Cherny A., Englebert H. - Singular Stochastic Differential Equations.pdf

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Lecture Notes in Mathematics
1858
Editors:
J.--M. Morel, Cachan
F. Takens, Groningen
B. Teissier, Paris
Alexander S. Cherny
Hans-Jurgen Engelbert
Singular Stochastic
Differential Equations
123
Authors
Alexander S. Cherny
Department of Probability Theory
Faculty of Mechanics and Mathematics
Moscow State University
Leninskie Gory
119992
, Moscow
Russia
e-mail: cherny@mech.math.msu.su
Hans-Jurgen Engelbert
Institut fur Stochastik
Fakultat fur Mathematik und Informatik
Friedrich-Schiller-Universitat Jena
Ernst-Abbe-Platz
1-4
Jena
Germany
e-mail: engelbert@minet.uni-jena.de
LibraryofCongressControlNumber: 2004115716
Mathematics Subject Classification (2000):
60-02, 60G17, 60H10, 60J25, 60J60
ISSN
0075-8434
ISBN
3-540-24007-1
Springer Berlin Heidelberg New York
DOI:
10.1007
/b
104187
,
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9
,
1965
Springer-Verlag Berlin Heidelberg
2005
Printed in Germany
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Preface
We consider one-dimensional homogeneous stochastic differential equations
of the form
dX t = b ( X t ) dt + σ ( X t ) dB t , X 0 = x 0 ,
∗
)
=0.
There is a rich theory studying the existence and the uniqueness of solu-
tions of these (and more general) stochastic differential equations. For equa-
tions of the form (
∗
), one of the best sucient conditions is that the function
) /σ 2 should be locally integrable on the real line. However, both in
theory and in practice one often comes across equations that do not satisfy
this condition. The use of such equations is necessary, in particular, if we want
a solution to be positive. In this monograph, these equations are called sin-
gular stochastic differential equations . A typical example of such an equation
is the stochastic differential equation for a geometric Brownian motion.
Apoint d
b
|
) /σ 2 is not locally integrable,
is called in this monograph a singular point. We explain why these points
are indeed “singular”. For the isolated singular points , we perform a complete
qualitative classification. According to this classification, an isolated singular
point can have one of 48 possible types. The type of a point is easily computed
through the coecients b and σ . The classification allows one to find out
whether a solution can leave an isolated singular point, whether it can reach
this point, whether it can be extended after having reached this point, and
so on.
It turns out that the isolated singular points of 44 types do not disturb
the uniqueness of a solution and only the isolated singular points of the
remaining 4 types disturb uniqueness. These points are called here the branch
points . There exists a large amount of “bad” solutions (for instance, non-
Markov solutions) in the neighbourhood of a branch point. Discovering the
branch points is one of the most interesting consequences of the constructed
classification.
The monograph also includes an overview of the basic definitions and facts
related to the stochastic differential equations (different types of existence and
uniqueness, martingale problems, solutions up to a random time, etc.) as well
as a number of important examples.
We gratefully acknowledge financial support by the DAAD and by the
European Community’s Human Potential Programme under contract HPRN-
CT-2002-00281.
∈ R
, at which the function (1 +
b
|
Moscow, Jena,
Alexander Cherny
October 2004
Hans-Jurgen Engelbert
(
where b and σ are supposed to be measurable functions and σ
(1 +
|
|

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