Positivity of solution of nonhomogeneous stochastic differential equation with non-Lipschitz diffusion
We give a sufficient condition on coefficients of a nonhomogeneous stochastic differential equation with non-Lipschitz diffusion for a solution starting from arbitrary nonrandom positive point to stay positive. Some examples of application of the condition mentioned above are considered.
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| Цитувати: | Positivity of solution of nonhomogeneous stochastic differential equation with non-Lipschitz diffusion / Y. Mishura, S. Posashkova // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 77-88. — Бібліогр.: 6 назв.— англ. |
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irk-123456789-45702009-12-08T12:00:31Z Positivity of solution of nonhomogeneous stochastic differential equation with non-Lipschitz diffusion Mishura, Y. Posashkova, S. We give a sufficient condition on coefficients of a nonhomogeneous stochastic differential equation with non-Lipschitz diffusion for a solution starting from arbitrary nonrandom positive point to stay positive. Some examples of application of the condition mentioned above are considered. 2008 Article Positivity of solution of nonhomogeneous stochastic differential equation with non-Lipschitz diffusion / Y. Mishura, S. Posashkova // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 77-88. — Бібліогр.: 6 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4570 en Інститут математики НАН України |
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We give a sufficient condition on coefficients of a nonhomogeneous stochastic differential equation with non-Lipschitz diffusion for a solution starting from arbitrary nonrandom positive point to stay positive. Some examples of application of the condition mentioned above are considered. |
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Article |
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Mishura, Y. Posashkova, S. |
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Mishura, Y. Posashkova, S. Positivity of solution of nonhomogeneous stochastic differential equation with non-Lipschitz diffusion |
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Mishura, Y. Posashkova, S. |
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Mishura, Y. |
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Positivity of solution of nonhomogeneous stochastic differential equation with non-Lipschitz diffusion |
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Positivity of solution of nonhomogeneous stochastic differential equation with non-Lipschitz diffusion |
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Positivity of solution of nonhomogeneous stochastic differential equation with non-Lipschitz diffusion |
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Positivity of solution of nonhomogeneous stochastic differential equation with non-Lipschitz diffusion |
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Positivity of solution of nonhomogeneous stochastic differential equation with non-Lipschitz diffusion |
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positivity of solution of nonhomogeneous stochastic differential equation with non-lipschitz diffusion |
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Інститут математики НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/4570 |
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Positivity of solution of nonhomogeneous stochastic differential equation with non-Lipschitz diffusion / Y. Mishura, S. Posashkova // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 77-88. — Бібліогр.: 6 назв.— англ. |
| work_keys_str_mv |
AT mishuray positivityofsolutionofnonhomogeneousstochasticdifferentialequationwithnonlipschitzdiffusion AT posashkovas positivityofsolutionofnonhomogeneousstochasticdifferentialequationwithnonlipschitzdiffusion |
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2025-07-02T07:46:53Z |
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2025-07-02T07:46:53Z |
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1836520480399425536 |
| fulltext |
Theory of Stochastic Processes
Vol.14 (30), no.3-4, 2008, pp.77-88
YULIYA MISHURA AND SVITLANA POSASHKOVA
POSITIVITY OF SOLUTION OF
NONHOMOGENEOUS STOCHASTIC
DIFFERENTIAL EQUATION WITH
NON-LIPSCHITZ DIFFUSION
We give a sufficient condition on coefficients of a nonhomogeneous
stochastic differential equation with non-Lipschitz diffusion for a so-
lution starting from arbitrary nonrandom positive point to stay posi-
tive. Some examples of application of the condition mentioned above
are considered.
1. Introduction
The main goal of this paper is to investigate the question about positivity
of solution of nonhomogeneous stochastic differential equation with non-
Lipschitz diffusion. Such stochastic differential equations arise in modelling
asset prices and interest rates on financial markets.
