Long-term returns in stochastic interest rate models
We consider the behavior of integral functional of the solution of stochastic differential equation with coefficients contained small parameter. The dependence on the order of small parameter in every term of equation with Wiener process and Poisson measure term is studied. We observe the convergenc...
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Інститут математики НАН України
2007
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| Cite this: | Long-term returns in stochastic interest rate models / V. Zubchenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 247–261. — Бібліогр.: 11 назв.— англ. |
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| author | Zubchenko, V. |
| author_facet | Zubchenko, V. |
| citation_txt | Long-term returns in stochastic interest rate models / V. Zubchenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 247–261. — Бібліогр.: 11 назв.— англ. |
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| description | We consider the behavior of integral functional of the solution of stochastic differential equation with coefficients contained small parameter. The dependence on the order of small parameter in every term of equation with Wiener process and Poisson measure term is studied. We observe the convergence of the long-term return, using an extension of the Cox-Ingersoll-Ross stochastic model of the short interest rate. Obtained results are applied for studying of two-factor stochastic interest rate model.
|
| first_indexed | 2025-12-07T17:08:03Z |
| format | Article |
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Theory of Stochastic Processes
Vol.13 (29), no.4, 2007, pp.247–261
VLADIMIR ZUBCHENKO
LONG-TERM RETURNS IN STOCHASTIC
INTEREST RATE MODELS
We consider the behavior of integral functional of the solution of
stochastic differential equation with coefficients contained small pa-
rameter. The dependence on the order of small parameter in every
term of equation with Wiener process and Poisson measure term is
studied. We observe the convergence of the long-term return, using
an extension of the Cox-Ingersoll-Ross stochastic model of the short
interest rate. Obtained results are applied for studying of two-factor
stochastic interest rate model.
1. Introduction
Controlling the risk induced by interest rate fluctuation is of crucial
importance for banks and insurance companies. In this light, we think it is
interesting to study and to model the long-term return in a mathematical
way.
Interest to the behavior of long-term return leads to the investigation
of limit behavior of integral functionals of solution of stochastic differen-
tial equation. In [1] properties of integral functionals of Brownian motion
with drift are considered. In [2] the boundary classification of diffusion is
used in order to derive a criterion for the convergence of perpetual integral
functionals of transient real-valued diffusion. In [3] the behavior of integral
functional of the solution to stochastic differential equation with Wiener
process and Poisson measure term, and with coefficients containing small
parameter is studied. The first part of this paper contains some generaliza-
tion of results from [3].
In the Cox-Ingersoll-Ross stochastic model [4] the dynamics of the short
interest rate (rt)t≥0 is expressed by stochastic differential equation
drt = k(γ − rt)dt + σ
√
rtdwt,
2000 Mathematics Subject Classifications. 60H10
Key words and phrases. Stochastic differential equation, integral functional, long-
term return, limit behavior, small parameter.
247
248 VLADIMIR ZUBCHENKO
with (wt)t≥0 one-dimensional Wiener process and k, γ and σ positive con-
stants. In this model rt never becomes negative and converges to the long-
term constant value γ [5]. Further, it is reasonable to conjecture that the
market will constantly change this level γ and the volatility σ. Therefore
it is natural to consider γ as the stochastic process and to generalize the
volatility.
In mentioned models the short interest rate satisfies the stochastic dif-
ferential equation of diffusion type. In this paper we consider the stochastic
differential equation for the processes with discontinuities. We observe the
convergence of the long-term return 1
t
∫ t
0
rs ds, where (rt)t≥0 satisfies the
stochastic differential equation with Wiener process and Poisson measure
term, which is the generalization of the Cox-Ingersoll-Ross stochastic model
of the short interest rate.
2. The limit behavior of integral functional of the solution
of stochastic differential equation
We study the behavior, as ε → 0, of the integral functional ηε(t) =
(εk/t)
t/εk∫
0
d(s, ξ(s))ds, where ξ(t) is the solution of stochastic differential
equation
dξ(t) = εk1f(t, ξ(t))dt+εk2g(t, ξ(t))dw(t)+εk3
∫
Rd
q(t, ξ(t), y)ν̃(dt, dy), (1)
ξ(0) = ξ0;
ε > 0 is the small parameter; k > 0, ki > 0, i = 1, 2, 3; d(t, x) is non-random
function; f(t, x) = {fi(t, x), i = 1, d}, q(t, x, y) = {qi(t, x, y), i = 1, d} are
non-random vector-valued functions; g(t, x) = {gij(t, x), i, j = 1, d} is non-
random matrix-valued function; t ∈ [0, T ], x, y ∈ Rd; w(t) is d-dimensional
Wiener process; ν̃(dt, dy) = ν(dt, dy) − Π(dy)dt, ν(dt, dy) is the Poisson
measure independent on w(t), Eν(dt, dy) = Π(dy)dt; Π(·) is a sigma-finite
measure on the σ-algebra of Borel sets in Rd; ξ0 is the random vector inde-
pendent on w(t) and ν̃(t, ·).
