above
preserves the strong Markov property for the time-delayd
process X whereas Example (ii) not.
The following theorem shows that every solution X , F of Eq.
(5) can be obtained from the fundamental solution (X , F) by
time delay.
Theorem
Let X , F be an arbitrary solution of Eq. (5). Then there exist a
fundamental solution (X , F) defined, possibly, on an enlarged
probability space, an (X , F) -delay
V , and a filtration G with
F t ⊆ Gt ⊆ F ∞ such that X , G is obtained from (X , F) by the
delay V .
Proof. Skipped.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Time Delay
Time Delay and Local Time
Particular important time delays are constructed by local time.
Suppose that (X , F) is a fundamental solution of Eq. (5). Let µ
be a nonnegative measure on (R, B (R)). We define
Z
(6)
Vt =
LX (t, y ) µ (dy ) ,
t ≥ 0.
R
Since LX (t, y ) = 0 on Eb , without loss of generality we can
assume µ (Eb ) = 0. Then V is an (X , F) -time delay iff
(i) µ is carried by M = Ebc ∩ Nb .
(ii) Eµ ∩ Ebc ⊆ Nb where
Eµ = {x ∈ R : µ (G) = +∞ ∀ open G 3 x}.
We call a measure µ with the properties (i) and (ii) a delay
measure.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Time Delay
Example
Let a ∈ Ebc ∩ Nb and define µ = γδa . Then
Vt = γLX (t, a) , t ≥ 0 ,
is a time delay.
Remark
Let X , F be the solution obtained by the time delay V from a
fundamental solution (X , F). Then X is a continuous strong
Markov family (w.r.t. the different initial values x0 ∈ R) if and
only if the time delay V is of “local time form" (8) for a delay
measure µ. The proof of this (deep) result is beyond the scope
of these lectures.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
An Equation with Pre-fixed Occupation Time
As before we consider a diffusion coefficient b and a drift
measure ν satisfying the ’atom condition’. Let M = Ebc ∩ Nb .
Suppose that we are given a delay measure µ:
(i) µ is carried by M = Ebc ∩ Nb .
(ii) Eµ ∩ Ebc ⊆ Nb where
Eµ = {x ∈ R : µ (G) = +∞ ∀ open G 3 x}.
Now we consider the following SDE
(7)

Z
Z t

X

 X t = x0 +
L (t, y ) ν (dy ) +
b X u dBu , t < S∞ X ,
R
0
Z


λ 0 ≤ u ≤ t : Xu ∈ M =
LX (t, y ) µ (y ) .

M
Theorem
Suppose that Eµ ∪ Eb ⊆ Nb . Then, for all initial values x0 ∈ R,
the SDE (7) has a solution unique in law.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
An Equation with Pre-fixed Occupation Time
Remark
The conclusions for pathwise uniqueness and the existence of
strong solutions obtained in case of SDEs with generalized drift
remain valid for Eq. (7).
Idea of Proof. A solution X , F can be constructed by the time
delay V ,
Z
(8)
Vt =
LX (t, y ) µ (dy ) , t ≥ 0 .
R
from a fundamental solution (X , F) of the SDE with generalized
drift (the first equation above). The proof of uniqueness is a
little bit more demanding: By space transformation, Eq. (7) can
be reduced to an SDE without drift (b = σ and ν = 0). The
latter SDE can be equivalently formulated as a martingale
problem which has a unique solution.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
9. Reflecting SDEs
We again consider the SDE with generalized drift
Z
Z t
LX (t, y ) ν (dy ) +
b (Xu ) dBu
(9)
Xt = X0 +
0
R
but from now on we allow that ν (x) = 12 for some x ∈ R. Note
that these points do not accumulate because, on every
compact, ν is a finite signed measure. For the sake of simplicity
we assume X0 = 0 and:
(10)
ν ({0}) =
1
,
2
ν ({x}) <
1
, ∀ x 6= 0 .