For example, Cox-Ingersoll-Ross interest rate model has the form:
rt = r0 +
∫ t
0
a(b− rs)ds+
∫ t
0
σ
√
rsdWs, t ≥ 0,
where r0, a, σ are real positive constants. It is easy to see that for rt > b the
drift is negative and for rt < b it is positive, so the solution of this equation
is mean-reverting.
Positivity of solutions of the stochastic differential equation with homo-
geneous coefficients of the form
X(t) = X0 +
∫ t
0
b(X(s))ds+
∫ t
0
σ(X(s))dW (s), t ≥ 0 (1)
was studied in [5], Chapter VI. This property is quite important for pro-
cesses, which model interest rate dynamics on financial market, because the
interest rate must be positive.
2000 Mathematics Subject Classifications. 60H10.
Key words and phrases. Cox-Ingersoll-Ross model, stochastic differential equation,
positivity of solutions.
77
78 YULIA MISHURA AND SVITLANA POSASHKOVA
The paper is organized as follows. Section 2 is devoted to nonhomoge-
neous stochastic differential equations with non-Lipschitz diffusion and con-
tains the sufficient condition of positivity of a solution starting from a non-
random positive value. In the paper [2] such sufficient condition was proved
in a particular case, where the diffusion was of the form σ(t, x) = σ(t)
√
x.
Section 3 contains some examples of application of the sufficient condi-
tion mentioned above, in particular we consider a nonhomogeneous version
of Cox-Ingersoll-Ross model.
2. Positivity of solution of nonhomogeneous stochastic
differential equation with non-Lipschitz diffusion
Consider a stochastic differential equation
X(t) = X0 +
∫ t
0
b(s,X(s))ds+
∫ t
0
σ(s,X(s))dW (s), t ≥ 0, (2)
where the initial value X0 > 0 is nonrandom, the coefficients b, σ : R+ ×
R → R are measurable, {W (t), t ≥ 0} is a Wiener process with respect to
filtration {Ft, t ≥ 0} on a probability space (Ω,F , P ).
Assume that the coefficients of this equation satisfy the following Ya-
mada conditions (Y1)–(Y4) (see e.g. [5,6]):
(Y1) the functions b and σ are jointly continuous.
(Y2) the coefficients grow at most linearly in x
|b(t, x)| + |σ(t, x)| ≤ C(1 + |x|), t ≥ 0, x ∈ R.
(Y3) the drift is Lipschitz continuous
|b(t, x) − b(t, y)| ≤ C |x− y| , t ≥ 0, x, y ∈ R.
(Y4) there exists such an increasing function ρ : R+ → R+, that∫
0+
ρ−2(u)du = +∞, and
|σ(t, x) − σ(t, y)| ≤ Cρ(|x− y|), t ≥ 0, x, y ∈ R.
Remark 2.1. An example of such function ρ is ρ(x) = xα, 1
2
≤ α ≤ 1, that
is why this condition is a kind of Hölder condition for σ. Other examples
are: ρ(x) = x1/2(ln 1/x)1/2, ρ(x) = x1/2(ln 1/x)1/2(ln ln 1/x)1/2 etc.
Definition 2.1. We say that the pair (X,W ) is a strong solution of the
equation (2), if X is the process adapted to the filtration
{F̄W
t
}
generated
by process W .
POSITIVITY OF SOLUTION 79
Definition 2.2. The equation (2) has the property of uniqueness of tra-
jectories if any two solutions (X,W ) and (X̃,W ), adapted to the same
filtration, satisfy the equality
P (∀t > 0 : X(t) = X̃(t)) = 1.
Theorem 2.1.([6]) Under conditions (Y1)–(Y4) the equation (2) has the
unique (relatively to its trajectories) solution from the class L2(Ω×[0, T ],F⊗
B([0, T ]), P × λ) for all T > 0, there λ is Lebesgue measure on R+.
Assume that for all x > 0 and t > 0 σ2(t, x) > 0, and that for all z > 0
the following inequality holds:
inf
x∈[z,+∞)
inf
t>0
σ2(t, x) = d = d(z) > 0. (3)
Let also for some ε > 0 there exist a positive continuous function A(t), t > 0,
such that for all t > 0 and x ∈ (0, ε)
2b(t, x)
σ2(t, x)
≥ A(t)
x
. (4)
The main result of the paper is the following one.
