Download stohasti^ke diferencijalne jedna^ine koje zavise od malih

STOHASTIČKE DIFERENCIJALNE JEDNAČINE
KOJE ZAVISE OD MALIH PARAMETARA
STOCHASTIC DIFFERENTIAL EQUATIONS
DEPENDING ON SMALL PARAMETERS
Svetlana Janković and Miljana Jovanović
Faculty of Science, Department of Mathematics, University of Niš,
Ćirila i Metodija 2, 18000 Niš, Yugoslavia
Rezime: Ispituje se bliskost u (2k)-tom smislu između rešenja perturbovane stohastičke diferencijalne jednačine tipa
Itoa koje zavisi od malih parametara i rešenja odgovarajuće neperturbovane jednačine, na konačnim vremenskim
intervalima ili na intervalima čija dužina teži beskonačnosti kada mali parametri teže nuli .
KLJUČNE REČI: STOHASTIČKE DIFERENCIJALNE JEDNAČINE, PARAMETARSKE PERTURBACIJE,
BLISKOST U (2K)-TOM SMISLU
Abstract: We investigate the (2k)-th mean closeness between the solution of the perturbed stochastic differential
equation of the Itô type depending on small parameters and the solution of the corresponding unperturbed one, on
finite time-intervals or on intervals whose length tends to infinity as small parameters tend to zero.
KEY WORDS: STOHASTIC DIFERENTIAL EQUATIONS, PARAMETRIC PERTURBATIONS, CLOSENESS IN
THE (2K)-TH MEAN.
In many problems in almost all areas of science and
engineering there are real phenomena depending on the
effect of "white noise" random forces and on deterministic
and stochastic perturbations. Note that white noise is, at
least, a tolerable abstraction and is never a completely
faithful representation of a physical noise source. Specially,
during the last years stimulating research have been
undertaken in the field of descriptions of real systems
subjected to random excitations of a Gaussian white noise
type. Having in view that a Gaussian white noise is
mathematically described as a formal derivative of a
Brownian motion process, all such problems are essentially
based on stochastic differential equations of the Itô type [3].
The theory of stochastic differential equations of the Itô
type had a permanent development with a large number of
innovations. However, the Itô calculus remains essential
because several phenomena in technical, biological, social
sciences, and recently in economics (see [1], [2], [5], [7],
for example), can be modeled and described by this theory,
which proves the flexibility of its applications. We state the
following simple examples in connection with financial
mathematics:
(i) (B. Oksendal, [6, 1992]) The simple population growth
can be modeled by ordinary differential equation
dnt
 at nt ,
dt
n(0)  n0 ,
t  0,
where nt is the size of the population at time t and at is the
relative rate of growth at time t. If at is not completely
known, but subjected to some random noise effect, i.e.
independent of wt, a(t,x) and b(t,x) are given scalar real
functions satisfying
at  rt "noise",
where rt is assumed to be non-random, and if it is not
known the exact behavior of the noise term, then instead of
the previous differential equation we shall consider the Itô
type stochastic differential equation
dnt  rt nt dt  dwt ,
t  0,
n(0)  n0 ,
(ii) (B. Oksendal, [6, 1992]) Suppose a person has an assert
or resource (e.g. a house, stocks, oil, ...) that she is planning
to sell. The price pt at time t of her assert on the open market
subjected to "noise" forces, varies according to a stochastic
differential equation of the Itô type
t  0,
p(0)  p0 ,
(iii) There are a lot of evolution models which describe the
behavior of stochastic interest rate rt, t  0, subjected to
various financial influences. For example:
The non-linear model (F. Black and P. Karasinski, [1,
1991])
t  0,
r (0)  r0 ,
where t, t, t, are non-random functions describing
financial parameters;
The non-linear model (L. Chen, [ 2 , 1995]),
drt  (t  t )dt  ( t rt )1 / 2 dwt ,
t  0,
r (0)  r0 ,
where t, t, t, are diffusion processes depending on
different independent Brownian motion processes.