2
Then Eq. (23) can be written
(11)
Z t
Z
1 X
b (Xu ) dBu + L (t, 0)+
Xt =
LX (t, y ) ν (dy ) , t < S∞ (X ) ,
2
0
[0,+∞)
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Reflecting SDEs
with ν ({0}) = 0, an SDE with generalized drift reflecting at zero
to the right. For this reason, we assume that b is a real function
and ν is a locally finite signed measure defined (only) on
[0, +∞).
Similarly, we can introduce the SDE
Z
(12)
Yt =
0
t
1
σ (Yu ) dBu + LY (t, 0) , t ≥ 0 ,
2
a reflecting SDE without drift (except for the reflection part),
and we can introduce reflecting SDEs with ordinary drift:
(13)
Z t
Z t
1 X
a (Xu ) du , t < S∞ (X ) ,
Xt =
b (Xu ) dBu + L (t, 0) +
2
0
0
with coefficients σ, a and b defined on [0, +∞).
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Reflecting SDEs
Question: How to solve these SDEs ?
Answer: Existence, uniqueness, pathwise uniqueness,
existence of strong solutions, etc., can be treated in an
analogous way starting from a reflected Brownian motion |W |
(instead of the Brownian motion W itself).The conditions stated
earlier can easily be translated for reflecting SDEs and the
results are analogous. We omit further details.
Instead we will shortly discuss another approach to the
existence: Given b and ν on [0, +∞) such that ν ({y }) < 12 ,
ν ({0}) = 0 and Eb ⊆ Nb are satisfied, we extend b on the
whole R symmetrically and define a signed measure νe
antisymmetrically by ν on [0, +∞) and νe = −ν on (−∞, 0).
The condition Eb ⊆ Nb remains valid. We also have νe ({y }) <
for y ≥ 0 but νe ({y }) > − 12 for y < 0. This requires that in Eq.
(23) we use right local time if y ≥ 0 but left local time if y < 0
eX (t, y ).
which will be denoted by L
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
1
2
Reflecting SDEs
Let (Z , F) be a solution of Eq. (23) with Z0 = 0. Now we put
X = |Z |. Using Tanaka’s formula, we obtain
Xt = |Zt |
Z t
=
sgn− (Zu ) dZu + LZ (t, 0)
0
Z t
Z t
Z
eZ (u, y ) ν (dy )
=
b (Zu ) sgn− (Zu ) dBu +
sgn− (Zu ) du
L
|
{z
}
0
0
R
eu
=dB
Z
+ L (t, 0)
| {z }
(∗)= 21 LX (t,0)
Z
t
Z
eu +
b (Xu ) dB
=
0
|0
t
Z
sgn− (Zu ) du
(∗∗)=
Hans-Jürgen Engelbert, FSU Jena, Germany
R
1
LZ (u, y ) ν (dy ) + LX (t, 0)
2
{zR
}
[0,+∞)
LX (t,y ) ν(dy )
Lectures on One-dimensional SDEs
Reflecting SDEs
which yields
Z t
Z
e u + 1 LX (t, 0) +
Xt =
b (Xu ) dB
LX (t, y ) ν (dy ) ,
2
0
[0,+∞)
proving that (X , F) is a solution provided that (*) and (**) hold.
Proof of (*). Note that
(14)
LX (t, y ) = LZ (t, y ) + LZ (t, (−y )−) ∀ y ≥ 0 .
On the other side,
LZ (t, 0) − LZ (t, 0−) = 2
Z
t
1{0} (Zu ) dVu
Z t
Z
eZ (u, z) νe (dz)
1{0} (Zu ) du
L
= 2
0
0
(15)
R
= 2 LZ (t, 0) νe ({0}) = 0 ,
| {z }
=0
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Reflecting SDEs
Hence (14) implies that
LX (t, 0) = LZ (t, 0) + LZ (t, 0−) = 2 LZ (t, 0) ,
proving (*).