Theorem 2.2. Under the assumptions (Y1)–(Y4), (3)-(4) and the condi-
tion
∀t > 0 A(t) > 1 (equivalently ∀T > 0 inf
(0,T ]×(0,ε)
2b(t, x)x
σ2(t, x)
> 1)
the trajectories of the process {X(t), t ≥ 0} will be positive with probability 1.
Remark 2.2. For homogeneous Cox-Ingersoll-Ross model described by the
stochastic differential equation
drt = a(b− rt)dt+ σ
√
rtdWt
with a initial condition r0 > 0, the necessary and sufficient condition pro-
viding positivity of the solution is ab ≥ σ2
2
(see [5]), and in a nonhomo-
geneous case, if b(t, x) = a(t)(b(t) − x), σ(t, x) = σ(t)
√
x, then the coef-
ficient A(t) is A(t) = 2a(t)b(t)
σ2(t)
. So we obtain the condition 2a(t)b(t)
σ2(t)
> 1 or
a(t)b(t) > σ2(t)
2
, that is a generalization of the sufficient condition for the ho-
mogeneous case (a corresponding sufficient condition for nonhomogeneous
Cox-Ingersoll-Ross model also is proved in [2])
80 YULIA MISHURA AND SVITLANA POSASHKOVA
Proof. At first we prove that the solution of the equation (2) is a semimartin-
gale. Indeed, under condition of linear growth the process {Y (t), t ≥ 0} of
the form
Y (t) =
∫ t
0
σ(s,X(s))dW (s)
is a local martingale, because∫ t
0
σ2(s,Xs)ds ≤
∫ t
0
C2(1+Xs)
2ds ≤ 2C2t+2C2
∫ t
0
X2
sds < +∞ P −a.s.,
where the last inequality is a consequence of Theorem 2.1. So,
P{
∫ t
0
σ2(s, Ys)ds < +∞} = 1
means that {Y (t), t ≥ 0} is a local martingale. Now let us show that
Z(t) =
∫ t
0
b(s,X(s))ds, t ≥ 0 is the process of bounded variation. Indeed,
from the condition of linear growth (Y2) we can see that∫ t
0
|b(s,Xs)|ds ≤
∫ t
0
C(1 + |Xs|)ds = Ct+ C
∫ t
0
|Xs|ds < +∞,
where the last inequality follows by Theorem 2.1.
So, the process {X(t), t ≥ 0} is a sum of a local martingale and a process
of bounded variation, that is why it is a semimartingale,
It is sufficient to prove that the trajectories of the process X are positive
on any [0, T ]. So, let T > 0 be fixed.
Let ε > 0 be the constant from the condition (4). For x ∈ (ε,+∞) by
the condition (Y 2) there exists a constant K > 0 such that
|b(t, x)| ≤ K(1 + x).
We denote
p := K
(
1 +
1
ε
)
,
and then obtain the following inequality
∀x ∈ (ε,+∞) |b(t, x)| ≤ px.
We define
V (x) =
{
lnx, if 0 < x < ε,
1
ε
exp{−pε2
2d
} ∫ x
ε
exp{pu2
2d
}du+ ln ε, if ε ≤ x <∞.
It is easy to see that the function V (x) is continuously differentiable on
(0,+∞) and V ′(x) has the form
V ′(x) =
{
1
x
, if 0 < x < ε,
1
ε
exp{−pε2
2d
} exp{px2
2d
}, if ε ≤ x <∞.
POSITIVITY OF SOLUTION 81
We consider the differential operator
L =
1
2
σ2(t, x)
∂2
∂x2
+ b(t, x)
∂
∂x
.
