1. The document presents a new approach to proving comparison theorems for stochastic differential equations (SDEs) using differentiation of solutions with respect to initial data. 2. It proves that if the drift term of one SDE is always greater than or equal to the other, and their initial values satisfy the same relation, then the solutions will also satisfy this relation for all time. 3. Two methods are provided: the first uses explicit solutions, the second avoids this by showing the difference process cannot reach zero in finite time based on its behavior.

1. 1
NEW APPROCH TO COMPARASION THEOREMS.
Ilya I. Gikhman
6077 Ivy Woods Court,
Mason, OH 45040, USA
ph. 513-573-9348
email: ilyagikhman@mail.ru
Abstract.
Large number authors have developed technique and generalize the areas of comparison
theorems applications. We use standard stochastic methods to prove comparison theorems for the
solutions of the stochastic differential equations.
Comparison theorem for stochastic differential equations was first presented in [1]. We are using
differentiation of the solutions of the stochastic Ito equations with respect to initial data to prove
comparison theorems.
Let (  , F , P ) be a complete probability space. Denote w ( t ) , t  0 a standard Wiener
process and F w
t filtration generated by the values of Wiener process up to the moment t, i.e.
F w
t = σ { w ( s ) , s ≤ t }. Let x j ( 0 ) be two random variables independent on filtration F w
t
having finite moments of the second order E | x j ( 0 ) | 2
<  , j = 0, 1 and denote
G t = σ { F w
t , x j ( 0 ) , j = 0 , 1 }.
I. Consider two scalar stochastic differential equations (SDEs)
x j ( t ) = x j ( 0 ) + 
t
0
b j ( s , x j ( s )) ds + 
t
0
c ( s , x j ( s )) dw ( s ) (1)

2. 2
j = 0, 1 and t  [ 0 , T ] , T > 0. Suppose that coefficients of the equations (1) are random G t –
measurable functions, which with probability 1 satisfy standard conditions
| b j ( t , x ) | + | c ( t , x ) |  K ( 1 + | x | )
(SC)
| b j ( t , x ) – b j ( t , y ) | + | c ( t , x ) – c ( t , y ) |  L ( R ) | x – y |
when | x | + | y | < R. Then there exists a unique G t – measurable solutions x j ( t ) of the
equation (1) for which
Tt
sup

E | x ( t ) | n
< 
n = 1, 2, … .
Theorem 1. Suppose that x j ( t ) are solutions of the equations (1) with coefficients satisfying
conditions (SC) and let
x 0 ( 0 )  x 1 ( 0 ) , b 0 ( t , x )  b 1 ( t , x )
for ( t , x )  [ 0 , T ]  ( –  , +  ). Then for each t  [ 0 , T ]
P { x 0 ( t ) < x 1 ( t ) } = 1
Remark 1. The proof of the theorem is based on the fact that if the statement of the theorem is
true then the solution of the stochastic differential equation x  ( t ) with drift
b  ( s , x ) = b 0 ( s , x ) +  [ b 1 ( s , x ) – b 0 ( s , x ) ]
and diffusion c ( t , x ) is an increasing function of  ,   [ 0 , 1 ] with probability 1. In this case
it is sufficient to verify that
P {
α
)t(x α


> 0 } = 1
This equality can be verified directly by using explicit solution of the linear stochastic equation
for
α
)t(x α


. On the hand, we can avoid to use explicit form of the solution and present the
proof that the process
α
)t(x α


never intersects level 0 for a finite time period.
Proof. 1. Introduce auxiliary equation
x  ( t ) = x  ( 0 ) + 
t
0
b  ( s , x  ( s )) ds + 
t
0
c ( s , x  ( s )) dw ( s ) (2)

3. 3
where drift coefficient b  ( s , x ) is defined above and
x  ( 0 ) = x 0 ( 0 ) +  [ x 1 ( 0 ) – x 0 ( 0 ) ]
with coefficients depending on parameter   [ 0 , 1 ]. For each  solution of the equation (2)
is continuous with probability 1 function and bearing in mind continuous dependence of the
solution of the equation (2) on coefficients [1], we conclude that there exists separable
modification of the solution of the equation (2) for which we preserve its original notation
x  ( t ) such that
P {
Tt0
max
 0α
lim

| x  ( t ) – x 0 ( t ) | > 0 } = 0
P {
Tt0
max
 1α
lim

| x  ( t ) – x 1 ( t ) | > 0 } = 0
Coefficients of the equation (2) are differentiable in   ( 0 , 1 ) functions. Then solution of the
equation (2) is differentiable in mean square meaning function in  [1]. By formal differentiation
equation (2) with respect to  we arrive at the equation
α
)t(x α