Proof of (**). We calculate
Z t
Z
eZ (u, y ) νe (dy )
sgn− (Zu ) du
L
0
R
Z
Z
Z
=
L (t, y ) ν (dy ) −
LZ (t, y −) (−ν) (dy )
(0,+∞)
(−∞,0]
Z
Z
=
LZ (t, y ) ν (dy ) +
LZ (t, (−y )−) ν (dy )
(0,+∞)
[0,+∞)
Z
LX (t, y ) ν (dy ) ,
=
[0,+∞)
because of (14).
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
10. The Predictable Representation Property
We now consider the SDE
Z
(16)
Yt = y0 +
t
σ (Yu ) dBu
0
without drift and diffusion coefficient σ. We assume that the
existence condition Eσ ⊆ Nσ is satisfied. In the following we
always consider the fundamental solution (Y , F) of Eq. (16):
(17)
σ 2 (Y ) > 0 on [0, DEσ (Y )) P-a.s.
Definition (Predictable Representation Property)
A (continuous) local martingale (Z , F) is said to satisfy the
predictable representation property (PRP)if any local martingale
(M, F) can be represented as stochastic integral
Z t
Mt = M0 +
Hu dZu , t ≥ 0 ,
0
where H is an F-predictable process.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
The Predictable Representation Property
Theorem (Integral Representation)
Let (Z , F) be a continuous local martingale, A = hZ , Z i and
Tt = inf {s ≥ 0 : As > t} , t ≥ 0 .
Suppose that there is a Brownian motion (W , G) such that
Z = W ◦ A and (Tt ) is FW -adapted. Then Z , FZ satisfies the
(PRP).
Proof. This is a simplified version of a theorem proven in E.&
Hess (1980). We give a sketch of the proof.
Lemma (Exercise)
Under the assumptions of the theorem we have
W
FTZt = Ft∧A
, t ≥ 0.
∞
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Proof of the Theorem
1) Let M, F Z be a uniformly integrable martingale:
h
i
Mt = E M∞ |FtZ , t ≥ 0 .
By Doob’s optional sampling theorem we get
h
i
h
i
W
MTt = E M∞ |FTZt = E M∞ |Ft∧A
, t ≥ 0,
∞
the last equality
being true by the Lemma. Hence
W
M ◦ T , F·∧A∞ is a martingale. It is well-known that W , FW
possesses the (PRP). Since A∞ is a (predictable) stopping time
of FW this yields that W A∞ , FW
·∧A∞ also possesses the (PRP).
W
Hence there is a Ft∧A
-predictable
process G such that
∞
Z
MTt = M0 +
t
0
Hans-Jürgen Engelbert, FSU Jena, Germany
Gu dWuA∞ , t ≥ 0 ,
Lectures on One-dimensional SDEs
Proof of the Theorem
and by time change (M is constant on [t, TAt ])
Z
Mt = MTAt
At
= M0 +
0
Z
t
= M0 +
0
Z
= M0 +
Gu dWuA∞
(G ◦ A)u dWAAu∞
t
Hu dZu ,
0
proving the (PRP).
2) Let M, FZ be a general martingale. Using Doob’s
inequality, by standard approximation it can be shown that M is
continuous. The integral representation is then obtained by
localization.
3) The case of general local martingales is treated by
localization.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Predictable Representation Property
Theorem (PRP)
Let (Y ,F) be the fundamental solution of Eq. (16). Then
Y , FY possesses the (PRP).
Proof. Let A = hY , Y i and Tt = inf {s ≥ 0 : As > t}, t ≥ 0.
Consider a Dambis–Dubins–Schwarz Brownian motion (W , G):
Yt = WAt , t ≥ 0 .
We have proven in Section 3 (Theorem on the uniqueness in
law of the fundamental solution):
(Tt ) is FW -adapted .