Under condition (4) we have that for all x ∈ (0, ε) the following holds
LV =
1
2
σ2(t, x)(− 1
x2
) + b(t, x)
1
x
=
1
2x
σ2(t, x)(
2b(t, x)
σ2(t, x)
− 1
x
) ≥
≥ 1
2x
σ2(t, x)(
A(t)
x
− 1
x
) =
1
2x
σ2(t, x)
A(t) − 1
x
> 0.
For x ∈ (ε,+∞) the following inequality takes place
LV =
1
2
σ2(t, x) exp{−pε
2
2d
}px
dε
exp{px
2
2d
} + b(t, x)
1
ε
exp{−pε
2
2d
} exp{px
2
2d
} ≥
≥ d
px
dε
exp{−pε
2
2d
} exp{px
2
2d
} − px
1
ε
exp{−pε
2
2d
} exp{px
2
2d
} = 0.
Thus LV ≥ 0 for all x > 0, x
= ε.
Let 0 < m < X0 < M be fixed constants. We consider the random
variable
τm,M = inf{t : X(t) = m X(t) = M}
We use the following generalization of Ito’s formula from the article [3]. Let
X = X(t), t ≥ 0 be a continuous semimartingale and let f : R+ → R+ be a
continuous function of bounded variation.
We define lfs (X) as the local time of the process X on the curve f of the
form:
lfs (X) = P − lim
δ↓0
1
2δ
∫ s
0
I(f(r) − δ < Xr < f(r) + δ)d < X,X >r .
We make a remark that in our case the curve is given by the equation
f(s) = ε, because the second derivative of the function V has a break
only at this point. The process {X(t), t ≥ 0} is a semimartingale from
the proof above. That is why for any semimartingale, also for the process
{X(t), t ≥ 0}, the local time exists at any point, it follows from Theorem
5.52 on the page 186 in [1]. So, lfs (X) = lεs(X) exists.
We have as a consequence of Theorem 2.1 from the paper [3] that if we
set C = [0, ε] and D = [ε,+∞) (and taking into account that the function
V does not depend on t and V ′(x) is continuous on interval [0,+∞) in our
case), we obtain
V (X(τm,M ∧ T )) = V (X0) +
∫ τm,M∧T
0
V ′(X(s))dX(s)+
82 YULIA MISHURA AND SVITLANA POSASHKOVA
+
1
2
∫ τm,M∧T
0
V ′′(X(s))I(X(s)
= ε)d < X,X >s= V (X0)+
+
∫ τm,M∧T
0
V ′(X(s))b(s,X(s))ds+
∫ τm,M∧T
0
V ′(X(s))σ(s,X(s))dW (s)+
+
1
2
∫ τm,M∧T
0
V ′′(X(s))I(X(s)
= ε)σ2(s,X(s))ds,
where we have used the equality
d < X,X >s= σ2(s,X(s))ds.
Functions b(t, x) and σ(t, x) satisfy Yamada conditions. The function
b(t, x) is bounded for x ∈ [m,M ] and separated from 0 and from ∞; σ(t, ·)
is uniformly at Hölder continuous. Then it follows from Theorem 3.2.1 and
Corollary 3.2.2 in [4] that the transition probability function P (s, x; t,Γ) for
the process {X(t), t ≥ 0} exists and has the density
P (s, x; t,Γ) =
∫
Γ
p(s, x; t, y)dy,
where p(s, x; t, y), 0 ≤ s < t, x, y ∈ R is a positive function, which is
continuous in all variables. We set s := 0, x := X0 and obtain that the
distribution function of the process {X(t), t ≥ 0} has the form
P1(t,Γ) =
∫
Γ
p1(t, y)dy.
Thus, the process {X(t), t ≥ 0} has a density, so
P (X(t) = ε) = 0, ∀t > 0.
Further,
0 ≤ |E
∫ τm,M∧T
0
(V ′(X(s))b(s,X(s))I(X(s) = ε)ds| ≤
≤ |V ′(ε)| max
s∈[0,T ]
|b(s, ε)|
∫ T
0
P (X(s) = ε)ds = 0,
so,
E
∫ τm,M∧T
0
(V ′(X(s))b(s,X(s))I(X(s) = ε)ds = 0.