= x 1 ( 0 ) – x 0 ( 0 ) + 
t
0
{ [ b 1 ( s , x  ( s )) – b 0 ( s , x  ( s )) ] +
+ [
x
))s(x,s(b α0


+  (
x
))s(x,s(b α1


–
x
))s(x,s(b α0


) ]
α
)s(x α


ds + (3)
+ 
t
0
x
))s(x,s(c α


α
)s(x α


dw ( s )
The solution of the SDE (3) is a continuous in  with probability 1. It can be written in the form
α
)t(x α


= [ x 1 ( 0 ) – x 0 ( 0 ) ] exp  ( 0 , t ) +
(4)
+ 
t
0
[ b 1 ( s , x  ( s )) – b 0 ( s , x  ( s )) ] exp  ( s , t ) ds
where
 ( s , t ) = 
t
s
{
x
))u(x,u(b α0


+  [
x
))u(x,u(b α1


–
x
))u(x,u(b α0


] –

4. 4
–
2
1
[
x
))u(x,u(c α


] 2
} du + 
t
s
x
))u(x,u(c α


dw ( u )
From the formula (4) it follows that
α
)t(x α


 0 with probability 1 for each t and   ( 0 , 1 ).
Therefore function x  ( t ) is a monotonic increasing with probability 1 function in  and
therefore
x 1 ( t ) – x 0 ( t ) = 
1
0
α
)t(x α


d  0 (5)
Theorem 1 is proved for the case when coefficients of the equations (1) are continuously
differentiable functions. Note that to prove that
α
)t(x α


 0 we used explicit form of the
solution of the equation (3). Now let us consider other approach, which does not use explicit
representation of the solution of the equation (3).
Let x 1 ( 0 ) – x 0 ( 0 ) > 0 and  > 0 be a small number . Denote
  (  ) = min { s :
α
)s(x α


=  } ,   (  )  t = min {  (  ) , t }
Denote a  b = min { a, b } and let N be a constant such that
x,t
max { |
x
)x,t(b 0


| + |
x
)x,t(b 1


| + |
x
)x,t(c


| 2
} < N
Assume that x 1 ( 0 ) > x 0 ( 0 ). Applying Ito formula to the process
α
)t(x α


we note that
[
α
)τt(x α


] – 1
= [ x 1 ( 0 ) – x 0 ( 0 ) ] – 1
– 
t
0
[
α
)τs(x α


] – 2
{ b 1 – b 0 +
– [
x
)u(b 0


+  (
x
)u(b 1


–
x
)u(b 0


) ]} u = ( s   , x  ( s   ))
α
)τs(x α


ds –
– 
t
0
[
α
)τs(x α


] – 2
x
))τs(x,τs(c α


α
)τs(x α


dw ( s ) +
+ 
t
0
[
α
)τs(x α


] – 3
[
x
))τs(x,τs(c α


α
)τs(x α


] 2
ds 

5. 5
 [ x 1 ( 0 ) – x 0 ( 0 ) ] – 1
+ N 
t
0
[
α
)τs(x α


] – 1
ds +
– 
t
0
[
α
)τs(x α


] – 1
x
))τs(x,τs(c α


dw ( s )
Hence
E [
α
)τt(x α


] – 1
+ E 
t
0
[
α
)τs(x α


] – 2
[ b 1 ( s   , x  ( s   )) –
– b 0 ( s   , x  ( s   )) ] ds  [ x 1 ( 0 ) – x 0 ( 0 ) ] – 1
+ N 
t
0
E [
α
)τs(x α


] – 1
ds
where N is a constant which does not depend on  and . Applying Gronwall inequality, we note
that
E [
α
)τt(x α