We need the following result.
Lemma 1. Let
∫
Rd q(t, x, y)Π(dy) is bounded and uniformly continuous in
x with respect to t ∈ [0,∞) in every compact set |x| ≤ C. Let Π(·) be a
sigma-finite measure on the σ-algebra of Borel sets in Rd. For each x ∈ Rd
there exists limit
lim
T→∞
1
T
∫ T
0
∫
Rd
q(t, x, y) Π(dy)dt =
∫
Rd
q̄(x, y) Π(dy),
STOCHASTIC INTEREST RATE MODELS 249
where
∫
Rd q̄(x, y) Π(dy) is bounded and continuous. Then for any stochasti-
cally continuous process ξ(t) we have
P− lim
ε→0
∫ t
0
∫
Rd
q(s/ε, ξ(s), y) Π(dy)ds =
∫ t
0
∫
Rd
q̄(ξ(s), y) Π(dy)ds
for all arbitrary t ∈ [0, T ].
Proof. Since the process ξ(t) is stochastically continuous then [6, pp.218-
219] for any δ1 > 0 there exists such constant C > 0 that
sup
t∈[0,T ]
P{|ξ(t)| > C} ≤ δ1
and for arbitrary δ1 > 0 and δ2 > 0 there exists such δ3 > 0 that
P{|ξ(t1) − ξ(t2)| > δ2} ≤ δ1
for all |t1−t2| < δ3, t1, t2 ∈ [0, T ]. We choose δ2 such that | ∫
Rd
q(t, x1, y) Π(dy)
− ∫
Rd
q(t, x2, y) Π(dy)| < δ1 and | ∫
Rd q̄(x1, y) Π(dy)− ∫
Rd q̄(x2, y) Π(dy)| < δ1
for all t ∈ [0, T ], as |x1 − x2| ≤ δ2, |x1| ≤ C, |x2| ≤ C.
Let us consider partition 0 = t0 < t1 < . . . < tn = t, t ∈ [0, T ] such that
max
0≤k≤n−1
|tk+1 − tk| < δ3. For any δ > 0 we have
P
{∣∣∣
∫ t
0
∫
Rd
q(s/ε, ξ(s), y) Π(dy)ds−
∫ t
0
∫
Rd
q̄(ξ(s), y) Π(dy)ds
∣∣∣ > δ
}
≤
≤ P
{∣∣∣
n∑
k=1
∫ tk
tk−1
∫
Rd
[q(s/ε, ξ(s), y)− q(s/ε, ξ(tk−1), y)] Π(dy)ds
∣∣∣ > δ/3
}
+
+P
{∣∣∣
n∑
k=1
∫ tk
tk−1
∫
Rd
[q(s/ε, ξ(tk−1), y) − q̄(ξ(tk−1), y)] Π(dy)ds
∣∣∣ > δ/3
}
+
+P
{∣∣∣
n∑
k=1
∫ tk
tk−1
∫
Rd
[q̄(ξ(s), y)− q̄(ξ(tk−1), y)] Π(dy)ds
∣∣∣> δ/3
}
= P1+P2+P3.