All the previous examples are special cases of the scalar
stochastic differential equation of the Itô type
dxt  a(t , xt )dt  b(t , xt )dwt ,
x(0)  x0 ,
t  0,
 | a(t, x) | dt  ,  | b(t, x) |
0
dt  
2
0
x  R (under these conditions Lebesgue and Itô
integrals in (1) are well defined), and ( xt , t  [0, T] ) is a
for all
On the basis of classical theory of stochastic differential
equations of the Itô type one can prove that if the functions
a(t,x) and b(t,x) satisfy the global Lipschitz condition and
the usual linear growth condition on the last argument, and
if E | x0 |2k  , ( E | x0 |2k is a mathematical expectation of
| x0 |2k ) for any fixed integer k, then there exists a unique
solution ( xt , t  [0, T ] ) of Eq. (1), continuous with
probability one, satisfying E{supt[0,T ] | xt |2k }  .
where r,  are known non--random constants. If the
inflation rate is a known constant, the question is at what
time should a person decide to sell in order her choice of
time will turn out to be the best.
drt  rt ( t   t ln rt )dt   t rt dwt ,
T
scalar stochastic process adapted to {Ft , t  0} .
in which wt is a Brownian motion process.
dpt  rpt dt  pt dwt ,
T
(1)
in which (wt , t  0) is a scalar-valued normalized Brownian
motion with a natural filtration {Ft, t  0} (i.e. Ft is a
smallest -field on which all random variables ws, s  t are
Ft-measurable), the initial condition x0 is a random variable
In many problems the Itô's equation depends on nonrandom parameters. In the basic paper [8] the following
equation
dxt  [a( s, xt )  1 ( s, xt ,  2 )]dt
 [b( s, xt )   2 ( s, xt ,  2 )]dwt ,
t  0,
(2)
x 0 (0)  x0 0 ,
is studied, in which  0 , 1,  2 are small parameters from the
interval (0,1). Because the solution depends on them, we
adopt the shorter notational convention, introducing the
superscript  in xt and emphasizing that  also depends
on them. The initial value x0 0 , satisfying E | x0 0 |2k   , is
independent on the same Brownian motion w, and
i (t , x,  i ), i  1, 2, are given scalar real functions. In
accordance with papers [4] and [8], the functions  i are
called the perturbations, while Eq. (2) is logically called the
perturbed equation with respect to the unperturbed
equation (1).
If Eq. (2) is not effectively solvable, then, from the
theoretical point of view, and much more from the point of
view of various applications, it is important to study its
solution by comparing it, in some reasonable sense, with the
solutions of the corresponding unperturbed equation (1).
In paper [8] the following assumptions are introduced:
There exist a non-random value  0 ( 0 ) and bounded
functions 1 (t , 1 ) and  2 (t ,  2 ), such that
Therefore, for sufficiently small parameters  0 ,  1 and  2 ,
E | x0 0  x0 |2 k   0 ( 0 ),
sup | i (t , x,  i ) |  i (t ,  i ), i  1,2.
(3)
xR
Let us suppose that both equations (1) and (2) have unique
solutions xt and xt respectively, continuous with
probability one and satisfying
sup t[0,T ] E | xt |2k  , sup t[0,T ] E | xt |2k  
and let the conditions (3) be satisfied. Moreover, if the
values  ( 0 ),  1 (t ,  1 ),  2 (t ,  2 ) are small for small
 0 , 1 ,  2 , then we could expect that the solutions xt and xt
are close in the (2k)-th moment sense. In connection with
these requirements, the name small perturbations is
logically kept for the perturbations 1 (t , x,  1 ) and
the solutions xt and xt are close in the (2k)-th moment
sense on a fixed finite time-interval [0,T]. But, if the timeinterval is infinite, i.e. T   , then the previous assertion is
generally not valid. Because of that, our intention is to
construct finite time-intervals which depend on  and
whose length goes to infinity as  goes to zero, such that
the solutions xt and xt are close in the (2k)-th moment
sense on these intervals.
Theorem 2. Let the conditions of Theorem 1 be satisfied
for t  [0, ) and the functions 1 (t ,  ),  2 (t ,  ) be
bounded on [0, ) . Let us denote   max{  0 , 1,  2 } and
 i ( )  sup  i (t ,  ), i  1,2
and define
t[ 0,  )
 2 (t, x,  2 ) .