] – 1
+ E 
t
0
[
α
)τs(x α


] – 2
[ b 1 ( s   , x  ( s   )) –
– b 0 ( s   , x  ( s   )) ] ds  [ x 1 ( 0 ) – x 0 ( 0 ) ] – 1
exp N T
Using continuity of the process
α
)tτ(x αα


and applying Chebyshev inequality
P {   (  )  t < t } = P {
α
)tτ(x αα


=  } = P {
α
)tτ(x αα


< 2 } =
= P { [
α
)tτ(x αα


] – 1
 ( 2 ) – 1
}  2 E |
α
)tτ(x αα


| – 1

 2 [ x 1 ( 0 ) – x 0 ( 0 ) ] – 1
exp N T
From latter estimate taking limit when  tends to zero we note that level zero is unreachable by
the process
α
)t(x α


from its initial value x 1 ( 0 ) – x 0 ( 0 ) > 0. Therefore applying formula
(5) we prove comparison theorem.
Let us now assume that x 1 ( 0 ) = x 0 ( 0 ). Then for arbitrary δ > 0 we consider auxiliary
equation with respect to the process x 1, δ ( t ) which satisfy the same equation as x 1 ( t ) with
initial data x 1, δ ( 0 ) = x 0 ( 0 ) + δ. Choosing  < δ similar to latter constructions that

6. 6
P {  δ
α (  )  t < t }  2 δ – 1
exp N T
Here  δ
α (  ) is the first moment when the process
α
)t(x δ
α


which underling processes are
x 0 ( t ) and x 1, δ ( t ) . For the fixed δ > 0 taking limit when  tends to zero we note that
0ε
lim

P {  δ
α (  )  t < t } = 0
Then putting δ → 0 we conclude that x 1 ( t ) – x 0 ( t )  0
2. Now assume that coefficients (1) are not differentiable functions, satisfy only Lipschitz
condition in x. Denote
Δ x ( t ) = x 1 ( t ) – x 0 ( t ) , A ( t ) = 
t
0
[ b 1 ( s , x 0 ( s )) – b 0 ( s , x 0 ( s )) ] ds
)t(x)t(x
))t(x,t(b))t(x,t(b
)ω,t(B
01
0111


 ,
)t(x)t(x
))t(x,t(c))t(x,t(c
)ω,t(C
01
01



Then
Δx ( t ) = Δx ( 0 ) + 
t
0
[ b 1 ( s , x 1 ( s )) – b 0 ( s , x 0 ( s )) ] ds +
+ 
t
0
[ c ( s, x 1 ( s )) – c ( s , x 1 ( s )) ] dw ( s ) =
= A ( t ) + 
t
0
B ( s ) Δx ( s ) ds + 
t
0
C ( s ) Δx ( s ) dw ( s )
Bearing in mind Lipschitz condition it follows that coefficients of the latter equation are
uniformly bounded functions and
| B ( t , ω ) | + | C ( t , ω ) | ≤ L
The solution of the equation admits representation
Δx ( t ) = exp  ( 0 , t ) { Δx ( 0 ) +
(6)

7. 7
+ 
t
0
exp –  ( 0 , s ) [ b 1 ( s , x 1 ( s )) – b 0 ( s , x 0 ( s )) ] ds }
where
 ( 0 , t ) = 
t
0
{
)s(x)s(x
))s(x,s(b))s(x,s(b
01
0011


–
–
2
1
[
)s(x)s(x
))s(x,s(c))s(x,s(c
01
01


] 2
ds + 
t
0
)s(x)s(x
))s(x,s(c))s(x,s(c
01
01


dw ( s )
Then it follows that
Δ x ( t ) = x 1 ( t ) – x 0 ( t ) > 0
for each t  [ 0 , T ] with probability 1. By using continuity of the random functions one can
choose separable modifications for the processes x j ( t ) , j = 0, 1 with common separability set
for which inequality Δ x ( t ) > 0 holds for all t at once.
To complete the proof of the theorem let us assume that coefficients of the equation (1) satisfy
local Lipschitz condition. Introduce a reduction operation
[ x ] N =
|x|N
xN

where u  v = max ( u , v ) and define cut-off coefficients
b j N ( t , x ) = b j ( t , [ x ] N ) , c N ( t , x ) = c ( t , [ x ] N )
j = 0 , 1. Then there exist the unique solution of the equation (1) with the drift coefficients
b j N ( t , x ) and diffusion c N ( t , x ). Define Markov stopping time moments
 j N (  ) = T  inf ( t : | x j ( t ) | > N ) ,  N (  ) =  0 N (  )   1 N (  )
For arbitrary number N > 0 processes xˆ j N ( t ) = x j ( t   N ) are solutions of the equations
xˆ j N ( t ) = x j N ( 0 ) + 
t
0
 (  N > s ) b j N ( s , xˆ j N ( s )) d s +
(7)
+ 
t
0
 (  N > s ) c N ( s , xˆ j N ( s )) d w ( s )