Hence the assumptions of the Theorem on the Integral
Representation are satisfied and the proof is finished.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
11. Financial Markets: An Example
Let us now consider a financial market on [0, T ] (trading period)
with one risky asset (stock) S and a bank account C. For
simplicity we assume that C ≡ 1. Suppose that the stock price
S follows a solution of an SDE with ordinary drift
Z
(18)
S t = s0 +
t
Z
a (Su ) du +
0
t
b (Su ) dBu
0
(linear stock) or that S is the stochastic exponential of such a
solution:
Z
1 t 2
e
(19)
St = s0 exp St −
b (Su ) du ,
2 0
where (S, F) is a solution of the SDE (18) (geometrical stock).
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
No-Arbitrage
Problem 1: Is the market arbitrage free?
It is well-known that this is equivalent to asking for a probability
measure Q which is equivalent to P on FT such that (S, F) is a
local martingale w.r.t. Q. (Q is then called an equivalent
martingale measure.)
Q is an equivalent martingale measure for the linear stock price
if and only if it is one for the geometrical stock price: Use Itô’s
formula to verify this fact.
Henceforth we shall consider linear stock prices. To construct
an equivalent martingale measure Q, we have to remove the
drift by Girsanov’s theorem. Define
(20)
Z t
Z
a (Su )
1 t a2 (Su )
du , t ∈ [0, T ] ,
Zt = exp −
dBu −
2 0 b2 (Su )
0 b (Su )
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Equivalent Martingale Measures
T
a2 (Su )
du must be finite for defining the
2
0 b (Su )
stochastic exponential above.
Z
Note that
This can be ensured by the local square integrability of
Then (Z , F) is a continuous local martingale .
a2
.
b2
Question: Is (Z , F) a true martingale?
For this, necessary and sufficient conditions on the coefficients
a and b can be given. We will not go into further details here.
Suppose that (Z , F) is a martingale. Then we can introduce the
probability measure
Q by dQ = ZT dP. By Girsanov’s theorem
e
we obtain that B, F with
Z t
a (Su )
e
Bt = Bt +
du , t ∈ [0, T ] ,
0 b (Su )
is a Brownian motion w.r.t. Q.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Equivalent Martingale Measures
Hence
Z
St
t
= s0 +
Z
t
a (Su ) du +
0
Z
b (Su ) dBu
0
t
= s0 +
Z
Z
= s0 +
Z
eu −
b (Su ) dB
a (Su ) du +
0
t
0
t
a (Su ) du
0
t
eu ,
b (Su ) dB
0
which is a continuous local martingale w.r.t. Q. Obviously, Q is
equivalent to P.
As a result, Problem 1 can be completely solved by giving
necessary and sufficient conditions on the coefficients a and b.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Complete Markets
Problem 2: Is the financial market complete?
In other words, can every (integrable) contingent claim, say, C
be hedged? As C being an integrable FTS -measurable random
variable, is there an (FS -predictable) trading strategy H such
that
Z
T
C = P-
Hu dSu ?
0
Since the Itô integral does not change by the passage to an
equivalent martingale measure Q it is equivalent to
Z
(21)
C = Q-
T
Hu dSu
0
where Q is the equivalent martingale measure constructed
above.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Complete Markets
Definition (Martingale Hedge)
H is called a martingale hedge if
(22)
h
EQ C|FtS
i
Z
= Q-
t
Hu dSu .
0
(Note that the left hand side is a martingale because C is
integrable.) To find a martingale hedge for every integrable
to the (PRP) of the continuous
contingent claim C is equivalent
local martingale S, FS under Q.
Theorem (Completeness)
Let the stock price (S, F) be a fundamental solution of the SDE
(18). Suppose that there exists an equivalent martingale
measure Q. Then the market is complete. Moreover, for any
integrable contingent claim C, there exists a martingale hedge
for C.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Uniqueness of Q
Proof. This follows from the (PRP) of S, FS under the
equivalent martingale measure Q proven in the last section and
from the discussion above.