For estimation of P1 and P3 we use Chebyshev inequality, properties of
chosen partition and above mentioned inequalities. Let us estimate P1 :
P1 ≤ 3
δ
E
( n∑
k=1
∫ tk
tk−1
∣∣∣
∫
Rd
[q(s/ε, ξ(s), y)− q(s/ε, ξ(tk−1), y)] Π(dy)
∣∣∣ds
)
≤
≤ 3
δ
E
( n∑
k=1
∫ tk
tk−1
(∣∣∣
∫
Rd
[q(s/ε, ξ(s), y)− q(s/ε, ξ(tk−1), y)] Π(dy)
∣∣∣×
250 VLADIMIR ZUBCHENKO
×χ{|ξ(s) − ξ(tk−1)| ≤ δ2} · χ{|ξ(s)| ≤ C, |ξ(tk−1)| ≤ C}+
+C1(χ{|ξ(s) − ξ(tk−1)| > δ2} + χ{|ξ(s)| > C} + χ{|ξ(tk−1)| > C})
)
ds
)
≤
≤ 3
δ
[
E
( n∑
k=1
∫ tk
tk−1
δ1 ds
)
+ C1
( n∑
k=1
∫ tk
tk−1
P{|ξ(s) − ξ(tk−1)| > δ2} ds+
+
n∑
k=1
∫ tk
tk−1
P{|ξ(s)| > C} ds +
n∑
k=1
∫ tk
tk−1
P{|ξ(tk−1)| > C} ds
)]
≤ Ctδ1
δ
.
Similarly we obtain P3 ≤ Ctδ1/δ, where we use notation C for any constant
independent on ε. For each k = 1, n from conditions of lemma we have
lim
ε→0
∫ tk
tk−1
∫
Rd
[q(s/ε, ξ(tk−1), y) − q̄(ξ(tk−1), y)] Π(dy)ds = 0 a.s.
Therefore lim
ε→0
P2 = 0, and for arbitrary δ1 > 0, δ > 0
lim
ε→0
P
{∣∣∣
∫ t
0
∫
Rd
[q(s/ε, ξ(s), y)− q̄(ξ(s), y)] Π(dy)ds
∣∣∣ > δ
}
≤ Ctδ1
δ
,
whence we obtain the statement of the lemma. �
Lemma 2. Let for each x ∈ Rd there exists lim
T→∞
1
T
∫ T
0
b(t, x) dt = b̄(x). The
function b̄(x) is bounded and continuous, function b(t, x) is bounded and con-
tinuous in x uniformly with respect to (t, x) in any region t ∈ [0,∞), |x| ≤ C,
and stochastic process ξ(t) is stochastically continuous, then
P− lim
ε→0
∫ t
0
b(s/ε, ξ(s)) ds =
∫ t
0
b̄(ξ(s)) ds
for all arbitrary t ∈ [0, T ].
The proof of this statement is similar to the proof of lemma 1.
We suppose that coefficients of equation (1) satisfy the following condi-
tions:
1) |f(t, x)|2 + ‖g(t, x)‖2 +
∫
Rd |q(t, x, y)|2Π(dy) ≤ C, where |f |2 =
d∑
i=1
f 2
i ,
‖g‖2 =
d∑
i,j=1
g2
ij;
2) For any N > 0 there exists LN > 0 such that
|f(t, x1) − f(t, x2)|2 + ‖g(t, x1) − g(t, x2)‖2+
STOCHASTIC INTEREST RATE MODELS 251
+
∫
Rd
|q(t, x1, y) − q(t, x2, y)|2 Π(dy) ≤ LN |x1 − x2|2,
for all xi ∈ Rd, i = 1, 2 such that |xi| ≤ N, i = 1, 2.
3) Functions f(t, x) and g(t, x) are continuous in x uniformly with respect
to t ∈ [0,∞) and x in every set |x| ≤ C. For each x ∈ Rd there exist
the following limits
lim
T→∞
1
T
∫ T
0
f(t, x) dt = f̄(x), lim
T→∞
1
T
∫ T
0
g(t, x)g∗(t, x) dt = Ḡ(x),
lim
T→∞
1
T
∫ T
0
∫
Rd
q(t, x, y)q∗(t, x, y)Π(dy)dt =
∫
Rd
Q̄(x, y)Π(dy).
Here g∗ is the matrix (vector) transpose to g, therefore for vector-
valued function q(t, x, y) the product q(t, x, y)q∗(t, x, y) is the d × d-
matrix-valued function.
4) The functions f̄(x), Ḡ(x),
∫
Rd Q̄(x, y) Π(dy) are bounded, continuous
in x. Matrix B̄(x) = Ḡ(x) +
∫
Rd Q̄(x, y) Π(dy) is uniformly parabolic.
5)
∫
Rd q(t, x, y)q∗(t, x, y) Π(dy) is bounded, continuous in x uniformly
with respect to t ∈ [0,∞) in every compact set |x| ≤ C.∫
Rd |q(t, x, y)|i Π(dy) ≤ C, i = 1, 6.