8. 8
j = 0 , 1 where solution of equations (7) are interpreted as continuous with probability 1
separable functions with the same separability set of for all N at once. In (7)  ( A ) denotes
indicator of an event A. Then we note that for each N > 0 with probability 1
xˆ 1 N ( t ) – xˆ 0 N ( t ) = x 1 ( t   N ) – x 0 ( t   N ) > 0
Applying Chebyshev inequality, we can verify that for any  > 0
P {  N (  ) < T }  P {
Tt
sup

| x 0 N ( t ) | > N } + P {
Tt
sup

| x 1 N ( t ) | > N }  
Then there exists limit
N
lim xˆ j N ( t ) = x j ( t )
j = 0, 1 which are solutions of the equations (1). In order to prove the latter statement one should
verify possibility of limit transition in each term of the equation (6) when N  + .
The theorem 1 has been proved.
II. Now let us consider generalization of the Theorem 1 for more general case. Let ν ( t , A ) ,
A  ( –  , +  ) be Poisson measure and E ν ( t , A ) = t Π ( A ). Introduce centered Poisson
measure
ν~ ( t , A ) = ν ( t , A ) – t Π ( A )
Consider diffusion equations with jumps
ξ j ( t ) = ξ j ( 0 ) + 
t
0
b j ( s , ξ j ( s )) ds + 
t
0
c ( s , ξ j ( s )) dw ( s ) +
(8)
+  
t
0 R *
f ( s , ξ j ( s ) , u ) ν~ ( ds , du )
j = 0 , 1 and R* = ( –  , +  ) {0}. Assume that random variables x j ( 0 ) are independent on
w ( t ), ν~ ( t , A ) and E | ξ j ( 0 ) | 2
<  and coefficients of the equations (8) satisfy standard
conditions that provide existence and uniqueness of the  t = σ { F w
t , F ν
t , ξ j ( 0 ), j = 0, 1 }
measurable solution
| b j ( t , x ) | + | c ( t , x ) | + 
*R
| f ( t , x , u ) | Π ( du ) < K ( 1 + | x | )
| b j ( t , x ) – b j ( t , y ) | + | c ( t , x ) – c ( t , y ) | +

9. 9
+ 
*R
| f ( t , x , u ) – f ( t , y , u ) | Π ( du ) < L | x – y |
of the equation (7) for which
Tt
sup

E | ξ j ( t ) | 2
.
Theorem 2. Suppose that drift coefficients satisfy inequality b 0 ( t , x ) < b 1 ( t , x ) ,
( t , x )  [ 0 , T ]  ( –  , +  ). Then for each t  [ 0 , T ]
P { ξ 0 ( t ) < ξ 1 ( t ) } = 1
Proof. Consider auxiliary equation with drift coefficient depending on parameter   [ 0 , 1 ]
ξ  ( t ) = ξ  ( 0 ) + 
t
0
{ b 0 ( s , ξ  ( s )) +  [ b 1 ( s , ξ  ( s )) – b 0 ( s , ξ  ( s )) ] } ds +
+ 
t
0
c ( s , ξ  ( s )) dw ( s ) +  
t
0 R *
f ( s , ξ  ( s ) , u ) ν~ ( ds , du )
t  [ 0 , T ]. The solution of the latter equation is differentiable with respect to parameter  in
square meaning [2]
0αΔ
lim
 ]T,0[t
sup

E |
αΔ
)t(ξ)t(ξ ααΔα 
–
α
)t(ξ α


| 2
= 0
where
α
)t(ξ α


= ξ 1 ( 0 ) – ξ 0 ( 0 ) + 
t
0
{ [ b 1 ( s , ξ  ( s )) – b 0 ( s , ξ  ( s )) ] +
+ [
x
))s(ξ,s(b α0