Take an expectation of both sides
V (X(τm,M ∧ T )) = V (X0) +
∫ τm,M∧T
0
V ′(X(s))b(s,X(s))ds+
POSITIVITY OF SOLUTION 83
+
∫ τm,M∧T
0
V ′(X(s))σ(s,X(s))dW (s)+
+
1
2
∫ τm,M∧T
0
V ′′(X(s))I(X(s)
= ε)σ2(s,X(s))ds,
and receive:
E(V (X(τm,M ∧ T ))) = V (X0) + E
∫ τm,M∧T
0
V ′(X(s))b(s,X(s))ds+
+E
∫ τm,M∧T
0
1
2
V ′′(X(s))I(X(s)
= ε)σ2(s,X(s))ds =
= V (X0) + E
∫ τm,M∧T
0
V ′(X(s))b(s,X(s))I(X(s) = ε)ds+
+E
∫ τm,M∧T
0
(V ′(X(s))b(s,X(s)) +
1
2
V ′′(X(s))σ2(s,X(s))I(X(s)
= ε)ds =
= V (X0) + E
∫ τm,M∧T
0
(V ′(X(s))b(s,X(s))+
+
1
2
V ′′(X(s))σ2(s,X(s))I(X(s)
= ε)ds.
The function V (x) is bounded from above on [0,M ], so we have
E(V (X(τm,M ∧ T ))) ≤ max
x∈[0,M ]
V (x)P (τ = τM ∧ T ) + V (m)P (τ = τm) ≤
≤ max
x∈[0,M ]
V (x) + V (m)P (τ = τm),
where τM (τm) is the first moment that the boundary m (boundary M) is
reached and τ = τm ∧ τM ∧ T .
Thus,
max
x∈[0,M ]
V (x) + V (m)P (τ = τm) ≥ V (X0)+
+E(
∫ τm,M∧T
0
(V ′(X(s))b(s,X(s)) +
1
2
V ′′(X(s))σ2(s,X(s)))I(X(s)
= ε)ds)
= V (X0) + E(
∫ τm,M∧T
0
LV (X(s))I(X(s)
= ε)ds) ≥ 0.
Let m tends to 0 and we obtain from above that
V (m) → −∞, m→ 0,
so the left-hand side of the inequality tends to −∞ and at the same time
the right-hand side of the inequality is nonnegative. Thus,
P (τ = τ0) = 0. (5)
84 YULIA MISHURA AND SVITLANA POSASHKOVA
So, for any fixed M and T the equality(5) holds. The proof follows when
M , T tend to +∞. �
4. Examples
Consider examples of application of Theorem 2.2 to some stochastic dif-
ferential equations.
Example 3.1.
Consider a stochastic differential equation
X(t) = X0 +
∫ t
0
(p(s) − q(s)X(s))ds+
∫ t
0
c(t)(X(s))αdW (s), t ≥ 0, (6)
where the initial value X0 > 0 is nonrandom, the functions p(t), q(t) and
c(t) are positive, continuous and inft>0 c(t) = d > 0, a constant 1
2
≤ α < 1.
Then in our notations
b(t, x) = p(t) − q(t)x,
σ2(t, x) = c2(t)x2α.
Thus, the left-hand part of inequality (4) can be rewritten in the form
2b(t, x)
σ2(t, x)
=
2(p(t) − q(t)x)
c2(t)x2α
=
2p(t)
c2(t)x2α
− 2q(t)
c2(t)
x1−2α ≥ 2p(t)
c2(t)x2α
Then, we put in (4) ε = 1, A(t) = 2p(t)
c2(t)
. For 1
2
≤ α < 1 we have that
1
x2α−1
A(t)
x
≥ A(t)
x
,
because 2α− 1 > 0 and x ∈ (0, ε) = (0, 1). So, by Theorem 2.2 we receive,
under the condition A(t) > 1 for all t > 0, which in our case has the form
∀t > 0 : p(t) >
c2(t)
2
,
that the trajectories of the process {X(t), t ≥ 0} are positive with proba-
bility 1.