Theorem 1. Let conditions 1)-5) be fulfilled, k = min(k1, 2k2, 2k3) and the
function d(t, x) is bounded, continuous in x uniformly with respect to (t, x) in
any region t ∈ [0,∞), |x| ≤ C. For each x ∈ Rd there exists
lim
T→∞
1
T
∫ T
0
d(t, x) dt = d̄(x). The function d̄(x) is bounded and continuous.
Let us consider ηε(t) = (εk/t)
t/εk∫
0
d(s, ξ(s))ds, where ξ(t) is the solution of
equation (1).
1. If k1 = 2k2 = 2k3, then stochastic process ηε(t) converges in law, as
ε → 0, to stochastic process η̄(t) = 1
t
∫ t
0
d̄(ξ̄(s)) ds, where process ξ̄(t) is the
solution of stochastic differential equation
dξ̄(t) = f̄(ξ̄(t))dt + σ̄(ξ̄(t))dw̄(t), ξ̄(0) = ξ0, (2)
σ̄(x) = B̄1/2(x); w̄(t) is some d-dimensional Wiener process.
2. If k < k1, then in equation (2) the drift coefficient f̄(x) is absent; if
k < 2k2, then in equation (2) the diffusion matrix B̄(x) does not depend on
Ḡ(x); and if k < 2k3, then B̄(x) does not contain the term
∫
Rd Q̄(x, y) Π(dy).
Proof. We can rewrite ηε(t) in the form ηε(t) = (1/t)
∫ t
0
d(s/εk, ξ(s/εk)) ds.
Let us denote ξε(t) = ξ(t/εk), wε(t) = εk/2w(t/εk), ν̃ε(t, ·) = ν(t/εk, ·) −
252 VLADIMIR ZUBCHENKO
(t/εk)Π(·). It worth to note that for any ε > 0 wε(t) is the Wiener process
and ν̃ε(t, ·) is the centered Poisson measure. With these notations from
equation (1) we obtain
ξε(t) = ξ0 + εk1−k
∫ t
0
f(s/εk, ξε(s)) ds + εk2−k/2
∫ t
0
g(s/εk, ξε(s)) dwε(s)+
+εk3
∫ t
0
∫
Rd
q(s/εk, ξε(s), y) ν̃ε(ds, dy). (3)
It follows from conditions 1), 2) that the solution of equation (3) exists and
unique for each ε > 0.
Let us check that following conditions are fulfilled:
a) lim
h↓0
lim
ε→0
sup
|t−s|<h
P{|ξε(t) − ξε(s)| > δ} = 0 for any δ > 0, t, s ∈ [0, T ];
b) lim
N→∞
lim
ε→0
sup
t∈[0,T ]
P{|ξε(t)| > N} = 0.
Using properties of stochastic integrals, we can obtain the estimates
E|ξε(t)|2 ≤ C[E|ξ0|2 + (ε2(k1−k)T + ε2k2−k + ε2k3−k)t],
E|ξε(t) − ξε(s)|2 ≤ C[ε2(k1−k)|t − s| + ε2k2−k + ε2k3−k]|t − s|.
From Chebyshev inequality and obtained estimates we have fulfillment of
conditions a) and b). Similarly we can check conditions a) and b) for
stochastic process
ζε(t) = εk2−k/2
t∫
0
g(s/εk, ξε(s)) dwε(s)+εk3
t∫
0
∫
Rd
q(s/εk, ξε(s), y) ν̃ε(ds, dy).
Therefore [7, pp.13-18], for any sequence εn → 0, n = 1, 2, . . . there exists
a subsequence εm = εnm → 0, m = 1, 2, . . ., probability space, stochastic
processes ξ̃εm(t), ζ̃εm(t), ξ̄(t), ζ̄(t) defined on this space, such that ξ̃εm(t) →
ξ̄(t), ζ̃εm(t) → ζ̄(t) in probability, as εm → 0, and finite-dimensional dis-
tributions of ξ̃εm(t), ζ̃εm(t) coincide with finite-dimensional distributions of
ξεm(t), ζεm(t). Since we are interested in limit behavior of distributions, we
can consider processes ξεm(t) and ζεm(t) instead of ξ̃εm(t), ζ̃εm(t). From (3)
we obtain equation
ξεm(t) = ξ0 + εk1−k
m
∫ t
0
f(s/εk
m, ξεm(s)) ds + ζεm(t). (4)
From this point we will omit the sub-index m in εm for simplicity of notation.