Theorem (Uniqueness)
Let (S, F) be a fundamental solution of the SDE (18). There is
at most one equivalent martingale measure Q.
Proof. Let Q and Q1 be equivalent martingale measures on
FTS . Then Q ∼ Q1 on FTS . Let
Zt =
dQ1 S Ft , t ∈ [0, T ] .
dQ
Then Z , FS is a Q-martingale. Note that under Q the
continuous local martingale S, FS is a fundamental solution of
the SDE (4) without drift and by the last section it possesses the
(PRP). Hence Z is continuous and therefore locally bounded.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Uniqueness of Q
Moreover, S is a Q1 -local
martingale which is equivalent to the
S
property that SZ , F is a Q-martingale. Thus S and Z are
strongly orthogonal locally bounded local martingales which
implies hS, Z i ≡ 0. But Z has the representation
Z
Zt = 1 +
t
Z
Hu dSu = 1 +
0
t
Hu b (Su ) dBu ,
0
by the (PRP) from which follows
Z
0 = hS, Z it =
t
Hu b2 (Su ) du , ∀ t ∈ [0, T ] .
0
This implies H = 0 on 0, T ∧ DEb (S) P-a.s. and therefore
Z = 1 on this set. This means that Q1 and Q coincide on
FTS∧DE (S) but the latter σ-field is equal to FTS , proving that Q1
b
and Q coincide on FTS .
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
12. On the Yamada–Watanabe Theorem
The most powerful criterion for the existence of strong solutions
is the Yamada–Watanabe theorem. As an example, we shall
consider the SDE
Z
Z t
X
(23)
X t = x0 +
L (t, y ) ν (dy ) +
b (Xu ) dBu
R
0
with generalized drift. However, as a general principle, the
Yamada–Watanabe theorem applies for many types of SDEs
(including multidimensional ones) driven by a Brownian motion.
Theorem (Yamada–Watanabe)
(i) The pathwise uniqueness of the solution of Eq. (23) implies
uniqueness in law.
(i) If the pathwise uniqueness of the solution of Eq. (23) holds
and if, for every starting point x0 , there exists a (weak) solution
of Eq. (23) then, for every starting point, there exists a pathwise
unique strong solution of Eq. (23).
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Joint Uniqueness in Law
Definition
The solution of Eq. (23) is called jointly unique in law if, for
given starting point x0 , the distribution of (X , B) is the same for
any solution pair (X , B).
Obviously, we have:
Joint Uniqueness in Law ((JUiL)) =⇒ Uniqueness in Law
((UiL)).
Remark
Pathwise uniqueness implies (JUiL). This was actually proven
by Yamada–Watanabe but not explicitly stated in their theorem.
We now a come to an extension of the Yamada–Watanabe
theorem.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Extension of the Yamada–Watanabe Theorem
Theorem
The following conditions (i) and (ii) are equivalent:
(i) (a) For every starting point x0 , there exists a solution.
(b) The solution is pathwise unique.
(ii) (a) For every starting point x0 , there exists a strong solution.
(b) The solution is jointly unique in law.
If one (and therefore both) of these conditions is satisfied then
every solution of Eq. (23) is strong.
Proof. The implication (i)⇒(ii) is just the Yamada–Watanabe
theorem.
(ii)⇒(i) Of course, (i)(a) is satisfied.
We haveto show (i)(b):
pathwise uniqueness. Let X 1 , F and X 2 , F defined on the
same probability space with the same Brownian motion (B, F)
such that X01 = X02 = x0 .
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
The Proof
e, F
e0 = x0
e with X
In view of (ii)(a), there is a strong solution X
e F
e on some probability space.
and driving Brownian motion B,
Hence there is a functional F such that
e P-a.s.
e =F B
X
We define X = F (B). Then it can be easily verified that (X , F)
is a strong solution with X0 = x0 and with driving Brownian
motion (B, F). Moreover, by (ii)(b), the distribution of (X , B)
coincides with that of X 1 , B . By the construction of X , the
conditional distribution PX (·|W ) is equal to δF (B) . But by (ii)(b),
the distribution of (X , B) coincides with that of X 1 , B .