+  (
x
))s(ξ,s(b α1


–
x
))s(ξ,s(b α0


) ]
α
)s(ξ α


ds + (9)
+ 
t
0
x
))s(ξ,s(c α


α
)s(ξ α


dw ( s ) +  
t
0 R *
x
)u,)s(ξ,s(f α


α
)s(ξ α


ν~ ( ds , du )
The solution of linear SDE with jumps can be presented in closed form [2]. Assume that
stochastic process ξ ( t ) is a solution of the equation
d ξ ( t ) = [  0 +  1 ξ ( t ) ] d t + [  0 +  1 ξ ( t ) ] d w ( t ) +
(10)

10. 10
+ 
*R
[  0 ( t , u ) +  1 ( t , u ) ξ ( t ) ] ν~ ( ds , du )
Coefficients of the equation (10) are bounded with probability 1 F w
t - measurable random
functions in t. In [2] jump term on the right hand side (10) is written with respect to standard
Poisson measure ν ( dt , du ). Bearing in mind relationship
ν~ ( ds , du ) = ν ( dt , du ) + dt  ( du )
can rewrite equation (10) as following
d ξ ( t ) = [ (  0 +  0 ) + (  1 +  1 ) ξ ( t ) ] d t + [  0 +  1 ξ ( t ) ] d w ( t ) +
(11)
+ 
*R
[  0 ( t , u ) +  1 ( t , u ) ξ ( t ) ] ν ( ds , du )
where functions  j =  j ( t ) are defined by formula
 j ( t ) = 
*R
 j ( t , u )  ( du )
j = 0 , 1. Applying formula presenting close form solution of the liner equation (11) we arrive at
the formula
ξ ( t ) = { ξ ( 0 ) + 
t
0
[  0 ( s ) +  0 ( s ) –  0 ( s )  1 ( s ) ]  – 1
( s ) ds +
(12)
+ 
t
0
 0 ( s )  – 1
( s ) d w ( s ) +  
t
0 R *
)u,s(γ1
)u,s(γ
1
0

 – 1
( s ) ν ( ds , du ) }  ( t )
where
 ( t ) = exp { 
t
0
[  1 ( s ) +  1 ( s ) –
2
1
 2
1 ( s ) ] ds + 
t
0
 1 ( s ) d w ( s ) +
+  
t
0 R *
ln [ 1 +  1 ( s , u ) ] ν ( ds , du ) }
Applying formula (12) equation (9) can be written as

11. 11
α
)t(ξ α


= ξ 1 ( 0 ) – ξ 0 ( 0 ) + 
t
0
{ [ b 1 ( s , ξ  ( s )) – b 0 ( s , ξ  ( s )) +  0 ( t ) ] +
+ [
x
))s(ξ,s(b α0


+  1 ( t ) +  (
x
))s(ξ,s(b α1


–
(9’)
–
x
))s(ξ,s(b α0


) ]
α
)s(ξ α


ds + 
t
0
x
))s(ξ,s(c α


α
)s(ξ α


dw ( s ) +
+  
t
0 R *
x
)u,)s(ξ,s(f α


α
)s(ξ α


ν ( ds , du )
Solution of the equation (9’) is
α
)t(ξ α


= { ξ 1 ( 0 ) – ξ 0 ( 0 ) +
+ 
t
0
 1
α

( s ) [ b 1 ( s , ξ  ( s )) – b 0 ( s , ξ  ( s )) ] ds }   ( t )
where
  ( t ) = exp { 
t
0
[
x
))s(ξ,s(b α0


+  (
x
))s(ξ,s(b α1


–
x
))s(ξ,s(b α0


) +
+  
t
0 R *
x
)u,)s(ξ,s(f α


ν ( ds , du ) –
2
1
|
x
))s(ξ,s(c α


| 2
] ds +
+ 
t
0
x
))s(ξ,s(c α


dw ( s ) +  
t
0 R *
ln [ 1 +
x
)u,)s(ξ,s(f α


] ν ( ds , du ) }
From this formula it follows that
α
)t(ξ α


> 0 with probability 1 for each t and   ( 0 , 1 ).
Then bearing in mind stochastic continuity of the random functions, we note that
ξ 1 ( t ) – ξ 0 ( t ) > 0
with probability 1 for all t  [ 0, T ] at once.

