Download Optimal Stopping and Free-Boundary Problems Series

Some
Optimal
PESKIR GORAN, SHIRYAEV
ALBERT
Optimal Stopping and FreeBoundary Problems
Series: Lectures in Mathematics.
ETH Zürich
2006, XXII + 500 pages
ISBN-10: 0-8176-4434-2
ISBN-13: 978-0-8176-4434-5
Birkhäuser Verlag
Boston. Basel. Berlin
www.birkhauser.ch
stopping theory has its basic definitions
roots in classical calculus and
properties
of variations in some are
reviewed
problems formulated by about
Lagrange,
Bolza.
Mayer
Some
and martingales and
optimal related processes:
stopping
problems supermartingales,
emerged from work by submartingales,
Wald
in
relation
to local martingales
problems of sequential and
Markov
testing and represent a processes. Some
method
of
inference
statistical fundamental
(sequential theorems
probability
ratio
are
test) recalled:
the
where the number of optimal sampling
observations
is
not theorem,
determined in advance of martingale
the
experiment.
fundamental
A convergence
connection theorem
and
between optimal stopping maximal
and
free
boundary inequalities. The
problems was discovered stochastic
by
a
number
of integral and Ito’s
researchers (among them formula
is
is
Shiryaev) and this presented.
equivalence
deeply
The
penetrates solutions of some
into
the classical
fundamentals of modern problems of the
probability
their
theory
connection
analysis
and potential
with as
theory
Dirichlet
through problem for the
boundary value problems. Poisson equation,
Studies
of martingales lead to Markov various
problems
and Markov processes are chains and give a sequential
central
to
optimal probabilistic
stopping problems and method
stopping
and other
times
are for Markov processes and the
Markov
chains, by means of the smallest for
theoretical fields as: the processes
superharmonic
and function.
theory of probability for Levy processes.
mathematical
of The
of
statistics important
finance
equilibria
mathematical as:
for
a
Markov
function process, with the
the
gain value
V
is
equivalent to the
A huge stimulus problem
most to the development of finding
for sequential statistics transformations
and
optimal
characterized the solution stopping problem
classical Brownian
inequalities,
and free-boundary
formulated first chapter it
problems from important motion, diffusion dominating
sharp
and
Markov optima stopping problems was proved that
anticipation) are key to reviewed:
Some
stopping
Some Dynkin
(non- processes
real world applications.
analysis, stopping
of stochastic control. And problems. In the
the concepts of filtration solving.
(information)
optimal
in between optimal
of
the
optimal stopping theory smallest
was provided by option superharmonic
of pricing theory, developed function V
,
stochastic time, change of in the late 1960s and the which dominates
lead
to
change
an space, change of 1970s. According to the the
gain
optimal
stopping measure
are modern theory of finance,
function G .
problem, which can be considered
are pricing an American
This
reduced to a free- defined and these option in a complete
leads
to
boundary
problem. are used in order market is equivalent to
differential
or
Therefore the principal to
obtain solving
an
optimal
integroproblems which arise are solutions in a stopping problem.
differential
to reformulate these closed
form,
The authors, A.
equations which
reductions and to find the solving optimal N. Shiryaev, the disciple
the function V
solution of the initial stopping
of A. N. Kolmogorov,
solves.
problems.
problems.
one of the greatest
Some
In some previous
In
probabilists of all time, is
methods
of
papers,
Mikhalevich, Chapter III it is an eminent authority on
solutions
are
considered free-boundary described
the stochastic processes. He
presented in the
problems for solving connection
is
a
Russian
Chapter
IV
mathematician,
for
his
known Reduction
work
probability
statistics
to any n and it is a stopping The problem is to
time if    , P-a.s. A determine which
in free-boundary
theory, problem,
and
financial superharmonic
mathematics.
at
method
of
the
optimal of the given two
stopping problem is to values
Goran characterization,
Peskir is a well known the
professor
form
compute the value
V*  sup EG
of
a
true
mean is. In the
Bayesian
the time change, the and to exhibit an optimal formulation it is
of method of space stopping time  , where assumed that the
University
Manchester.
change,
the E is the expectation.
unknown
of
a
A first method is has
The first Chapter method
mean
given
stopping: measure change. to
use
backward distribution and
General facts offer a Chapter V is induction in order to in this chapter a
survey of basic results devoted to some construct a sequence of solution of the
Optimal
concerning
optimal stochastic
stopping in the case of analysis aspects
discrete time, considering which are closely
a
martingale
and
a related to sharp
Markov approach.
inequalities
as;
It is considered a those of Wald,
sequence
of
random Bessel,
variables,
denoted
Doob,
by Hardy-
Gn , n  N defined
filtered
on a Littlewood.
One
probability
space (, F , Fn , P), n  N
.
of
the objectives of
the chapter VI is
G
n
The variable
is the study of the
interpreted as the gain, if problem
of
the observation is stopped sequential testing
at time t and
Fn is the of
two
that problem for a
solves in a stochastic Wiener process
finite
sense, but in the case of with
horizon is made.
finite horizon N , which
A solution of the
choose the optimal time
Bayesian
and
among
those
which
variational
belong
to
the
problem for a
N
class M n  { | n    N }
Wiener process
. This method led to the
with
finite
general
principle
of
horizon is also
dynamic
programming
derived.
The
(the Bellman’s principle).
same problems
The method of
are
studied,
essential
supremum
replacing
the
solves the problem in the
Wiener processes
case of infinite horizon N by
Poisson
random
simple
information available up hypotheses about
.
to the time n. A random the mean value
variables
processes.
In the subsection
Some
variable  is a Markov of an observed
approach concrete
Wiener process. Markovian
time if {  n}  Fn , for
some basic results of problems
of
based on studies
optimal stopping when mathematical
of
the time is discrete and finance can be
and on modern
the process is Markovian. reformulated
as
and
For
of
research of some
the
case
continuous
of problems
time,
a optimal stopping
martingale approach is of
given.
stochastic
processes
Markov
kind
of
process,
authors
relevant
prestigious
mathematicians.
and
A Markov time is a solved
particular
the
by
a reduction to freefor boundary
The
book
addresses
to those readers
with a high level
which the strong Markov problems of real
of mathematical
property
information and
remains analysis.
These
preserved. Therefore a kinds
of
brief review of stochastic situations
processes is
Chapter II.
are
made in studied
in
the
understanding
needs
a
good
chapter VII and
background
VIII.
probability
Since
the
and
stochastic
the gain process
in
in
processes.
optimal
The
stopping problem
origin
depends on the
concepts is in the
future,
study of financial
an
of
the
optimal
engineering and
prediction
mathematical
problem appears
statistics
and
which is of a
problems
of
particular interest
general
in
can be solved in
financial
engineering.
an abstract, but
The
monograph
interest
very
is
manner.
general
The
book
may
be
as
an
viewed
ideal
compendium for
an
interested
reader
who
wishes to master
stochastic
calculus
via
fundamental
examples. Areas
of
application
where examples
are worked out in
full detail include
financial
mathematics,
financial
engineering,
mathematical
statistics,
and
stochastic
analysis.
Associated
Professor
Liliana Popa
Department of
Mathematics
"Gheorghe
Asachi"
Technical
University of
Iasi