It worth to note that processes ξε(t) and ζε(t) are stochastically continuous
STOCHASTIC INTEREST RATE MODELS 253
without discontinuity of second kind. Let us obtain some estimates for
processes ξε(t) and ζε(t):
E|ξε(t) − ξε(s)|4 = E
∣∣∣εk1−k
∫ t
s
f(τ/εk, ξε(τ)) dτ + ζε(t) − ζε(s)
∣∣∣4 ≤
≤ C[ε4(k1−k)|t − s|4 + E|ζε(t) − ζε(s)|4]. (5)
E|ζε(t) − ζε(s)|4 ≤ 8
(
ε4k2−2kE
∣∣∣
∫ t
s
g(τ/εk, ξε(τ)) dwε(τ)
∣∣∣4+
+ε4k3E
∣∣∣
∫ t
s
∫
Rd
q(τ/εk, ξε(τ), y) ν̃ε(dτ, dy)
∣∣∣4).
If we use Jensen’s inequality and properties of one-dimensional Wiener pro-
cess, we obtain
E
∣∣∣
∫ t
s
g(τ/εk, ξε(τ)) dwε(τ)
∣∣∣4 ≤ d
d∑
i=1
E
∣∣∣
d∑
j=1
∫ t
s
gij(τ/εk, ξε(τ)) dwj
ε(τ)
∣∣∣4 ≤
≤ d4
d∑
i,j=1
E
∣∣∣
∫ t
s
gij(τ/εk, ξε(τ)) dwj
ε(τ)
∣∣∣4 ≤ C(t − s)2.
Let us estimate
E
∣∣∣
∫ t
s
∫
Rd
q(τ/εk, ξε(τ), y) ν̃ε(dτ, dy)
∣∣∣2m
for m = 2, 3. (6)
Since
E
∣∣∣
∫ t
s
∫
Rd
q(τ/εk, ξε(τ), y) ν̃ε(dτ, dy)
∣∣∣2m
≤
≤ dm−1
d∑
i=1
E
∣∣∣
∫ t
s
∫
Rd
qi(τ/εk, ξε(τ), y) ν̃ε(dτ, dy)
∣∣∣2m
,
it is sufficient to estimate E| ∫ t
s
∫
Rd qi(τ/εk, ξε(τ), y) ν̃ε(dτ, dy)|2m, i = 1, d.
In view of this later on, estimating (6), we will consider one-dimensional
case. Therefore for simplicity of notations we will omit the sub-index i.
Let ξ̂ε(t) =
∫ t
s
∫
Rd q(τ/εk, ξε(τ), y) ν̃ε(dτ, dy). If we apply generalized Ito
formula to |ξ̂ε(t)|2m and take mathematical expectation, we get
E|ξ̂ε(t)|2m = E
∫ t
s
∫
Rd
{
|ξ̂ε(τ) + q(τ/εk, ξε(τ), y)|2m − |ξ̂ε(τ)|2m−
−2m|ξ̂ε(τ)|2m−1 sign ξ̂ε(τ) · q(τ/εk, ξε(τ), y)
} Π(dy)
εk
dτ.
254 VLADIMIR ZUBCHENKO
We obtain
E
∣∣∣
∫ t
s
∫
Rd
q(τ/εk, ξε(τ), y) ν̃ε(dτ, dy)
∣∣∣4 ≤
≤ C[(t − s)2ε−2k + (t − s)3/2ε−3k/2 + (t − s)ε−k],
E
∣∣∣
∫ t
s
∫
Rd
q(τ/εk, ξε(τ), y) ν̃ε(dτ, dy)
∣∣∣6 ≤
≤ C[(t − s)3ε−3k + (t − s)5/2ε−5k/2 + (t − s)2ε−2k + (t − s)ε−k].