Therefore,
PX 1 (·|W ) = PX (·|W ) = δF (B) .
Obviously, this implies X 1 = F (B) P-a.s.The same holds for
X 2 and hence X 1 = X 2 P-a.s.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Equivalence of Conditions
Symbolically, we can state the following equivalence:
Weak Existence
&
Pathwise Uniqueness
⇐⇒
Strong Existence
&
Joint Uniqueness in Law
Now we come to the problem of non-existence of strong
solutions.
Corollary (No Strong Solution)
Let be given a “general" SDE. Suppose that we have:
(i) Weak existence.
(ii) Joint Uniqueness in Law.
(iii) No pathwise uniqueness.
Then there does not exist a strong solution.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
No Strong Solutions
Assumption: (A) Eb = Nb = ∅
Proposition
The solution (X , F) of Eq. (23) is jointly unique in law.
Proof. We know that the solution (X , F) is unique in law if
Eb = Nb . On the other side,
Z t
Z t
Z
−1
−1
Bt =
b (Xu ) dXu −
b (Xu ) du
LX (u, y ) ν (dy )
0
0
R
from which follows that B and hence (X , B) is a
well-determined measurable functional of X . Hence the law of
(X , B) is unique.
Remark
Cherny (2003) proved that for general, in particular, multidimensional
SDEs with ordinary drift uniqueness in law is equivalent to joint
uniqueness in law.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
No Strong Solutions
Let us assume for simplicity X0 = 0.
Theorem (No Strong Existence)
Suppose that Assumption (A) is satisfied and ν ({x}) = 0 for all
x ∈ R. Furthermore, assume that b and ν are antisymmetric:
b (x) = −b (−x) , λ-a.e., ν(A) = −ν(−A), A ∈ B ([−N, N]) , N ≥ 1 .
Then there does not exist a strong solution.
Proof. We know that there is a unique solution (X , F) and by
the above Proposition it is jointy unique in law. Moreover,
pathwise uniqueness fails. With (X , F), also (−X , F) is a
solution:
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
No Strong Solutions
Z
−Xt
t
Z
X
= − L (t, y ) ν (dy ) −
b (Xu ) dBu
R
0
Z
Z t
X
=
L (t, y ) (−ν) (dy ) +
(−b) (Xu ) dBu
R
0
Z t
Z
b (−Xu ) dBu
LX (t, −y ) ν (dy ) +
=
{z }
0
R|
=L−X (t,y )
Z
−X
L
=
Z
(t, y ) ν (dy ) +
R
t
b (−Xu ) dBu ,
0
where we have used that ν is atomless which implies that
LX (t, y ) is continuous in y . Apply the Corollary about no strong
existence to conclude that there is no strong solution.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Tanaka’s Example
Example (Tanaka)
Let ν = 0 and b = sgn− . Then the SDE
Z
t
Xt =
0
sgn− (Xu ) dBu
has a jointly unique solution. Pathwise uniqueness fails
because (−X , F) is a solution whenever (X , F) is a solution.
Hence there does not exist a strong solution.
On Skew Brownian Motion
Consider
Xt = α LX (t, 0) + Bt ,
t ≥ 0,
the SDE for skew Brownian motion with parameter α < 12 . We
already know that this SDE has a pathwise unique strong
solution (X , F).
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Tanaka’s Example
Now we modify the SDE as follows:
Z t
(24)
Xt = α LX (t, 0) +
sgn− (Xu ) dWu ,
0
|
{z
}
t ≥ 0.
=Bt
For α = 0 we get Tanaka’s Example.
Every solution (X , F) is a Skew Brownian motion with
parameter α < 12 .