12. 12
III. Consider an application of the comparison problem to stochastic partial differential equation.
Let x j ( t ) = x j ( t ; s , y j ) , j = 0 , 1 be solution of the equation (1) on [ s , T ] , 0  s  t  T,
x j ( s ) = y j where y j are non-random constants . Introduce random functions
f j ( t , y j ,  ) =  j ( x j ( T ; t , y j )) exp 
T
t
V j ( s , x j ( s ; t , y j )) ds (13)
j = 0 , 1 and T < +  is a fixed parameter. Consider random function f ( t , y ,  ) that is defined
by the formula (14), i.e.
f ( t , y ,  ) =  ( x ( T ; t , y )) exp 
T
t
V ( s , x ( s ; t , y )) ds (14)
Here for writing simplicity the low index j = 0 , 1 is omitted . Note that the random function
f ( t , y ,  ) defined by (14) is adapted to backward directed filtration of -algebras

F t =  { w ( u ) – w ( v ) , t  u  v  T }
Theorem 3. Let us stochastic process x ( t ) is a solution of the equation (1) with drift coefficient
b ( t , x ). Suppose that coefficients of the equation (1) and functions  ( x ) and V ( t , x ) in
formula (14) are continuous in t , t  [ 0 , T ] and continuously two times differentiable with
respect to x. Assume that second derivatives are uniformly bounded and satisfy Hölder inequality
of the γ , γ  ( 0 , 1 ]. Then function f ( t , y ,  ) defined by equality (14) is a solution of the
stochastic Cauchy problem
t
)y,t(f


=
x
)y,t(f


b j ( t , y ) +
2
1
2
2
x
)y,(f

 t
c 2
( t , y ) +
(15)
+
x
)y,t(f


c ( t , y )
dt
)(twd
with terminal boundary condition at t = T
f ( T , y ,  ) =  ( y )
The equation (15) is interpreted in integral form in which stochastic integral is backward Ito
integral [3] formally defined as a limit for a

F t measurable separable random function g ( t )
such that
P { 
T
0
g 2
( s ) ds < +  }

13. 13
as

T
t
g ( s ) d

w ( s ) =
0λ
l.i.m


n
1j
g ( t k ) [ w ( t k – 1 ) – w ( t k ) ]
where  =
nk0
max

 t k .
Proof. Similar to [3] it is easy to show that random process f ( t , y ,  ) is twice continuously
differentiable with respect to x in mean square and expected value of the second derivative
satisfy Hölder condition of the γ order. Bearing in mind uniqueness of the solution of the
equation (1) and applying Taylor formula we note that
f ( t –  t , y ,  ) – f ( t , y ,  ) =
= [ f ( t , x ( t –  t ; t , y ) ,  ) – f ( t , y ,  ) ] exp 
t
t-t
V ( s , x ( s ; t , y )) ds +
+ f ( t , y ,  ) [ exp 
t
t-t
V ( s , x ( s ; t , y )) ds – 1 ] =
(15)
=
x
)y,t(f


[ x ( t ; t –  t , y ) – y ] +
2
1
2
2
x
)y,t(f


[ x ( t ; t –  t , y ) – y ] 2
+
+
2
1

1
0
( 1 –  ) [ 2
θ
2
x
))y,t-t;t(x,t(f


– 2
2
x
)y,t(f


] [ x ( t ; t –  t , y ) – y ] 2
d
where x  ( t ; t –  t , y ) =  y + ( 1 –  ) x ( t ; t –  t , y ). Let t = t 0 < t 1 < …< t n =
T be a partition of the interval [ t , T ] . Applying equality (16) on each [ t k , t k + 1 ] we note that
 ( y ) – f ( t , y ,  ) = 
n
0k
f ( t k + 1 , y ,  ) – f ( t k , y ,  ) = L  ( t k + 1 , y ) f + R
where
L  ( t k + 1 , y ) f = – 
n
0k x
)y,t(f 1k

 
[ x ( t k + 1 ; t k , y ) – y ] –
–
2
1
2
1k
2
x
)y,t(f

 
[ x ( t k + 1 ; t k , y ) – y ] 2
and remainder term is equal to

14. 14
R  = –
2
1

n
0k

1
0
( 1 –  k ) 
 [ 2
k1kθ
2
x
))y,t;t(x,t(f

 
– 2
k
2
x
)y,t(f


] [ x ( t k + 1 ; t k , y ) – y ] 2
d k
Similar to [3, Theorem 4.9] it can be proved that
0λ
l.i.m