Example 3.2.
Consider a stochastic differential equation
X(t) = X0 +
∫ t
0
(p(s) − q(s)X(s))ds+
POSITIVITY OF SOLUTION 85
+
∫ t
0
√
(X(s) ∨ 0)((c(t) −X(s)) ∨ 0)dW (s), t ≥ 0, (7)
where the initial value 0 < X0 < c(0) is nonrandom, the functions p(t), q(t)
and c(t) are positive, continuous and c(t) is also nondecreasing. Assume
that
p(t) < q(t)c(t) ∀t > 0.
Then we have in our notations
b(t, x) = p(t) − q(t)x,
σ(t, x) =
√
(x ∨ 0)((c(t) − x) ∨ 0).
Let us prove that such stochastic differential equation has a solution. It is
sufficient to verify the fulfillment of Yamada conditions on the coefficients.
We can consider the interval t ∈ [0, T ] and then tend T to +∞.
(Y1) From the initial condition on functions p(t), q(t) and c(t) we can
see that b(t, x) and σ(t, x) are jointly continuous.
(Y2)
|b(t, x)| + |σ(t, x)| ≤ p(t) + q(t)|x| + |x| + c(t) + |x| = p(t) + c(t)+
+|x|(1 + q(t)) ≤ max
t∈[0,T ]
(p(t) + c(t), 1 + q(t))(1 + |x|)
(Y3)
|b(t, x) − b(t, y)| = |q(t)(x− y)| ≤ max
t∈[0,T ]
q(t)|x− y|
(Y4)
|σ(t, x) − σ(t, y)| = |
√
x(c(t) − x) ∨ 0 −
√
y(c(t) − y) ∨ 0|
Let the function g : R+ × R × R → R be defined as
g(t, x, y) = |
√
x(c(t) − x) ∨ 0 −
√
y(c(t) − y) ∨ 0|.
We assume, that t ∈ [0, T ], and consider the next cases.
1) If (x < 0 or x > c(t)) and (y < 0 or y > c(t)) then
g(t, x, y) = 0 ≤
√
c(t)
√
|x− y|.
2) If 0 ≤ x ≤ c(t) and y < 0 then
g(t, x, y) = |
√
x(c(t) − x)| ≤ |
√
(x− y)(c(t) − x)| ≤ |
√
(x− y)c(t)| =
=
√
c(t)
√
|x− y|.
86 YULIA MISHURA AND SVITLANA POSASHKOVA
Note that under the symmetry the same estimation for g(t, x, y) holds when
x < 0 and 0 ≤ y ≤ c(t).
3) If 0 ≤ x ≤ c(t) and y > c(t) then
g(t, x, y) = |
√
x(c(t) − x)| ≤ |
√
x(y − x)| ≤ |
√
c(t)(y − x)| =
=
√
c(t)
√
|x− y|.
Note that under the symmetry the same estimation for g(t, x, y) holds when
x > c(t) and 0 ≤ y ≤ c(t).
4) If 0 ≤ x ≤ c(t) and 0 ≤ y ≤ c(t) then
g(t, x, y) = |
√
x(c(t) − x)−
√
y(c(t) − y)| ≤
√
|x(c(t) − x) − y(c(t) − y)| =√
|x(c(t) − x) − y(c(t) − x) + y(c(t) − x) − y(c(t) − y)| =
=
√
|(x− y)(c(t) − x) + y(c(t) − x− c(t) + y)| =
=
√
|(x− y)(c(t) − x− y)| ≤
√
c(t)|x− y| =
√
c(t)
√
|x− y|.
Thus, in all cases we have that
g(t, x, y) ≤
√
c(t)
√
|x− y|.
That is why
max
t∈[0,T ]
g(t, x, y) ≤ ( max
t∈[0,T ]
√
c(t))
√
|x− y|.