Starting from an arbitrary skew Brownian motion (X , F), we
can define
Z t
Wt =
sgn− (Xu ) dXu + α LX (t, 0)
0
Z t
Z t
=
sgn− (Xu ) dBu −
α du LX (u, 0) + α LX (t, 0)
0
0
Z t
=
sgn− (Xu ) dBu ,
0
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Skew Brownian Motion
which is a Brownian motion and (X , W ) satisfies Eq. (24):
Z t
sgn− (Xu ) dWu
0
Z t
Z t
=
sgn− (Xu ) sgn− (Xu ) dXu +
sgn− (Xu ) du αLX (u, 0)
0
0
X
= Xt − α L (t, 0) .
Hence there exists a unique solution (X , F) of Eq. (24).
From the above Proposition: (X , F) is jointly unique in law.
However: There does not exist a strong solution.
Proof. Integrating sgn− (Xu ) by X we observe
Z t
Z t
sgn− (Xu ) dXu = α
sgn− (Xu ) du LX (ds, 0) + Wt
0
|0
{z
}
=|Xt |−LX (t,0)
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Skew Brownian Motion
|Xt | − LX (t, 0) = −α LX (t, 0) + Wt ,
hence
Wt = |Xt | + (α − 1) LX (t, 0) = |Xt | −
1 |X |
L (t, 0) .
2
The last equality is true in view of
Z t
X
X
L (t, 0)−L (t, 0−) = 2
1{0} (Xu ) α du LX (u, 0) = 2 αLX (t, 0)
0
which yields
=(1−2α) LX (t,0)
z }| {
L|X | (t, 0) = LX (t, 0) + LX (t, 0−)
= LX (t, 0) + (1 − 2α) LX (t, 0) = 2 (1 − α) LX (t, 0) ,
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Skew Brownian Motion
hence
(α − 1) LX (t, 0) =
α−1
1
L|X | (t, 0) = − L|X | (t, 0) .
2 (1 − α)
2
As a result,
|X |
FtW ⊆ Ft
⊂ FtX .
This shows that (X , F) cannot be a strong solution.
Space Transformation
If Eq. (23) does not possess a strong solution and G is a space
transformation which is one-to-one and a difference of convex
functions as its inverse H then the SDE transforms into a new
SDE which also does not possess a strong solution.
This is a general principle which will now be illustrated by an
example.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Space Transformation
Let (X , F) be a solution of Eq. (24) and set ν = α δ0 . As we
already know, the corresponding solution g of the integral
equation

 1 − 2R
[0,x] g (y −) ν (dy ) for x ≥ 0 ,
g (x) =
R
 1+2
g (y −) ν (dy ) for x < 0 ,
(x,0)
is

 exp (−2ν ([0, x])) Q
0≤y ≤x (1 − 2ν ({y })) exp (2ν ({y }))
g (x) =
Q
 exp (2ν ((x, 0)))
(1 − 2ν ({y }))−1 exp (−2ν ({y }))
x<y <0
which yields

 1,
x < 0,
g (x) =
 1 − 2α, x ≥ 0 .
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Space Transformation
and the primitive G of g

 d x, x < 0,
G (x) =
 c x, x ≥ 0 .
From earlier sections we know that Y = G (X ) is a solution of
the SDE without drift
Z t
(25)
Yt =
σ (Yu ) dBu
0
where σ = (g ◦ H) sgn− ◦ H which is calculated as

 −1,
x < 0,
σ (x) =
 1 − 2α, x ≥ 0 .
The same procedure is possible if we multiply g by d > 0 and
set c := d (1 − 2α).
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Space Transformation
Then the space transformation yields a solution (Y , F) of SDE
(25) without drift, now with σ equal to Jump to second slide
(26)

 −d, x < 0,
σ (x) =
 c, x ≥ 0 .
where c, d > 0 are arbitrarily given positive constants. As the
transformation works also in the converse direction, we arrive at
Theorem (No Strong Solution)
The SDE (25) with σ given by (26) has no strong solution.