Taking into account obtained estimates of E| ∫ t
s
g(τ/εk, ξε(τ)) dwε(τ)|4 and
E| ∫ t
s
∫
Rd q(τ/εk, ξε(τ), y) ν̃ε(dτ, dy)|4, we have:
E|ζε(t) − ζε(s)|4 ≤ C[(ε4k2−2k + ε4k3−2k)|t − s|2+
+ε4k3−3k/2|t − s|3/2 + ε4k3−k|t − s|]. (7)
Similarly E|ξε(t)− ξε(s)|6 ≤ C(ε6(k1−k)|t−s|6 +E|ζε(t)− ζε(s)|6). Using the
estimate of E| ∫ t
s
∫
Rd q(τ/εk, ξε(τ), y) ν̃ε(dτ, dy)|6 and taking into considera-
tion that k = min(k1, 2k2, 2k3), we obtain:
E|ξε(t) − ξε(s)|6 ≤ C, E|ζε(t) − ζε(s)|6 ≤ C. (8)
Since ξε(t) → ξ̄(t), ζε(t) → ζ̄(t) in probability, as ε → 0, then, using (8),
from (5) and (7) we obtain estimates
E|ξ̄(t) − ξ̄(s)|4 ≤ C(|t − s|4 + |t − s|2), E|ζ̄(t) − ζ̄(s)|4 ≤ C|t − s|2.
Therefore processes ξ̄(t) and ζ̄(t) satisfy Kolmogorov’s continuity condition
[8, pp.235-237]. It should be noted that process ζε(t) is the vector-valued
square integrable martingale with matrix characteristic
〈ζε, ζε〉(t) = ε2k2−k
∫ t
0
g(s/εk, ξε(s))g
∗(s/εk, ξε(s)) ds+
+ε2k3−k
∫ t
0
∫
Rd
q(s/εk, ξε(s), y)q∗(s/εk, ξε(s), y) Π(dy)ds. (9)
For any δ > 0
P
{∣∣∣
∫ t
0
d(s/εk, ξε(s)) ds −
∫ t
0
d̄(ξ̄(s)) ds
∣∣∣ > δ
}
≤
≤ 2
δ
E
∣∣∣
∫ t
0
[d(s/εk, ξε(s)) − d(s/εk, ξ̄(s))] ds
∣∣∣+
+P
{∣∣∣
∫ t
0
d(s/εk, ξ̄(s)) ds −
∫ t
0
d̄(ξ̄(s)) ds
∣∣∣ > δ/2
}
=
2
δ
I1 + I2.
STOCHASTIC INTEREST RATE MODELS 255
Since the function d(t, x) is continuous in x uniformly with respect to (t, x)
in any region t ∈ [0,∞), |x| ≤ N , then for any δ1 > 0 there exists δ2 > 0
such, that supt≥0 |d(t, x) − d(t, y)| ≤ δ1 as |x − y| ≤ δ2, |x| ≤ N, |y| ≤ N .
Therefore from boundedness of d(t, x) we have
I1 ≤ E
∫ t
0
|d(s/εk, ξε(s)) − d(s/εk, ξ̄(s))|χ{|ξε(s) − ξ̄(s)| ≤ δ2}×
×χ{|ξε(s)| ≤ N, |ξ̄(s)| ≤ N} ds + C
(∫ t
0
P{|ξε(s) − ξ̄(s)| > δ2} ds +
+
∫ t
0
P{|ξε(s)| > N} ds +
∫ t
0
P{|ξ̄(s)| > N} ds
)
≤
≤ δ1t +
C
N2
+ C
∫ t
0
P{|ξε(s) − ξ̄(s)| > δ2} ds.
Since P− lim
ε→0
ξε(s) = ξ̄(s), δ1 > 0 and N > 0 are arbitrary, then lim
ε→0
I1 = 0.
The process ξ̄(s) is continuous and function d(t, x) satisfies the condi-
tions of lemma 2. Therefore lim
ε→0
I2 = 0 and
lim
ε→0
∫ t
0
d(s/εk, ξε(s)) ds =
∫ t
0
d̄(ξ̄(s)) ds (10)
in law (because the distributions of ξεm(t), ζεm(t) coincide with distribu-
tions of stochastic processes ξ̃εm(t), ζ̃εm(t) and in fact we have proved that
P− limεm→0
∫ t
0
d(s/εk
m, ξεm(s)) ds =
∫ t
0
d̄(ξ̄(s)) ds).
Let us consider the case k1 = 2k2 = 2k3. From (4) we obtain
ξε(t) = ξ0 +
∫ t
0
f(s/εk, ξε(s)) ds + ζε(t),
where martingale ζε(t) has a matrix characteristic
〈ζε, ζε〉(t) =
∫ t
0
g(s/εk, ξε(s))g
∗(s/εk, ξε(s)) ds+
+
∫ t
0
∫
Rd
q(s/εk, ξε(s), y)q∗(s/εk, ξε(s), y) Π(dy)ds.
