EXPLOSIONS AND ARBITRAGE IOANNIS KARATZAS

59 Pages·2015·0.37 MB·English

EXPLOSIONS AND ARBITRAGE
IOANNIS KARATZAS
Department of Mathematics, Columbia University, New York
INTECH Investment Management LLC, Princeton
Joint work with Daniel FERNHOLZ and Johannes RUF
Talk at the Steven-Shreve-Fest, CMU
Pittsburgh, June 2015
For Information Purposes Only
PART ONE: A CLASSICAL SETTING
DISTRIBUTION OF THE TIME-TO-EXPLOSION FOR
LINEAR DIFFUSIONS
I.1: STOCHASTIC DIFFERENTIAL EQUATION
dX(t) = s(X(t)) dW(t) + b(X(t))dt , X(0) = ⇠
2I
h i
The state-space is an open subinterval = (`, r)
R
I ✓
of the real line. Here W( ) is standard Brownian motion,
·
and b : , s : 0 are measurable functions.
R R
I! I! \ { }
2
Standing Assumption: The function 1/s ( ) and the local
·
mean/over/variance (or “signal-to-noise ratio”) function
b( ) b( ) s( )
f( ) := · = · ·
· s( ) s2( )
· ·
are locally integrable over .
I
. Under these conditions, there exists a weak solution of the
above SDE, defined up until the so-called “explosion time”
:= lim , = inf t 0 : X(t) / (` , r )
n n n n
S n "S S { � 2 }
!1
for ` ` , r r . This solution is unique in distribution.
n n
## ""
(ENGELBERT & SCHMIDT 1984, 1991.)
We know that ( = ) = 1 holds under the familiar
P
S 1
ˆ
linear growth conditions of the ITO theory, when = .
R
I
More generally, fixing a reference point c and introducing
2I
the “FELLER function”
x y y dz
v(x) := exp 2 f(u)du dy , x ,
c c � z s2(z) 2I
Z Z ✓ Z ◆
we have: ( = ) = 1 if and only if
P
S 1
v(`+) = v(r ) = .
� 1
This is the classical FELLER test for explosions.
QUESTION (posed to us by Marc YOR):
. If this condition fails and ( < ) > 0 ,
P
S 1
what can we say about the distribution function
( T), 0 < T < of the explosion time?
P
S 1
I.2: A GENERALIZED GIRSANOV / McKEAN IDENTITY
Let us consider the di↵usion in natural scale
o o o
dX (t) = s(X (t)) dW (t) , X(0) = ⇠
2I
o o
with explosion time ; clearly, ( = ) = 1 if = .
Q R
S S 1 I
o
Here W ( ) is Brownian motion under another probability
·
measure (possibly on a di↵erent probability space).
Q
Suppose that the mean/variance function f( ) is locally square-
·
integrable on , and define the exponential local martingale
Q
I �
1
L( ; Xo) := exp · b(Xo(t)) dWo(t) · b2(Xo(t)) dt
· 0 � 2 0
⇢Z Z �
1
= exp · f(Xo(t)) dXo(t) · b2(Xo(t)) dt on [0, o).
0 � 2 0 S
⇢Z Z �
Then for T (0, ) and bounded, measurable h :⌦ ,
R
T T
2 1 B � !
P h (X) 1 = Q L(T; Xo) h (Xo) 1 .
E T >T E T o>T
· ·
{S } {S }
h i h i
A couple of early lessons from this identity. Suppose X( ) is
·
non-explosive: ( = ) = 1 .
P
S 1
Then
P h (X) = Q L(T; Xo) h (Xo) 1 .
E T E T o>T
·
{S }
h i h i
o
In particular, the exponential process L( ; X ) 1 is then a
o>
·
{S ·}
true martingale; and for every T (0, ) we have
Q
� 2 1
1
P = o > T
E Q
L(T; X) S
!
⇣ ⌘
d
P
o
“ = L(T; X ) 1 . ”
o>T
d ·
Q � {S }
(T)
�
�F
�
�
. Please also note that, always under ( = ) = 1 , the expo-
P
S 1
nential process
1 1
2
= exp · f(X(t)) dX(t) + · b (X(t)) dt
L( ; X) � 0 2 0
⇢ Z Z �
·
1
2
= exp · b(X(t)) dW(t) · b (X(t)) dt
� 0 � 2 0
⇢ Z Z �
is a strictly positive local martingale (and supermartingale).
P
�
. It is a true martingale, if and only if we have, in addition,
P
�
o
( = ) = 1 .
Q
S 1
. When f( ) is actually continuous and continuously di↵eren-
·
tiable on , the above expression gives
I
o
X (T) T
( > T) = Q exp f(z) dz V (Xo(t))dt 1
P⇠ E o>T
S � ·
" ⇠ 0 {S }#
✓ Z Z ◆
where
1
2 2
V (x) := s (x) f (x) + f (x) .
0
2
⇣ ⌘
. And in a totally “symmetrical” fashion:
X(T) T
( o > T) = P exp f(z) dz + V (X(t))dt 1 .
Q E
⇠ >T
S � ·
" ⇠ 0 {S }#
✓ Z Z ◆
I.3: RESULTS: We have the following, general results.
PROPOSITION 1: Positivity, Full Support. The function
[0, ) (T,⇠) U(T,⇠) := ( > T) (0, 1]
P
⇠
1 ⇥I 3 7�! S 2
is (strictly positive and) continuous;
as well as strictly decreasing in T ( ), when ( < ) > 0 .
P
⇠
⇤ ⇤ ⇤ S 1
(***) Last result – strict decrease – needs the
2
local square-integrability of 1/s ( ) on
· I
(with the possible exception of finitely many points).
This assumption guarantees that “the di↵usion can
reach far away points fast, with positive probability”.
. It has been removed very recently, in work of
Cameron BRUGGEMAN and Johannes RUF.