This result was proven by Barlow using other methods.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Space Transformation
We can extend this theorem and the result for skew Brownian
motion in the following way. We consider the SDE
Z t
X
(27)
Xt = α L (t, 0) +
sgn− (Xu ) b (Xu ) dWu , t ≥ 0 .
0
where α < 21 and b is an even function (i.e., b (x) = b (|x|))
such that Eb = Nb = ∅ is satisfied. We also consider the SDE
without drift
Z t
(28)
Yt =
gc,d (Yu ) σ (Yu ) dBu
0
where gc,d = −d 1(−∞,0) + c 1[0,+∞) for arbitrary c, d > 0 and
σ is an even function such that Eσ = Nσ = ∅ is satisfied.
Theorem (No Strong Solution)
There is no strong solution (X , F) of Eq. (27).
There is no strong solution (Y , F) of Eq. (28).
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
13. Financial Bubbles
E NGELBERT–S ENF (1990)
We consider the one-dimensional SDE
Z t
(29)
Yt =
σ (Yu ) dBu , t ≥ 0 ,
Y0 = 0 ,
0
without drift where (B, F) is a Brownian motion and σ some
real Borel function. We recall
Z x+ε
Eσ = {x ∈ R :
σ −2 (y ) dy = ∞, ∀ε > 0}
x−ε
and assume (for simplicity) Eσ ⊆ Nσ . Then there exists a
fundamental solution (Y , F), i.e., a solution satisfying
Z DE
b
1{σ=0} (Yu ) du = 0 P-a.s. ,
0
where DEb is the first entry time of Y into Eσ . Note that every
x ∈ Eσ is an absorbing point for Y .
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Financial Bubbles
(Y , F) being a solution of the SDE (29), we observe
Rt
At = 0 σ 2 (Yu ) du, t ∈ [0, ∞] ,
and there is a Dambis–Dubins–Schwarz Brownian motion
(W , G) such that for the right inverse (Tt ) of A we have
Rt
Tt = 0 σ −2 (Wu ) du, t ∈ [0, ∞] .
The Brownian motion W and the solution Y can be given on the
space (C, C) of continuous functions.
We are interested in the exponential local martingale
Z
1 t 2
(30)
St = E(Y )t := exp Yt −
σ (Yu ) du
2 0
which we interprete as the stock price on a financial market
(assuming bank account ≡ 1) under the unique (local)
martingale measure, the given P.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Financial Bubbles
Problem: Is (S, F) a true martingale or not? In the latter case,
S is called a financial bubble.
Let Q be the distribution of the Brownian motion with drift on
(C, C).
Theorem
(S, F) is a true martingale if and only if
Q ({At < ∞ ∀ t ≥ 0}) = 1.
Remark
The condition of the theorem is equivalent to
Q ({T∞ = ∞}) = 1.
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Financial Bubbles
Hence we have to investigate the behaviour of
Z ∞
T∞ =
σ −2 (Wu ) du
0
for a Brownian motion (W , G) with drift.
Integral functionals of a Brownian motion with drift have
been studied well in the past.
The result is
Z
∞
Q({T∞ = ∞}) = 1 ⇐⇒
σ −2 (x) dx = ∞
∀ε > 0 .
−ε
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs
Financial Bubbles
Summarizing we obtain the following purely analytical criterion.
Theorem
The stochastic exponential (S = E(Y ), F) associated with the
solution of SDE (29)
Z is a martingale if and only if
∞
σ −2 (x) dx = ∞
∀ε > 0 .
−ε
Corollary
(E(X ), F) is a strict local martingale (what is called bubble in
mathematical finance) if and
Z only if
∞
∃ε > 0 :
σ −2 (x) dx < ∞ .
−ε
Hans-Jürgen Engelbert, FSU Jena, Germany
Lectures on One-dimensional SDEs