Using lemma 1 and lemma 2 it is easy to show that
P− lim
ε→0
∫ t
0
f(s/εk, ξε(s)) ds =
∫ t
0
f̄(ξ̄(s)) ds,
P− lim
ε→0
〈ζε, ζε〉(t) =
∫ t
0
B̄(ξ̄(s)) ds.
256 VLADIMIR ZUBCHENKO
Hence ζ̄(t) is a vector-valued continuous square integrable martingale with
matrix characteristic 〈ζ̄ , ζ̄〉(t) =
∫ t
0
B̄(ξ̄(s)) ds. It follows from conditions 4)-
5) and [9, pp.446-449] that there exists a d-dimensional Wiener process w̄(t)
such that ζ̄(t) =
∫ t
0
σ̄(ξ̄(s)) dw̄(s), where σ̄(x)σ̄∗(x) = B̄(x). Therefore the
process ξ̄(t) is the solution of stochastic differential equation
ξ̄(t) = ξ0 +
∫ t
0
f̄(ξ̄(s)) ds +
∫ t
0
σ̄(ξ̄(s)) dw̄(s). (11)
Furthermore, equation (11) has unique weak solution. Hence for any se-
quence εm → 0 the stochastic process ξεm(t) converges in probability to
the solution ξ̄(t) of equation (11). From this and (10) we have proof of
statement 1) of the theorem.
When k < k1 the boundedness of f(t, x) implies that
E
∣∣∣∫ t
0
f(s/εk, ξε(s)) ds
∣∣∣ ≤ C, therefore the second term in the right side
of (3) converges to 0 in probability, as ε → 0, and we obtain the first state-
ment in 2). From boundedness of g(t, x) and
∫
Rd q(t, x, y)q∗(t, x, y) Π(dy)
we obtain that either first or second term in the right side of (9) converges
to 0 in probability (respectively to the cases k < 2k2 or k < 2k3), as ε → 0.
Then we can complete the proof of statement 2) of the theorem as the proof
of statement 1). �
3. Long-term returns in stochastic interest rate models
Suppose that a stochastic process Xt satisfies the stochastic differential
equation
dXt = (2βXt + δt)dt + g(Xt)dwt +
∫
R
q(Xt, y)ν̃(dt, dy) ∀ t ∈ R+ (12)
β < 0; g(x), q(x, y) are non-random functions; x, y ∈ R; wt is one-dimensio-
nal Wiener process; ν̃(dt, dy) = ν(dt, dy)−Π(dy)dt, ν(dt, dy) is the Poisson
measure independent on wt, Eν(dt, dy) = Π(dy)dt ; Π(·) is a sigma-finite
measure on the σ-algebra of Borel sets in R; X0 is the random variable
independent on wt and ν̃(t, ·) and such that there is a constant c > 0 with
EX2
0 ≤ c.
We suppose that the following conditions are fulfilled:
1) g(0) = 0, q(0, y) = 0 ∀ y ∈ R;
2) there is a constant b > 0 with |g(x1) − g(x2)|2 ≤ b|x1 − x2| and∫
R
|q(x1, y) − q(x2, y)|2 Π(dy) ≤ b|x1 − x2| ∀x1, x2 ∈ R;
3) δt is non-random bounded function.
STOCHASTIC INTEREST RATE MODELS 257
It follows from these conditions that equation (12) has weak solution
[10, p.357]. If we use generalized Ito formula, we get the following rep-
resentation of the solution:
Xt = e2βt
(
X0 +
∫ t
0
δse
−2βs ds +
∫ t
0
g(Xs)e
−2βs dws+
+
∫ t
0
∫
R
q(Xs, y)e−2βs ν̃(ds, dy)
)
(13)
Theorem 2. Suppose that Xt satisfies the stochastic differential equa-
tion (12). Let conditions 1)-3) be fulfilled and 1
t
∫ t
0
δs ds → δ̄, as t → ∞.
Then 1
t
∫ t
0
Xsds → δ̄
−2β
in mean square, as t → ∞.
Proof. Using representation (13), let us estimate EX2
t , t ∈ R+.