 ( )  max{ 01 / k ( ),  1 ( ),  2 2 ( )},
In paper [8] the following estimation of the (2k)-th moment
closeness for the solutions xt and xt is obtained: for all
t [0, T ],
( E | xt  xt |2 k )1 / k 
t [ 0,T ( )]
t
δ (ε 0 )  exp{Mt  2   1 ( s, 1 )ds}
1/k
0
(4)
Clearly,  ( )  0 as   0 and T ( )  0 as   0 .
0
t
t
0
s
   ( s ) exp{ Mt  2   1 (u , 1 )du}ds ,
2
Theorem 1. Let the coefficients of Eq. (2) be Lipschitz
continuous and the conditions (3) be satisfied and let
 0 (t ,  ),  1 (t ,  ),  2 (t ,  ), monotonously tend to zero
as   0 , uniformly in [0,T]. Then
sup E | xt  xt |2 k  C  ( ( )) k (1 r )  0,   0.
t[ 0,T ( )]
Note that the proof of the last assertion gives us an
important result, the rate of closeness of the solutions xt
and xt for a fixed  on the time-interval [0, T ( )].
Namely, for a given small value   0 we can determine a
limit for  ( ) ,
 ( )  ( / C)1 / k (1 r )
as the size of the small perturbations, and after that T ( ) ,
such that supt[0,T ( )] E | xt  xt |2k   .
sup E | xte  xt |2 k  0 as   0 .
t[ 0 ,T ]
The proof follows immediately from (4) because
where C and K are generic constants.
T ( ) is determined from (5), taking
KT ( )  r ln  ( ) for any number r  (0,1) and 
sufficiently small, such that  ( )  1 . Thus,
The value
where  ( s)  21 ( s, 1 )  (2k  1) 2 ( s,  2 ) and M is a
generic constant. Starting from this estimation, some special
types of perturbations are considered in paper [8]. In the
present paper, similarly to [4], we shall observe a general
case of perturbations and we shall give conditions under
which these solutions are close on fixed finite intervals or
on intervals whose length tends to infinity when the small
parameters tend to zero.
E | xt  xt |2 k  C   k ( )  e Kt ,
Then, for an arbitrary number r  (0,1) and  sufficiently
small, there exists a number T ( )  0 , determined by
r
T ( )   ln  ( ),
K
where K is a generic positive constant, such that
sup E | xt  xt |2 k  0 as   0 .
t  [0, T ],
(5)
Specially  0 , 1,  2 could be treated as small disturbances,
which effect to a financial market. Therefore, under the
preceding conditions for the perturbations, it is reasonable
to believe that the response xt behaves, approximately in
the (2k)-th moment sense, as the corresponding undisturbed
xt . Then the researcher's interest could also be concentrated
on exploring the bifurcational behavior of the solution xt
and on conditions of its stability or instability with respect
to parameters.
[4] Janković Sv. and Jovanović M., "Convergence in (2m)th Mean for Perturbed Stochastic Integrodifferential
Equations", Publications de l'Institut Mathematique,
Belgrade, (2000) (accepted for publication).
LITERATURA
[5] Melnykov A.V., Financial market (Stochastic analysis),
Scientific Publishing, Moscow, 1997 (in Russian).
[1] Black F. and Karasinski P., "Bond and option pricing
when short rates are lognormal", Financial Analysts
Journal, (1991), 52--59.
[2] Chen L., "A Three-Factor Model of the Structure of
Interest Rates", Preprint, Washington, USA, Federal
Reserve Board, 1995.
[3] Itô, K., "On Stochastic Differential Equations",
Memorial Mathematical Society, 4, (1951), 1--51.
[6] Oksendal B.,
Stochastic Differential Equations,
Springer--Verlag, Berlin, 1992.
[7] Shiryaev A.N., Basis of Stochastic Financial
Mathematics, I, II, Phasys, Moscow,1998 (in Russian)
[8] Stoyanov, J. and Botev, D., "Quantitative Results for
Perturbed Stochastic Differential Equations", Journal of
Applied Mathematics and Stochastic Analysis, V. 9, 3,
(1996), 255--261.