R  = 0 and
–
0λ
l.i.m

L  ( t k + 1 , y ) f = 
T
t
x
)y,s(f


b ( s , y ) ds + 
T
t
x
)y,s(f


c ( s , y ) d

w ( s ) +
+
2
1

T
t
2
2
x
)y,s(f


c 2
( s , y ) ds
Here stochastic integral with respect to Wiener measure d

w is interpreted as backward Ito
integral [3]. Thus
f ( t , y ,  ) =  ( y ) – 
T
t
[
x
)y,s(f


b ( s , y ) +
2
1
2
2
x
)y,s(f


c 2
( s , y ) ] ds –
(16)
– 
T
t
x
)y,s(f


c ( s , y ) d

w ( s )
Next statement follows from the Theorem 3.
Theorem 4. Suppose that x j ( t ) , j = 0 , 1 satisfy conditions of the theorem 1 and suppose that
 0 ( x )   1 ( x ) , V 0 ( t , x )  V 1 ( t , x )
for x  ( –  , +  ) and t  0. Then random functions defined by equalities (13) are solutions of
the stochastic Cauchy problem (16) and
f 0 ( s , y ,  )  f 1 ( s , y ,  )
with probability 1 for s  [ 0 , T ].
Bearing in mind continuity of the all functions in equality (16) there exist separable
modifications of the solutions (16) for which statement takes place at for all ( s , y )  [ 0 , T ] 
( –  , +  ) at once.

15. 15
Now let us consider more general case than the system (1). We reduced comparison theorem to
analyzing solution of the general linear SDE. Based on explicit form of the linear SDE one can
consider a stochastic system. Let x 0 ( t ) , x 1 ( t ) be solution of the stochastic system
x 0 ( t ) = x 0 ( 0 ) + 
t
0
b 0 ( s , x 0 ( s ) , x 1 ( s )) ds + 
t
0
c ( s , x 0 ( s )) dw ( s )
(17)
x 1 ( t ) = x 1 ( 0 ) + 
t
0
b 1 ( s , x 0 ( s ) , x 1 ( s )) ds + 
t
0
c ( s , x 1 ( s )) dw ( s )
Theorem 5. Let coefficients of the system (17) satisfy conditions of the theorem 1 with respect
to variables ( t , x , y ) and assume that nonrandom initial values and drift coefficients are subject
to conditions
x 0 ( 0 )  x 1 ( 0 ) , b 0 ( s , x , y )  b 1 ( s , x , y )
Then x 0 ( t )  x 1 ( t ) with probability 1 for each t  [ 0 , T ].
Proof. Indeed, putting
Δ x ( t ) = x 1 ( t ) – x 0 ( t ) ,
B ( t ) = b 1 ( t , x 0 ( t ) , x 1 ( t )) – b 0 ( t , x 0 ( t ) , x 1 ( t )) ,
)t(x)t(x
))t(x,t(c))t(x,t(c
)ω,t(C
01
01



we note that random process Δ x ( t ) is a solution of the SDE
Δx ( t ) = Δx ( 0 ) + 
t
0
B ( s ) ds + 
t
0
C ( s ) Δx ( s ) dw ( s )
that admits closed form representation
Δx ( t ) = exp  ( 0 , t ) { Δx ( 0 ) + 
t
0
exp –  ( 0 , s ) [ b 1 ( s , x 0 ( s ) , x 1 ( s )) –
– b 0 ( s , x 0 ( s ) , x 1 ( t )) ] ds }
Here
 ( 0 , t ) = –
2
1

t
0
[
)s(x)s(x
))s(x,s(c))s(x,s(c
01
01


] 2
ds +

16. 16
+ 
t
0
)s(x)s(x
))s(x,s(c))s(x,s(c
01
01


dw ( s )
Therefore
Δ x ( t ) = x 1 ( t ) – x 0 ( t ) > 0
for each t  [ 0 , T ] with probability 1.

17. 17
References.
[1] Skorohod A. V. Issledovaniya po teorii sluchainykh protsessov. Izdat. Kiev, Univ., Kiev,
1961. Studies in the theory of random processes. Translated from the Russian, Addison-Wesley
Publishing Co., Inc., Reading, Mass., 1965.
[2] Gihman I.I., Skorohod A.V. Stochastic Differential Equations. Springer-Verlag, 1972, 354.
[3] Gikhman I. Stochastic differential Equations and Its Applications. Lambert Academic
Publishing. ISBN: 978-3-8454-0791-3, 2011, 242.