We obtain that
|σ(t, x) − σ(t, y)| = |
√
x(c(t) − x) ∨ 0 −
√
y(c(t) − y) ∨ 0| ≤
≤ ( max
t∈[0,T ]
√
c(t))
√
|x− y|.
Then ρ(|x− y|) =
√|x− y|, ρ(x) =
√
x increases and∫
0+
ρ−2(u)du =
∫
0+
1
u
du = +∞.
It means that Yamada’s conditions are fulfilled and the equation has the
unique solution.
Let us prove that X(t) ≥ 0 a.s. for all t ≥ 0. Define τ1 = inf{t : X(t) =
−δ}, where δ > 0 is some fixed constant. Assume that P{τ1 < ∞}. So, a
constant r < τ1 exists such that
X(t) < 0 t ∈ (r, τ1) a.s.
However, in this case
dX(t) = (p(t) − q(t)X(t))dt > 0.
POSITIVITY OF SOLUTION 87
So, the function t→ X(t) increases on (r, τ1) which is impossible.
Now we prove that X(t) ≤ c(t) a.s. for all t ≥ 0. Define τ2 = inf{t :
c(t) − X(t) = −δ}, where δ > 0 is some fixed constant. Assume that
P{τ2 <∞}. So, a constant R < τ2 exists such that
−δ < c(t) −X(t) < −δ + θ t ∈ (R, τ2) a.s.,
where θ is such constant that p(t)−q(t)c(t)
q(t)
+ θ < 0, t ∈ (R, τ2). Then
c(t) + δ − θ < X(t) < c(t) + δ t ∈ (R, τ2).
But
dX(t) = (p(t) − q(t)X(t))dt < (p(t) − q(t)(c(t) + δ − θ))dt =
= ((p(t) − q(t)c(t) + q(t)θ) − q(t)δ)dt < 0.
We obtain that dX(t) < 0, t ∈ (R, τ2), thus, the function t → X(t)
decreases on (r, τ1), but this is impossible (because the function c(t) − x(t)
must decrease and the function c(t) is nondecreasing). From this we obtain
that the initial stochastic differential equation can be rewritten in the form
X(t) = X0+
∫ t
0
(p(s)−q(s)X(s))ds+
∫ t
0
√
X(s)(c(t) −X(s))dW (s), t ≥ 0.
We know from above that b(t, x) = p(t) − q(t)x, σ2(t, x) = x(c(t) − x)
Thus, the inequality (4) can be rewritten in the form
2b(t, x)
σ2(t, x)
=
2(p(t) − q(t)x)
x(c(t) − x)
=
2p(t)
x(c(t) − x)
− 2q(t)
c(t) − x
.
As x ∈ (0, ε), where ε < c(0), then we can estimate the above expression as
2p(t)
x(c(t) − x)
− 2q(t)
c(t) − x
≥ 2p(t)
xc(t)
− 2q(t)
c(t) − ε
≥ 2p(t)
xc(t)
Denote A(t) = 2p(t)
c(t)
. Thus, by Theorem 2.2 the condition A(t) > 1 must
be satisfied for all t > 0, and in our case it has the form
∀t > 0 : p(t) >
c(t)
2
.
It is sufficient for trajectories of the process {X(t), t ≥ 0} be positive a.s.
So, if
c(t)
2
< p(t) < q(t)c(t) ∀t > 0,
then trajectories of the process {X(t), t ≥ 0} given by the stochastic differ-
ential equation (7) are positive with probability 1.
88 YULIA MISHURA AND SVITLANA POSASHKOVA
4. Conclusion
We declare the sufficient condition on coefficients which provides a.s.
positivity of the trajectories of the solution of the stochastic differential
equation with nonhomogeneous coefficients and non-Lipschitz diffusion. The
result of this paper is applied to some stochastic differential equations, in
particular for nonhomogeneous Cox-Ingersoll-Ross model.
References
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and non-Lipschitz diffusion, Theory of Probability and Math.Statistics,
79, (2008), submitted. (Ukrainian)
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Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: myus@univ.kiev.ua, revan1988@gmail.com
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