EX2
u ≤ 4Ee4βu
(
X2
0 +
(∫ u
0
δse
−2βs ds
)2
+
(∫ u
0
g(Xs)e
−2βs dws
)2
+
+
(∫ u
0
∫
R
q(Xs, y)e−2βs ν̃(ds, dy)
)2
)
= I.
∫ u
0
Eg2(Xs)e
−4βs ds ≤ be−4βu
∫ u
0
E|Xs| ds ≤ be−4βu
∫ u
0
(EX2
s )1/2 ds.
The same estimate we have for
u∫
0
∫
R
Eq2(Xs, y)e−4βs Π(dy)ds. It follows from
[10, p.370] that in conditions of the theorem EX2
s is bounded on [0, u],
therefore
u∫
0
Eg2(Xs)e
−4βs ds < ∞ and
u∫
0
∫
R
Eq2(Xs, y)e−4βs Π(dy)ds < ∞.
Then
I = 4Ee4βuX2
0 + 4e4βu
(∫ u
0
δse
−2βs ds
)2
+
+4e4βu
∫ u
0
Eg2(Xs)e
−4βsds + 4e4βu
∫ u
0
∫
R
Eq2(Xs, y)e−4βs Π(dy)ds.
e4βu
(∫ u
0
δse
−2βs ds
)2
≤ Ce4βu
(e−2βu − 1
−2β
)2
≤ Ce4βu · e−4βu = C.
∫ u
0
Eg2(Xs)e
−4βsds +
∫ u
0
∫
R
Eq2(Xs, y)e−4βs Π(dy)ds ≤
≤ 2b
∫ u
0
E|Xs|e−4βs ds ≤ 2b
∫ u
0
(EX2
s )1/2e−4βs ds.
sup
u≤t
EX2
u ≤ 4EX2
0 + C + 8b sup
u≤t
e4βu
∫ u
0
(EX2
s )1/2e−4βs ds ≤
258 VLADIMIR ZUBCHENKO
≤ C1 + C2 sup
u≤t
e4βu
∫ u
0
(sup
τ≤t
EX2
τ )1/2e−4βs ds ≤
≤ C1 + (sup
τ≤t
EX2
τ )1/2 · C2 sup
u≤t
e4βu e−4βu − 1
−4β
≤ C1 + C3(sup
u≤t
EX2
u)1/2.
Therefore sup
u≤t
EX2
u − C3(sup
u≤t
EX2
u)1/2 ≤ C1, whence sup
u≤t
EX2
u ≤ C, where
constant C is independent on t, and thus EX2
t ≤ C.
From the equation (12) we obtain
Xt − X0
t
=
1
t
∫ t
0
2βXs ds +
1
t
∫ t
0
δs ds+
+
1
t
∫ t
0
g(Xs) dws +
1
t
∫ t
0
∫
R
q(Xs, y) ν̃(ds, dy). (14)
Using that EX2
t ≤ C, we have
E
(Xt − X0
t
)2
≤ 2
(EX2
t
t2
+
EX2
0
t2
)
→ 0, t → ∞.
1
t2
E
(∫ t
0
g(Xs) dws
)2
=
1
t2
∫ t
0
Eg2(Xs) ds ≤ 1
t2
∫ t
0
bE|Xs| ds ≤
≤ 1
t2
∫ t
0
b(EX2
s )1/2 ds ≤ C
t
→ 0, t → ∞.
1
t2
E
(∫ t
0
∫
R
q(Xs, y)ν̃(ds, dy)
)2
=
1
t2
∫ t
0
∫
R
Eq2(Xs, y) Π(dy)ds ≤
≤ 1
t2
∫ t
0
bE|Xs| ds ≤ 1
t2
∫ t
0
b(EX2
s )1/2 ds ≤ C
t
→ 0, t → ∞.
1
t
∫ t
0
δs ds → δ̄, t → ∞.
Therefore 1
t
∫ t
0
Xs ds → δ̄
−2β
in mean square �
Remark. In conditions of theorem 2 there is a constant C > 0 with
EX2
t ≤ C.
Let us generalize theorem 2 to the case of the stochastic process δt.
Theorem 3. Suppose that the stochastic differential equation (12) has a
solution Xt such that EX2
t < ∞ and let conditions 1)-2) be fulfilled. Sup-
pose that δt is a stochastic process and that there is a constant k > 0 such
that
∫ t
0
Eδ2
s ds ≤ k(1 + t) ∀ t ∈ R+ and 1
t
∫ t
0
δs ds → δ̄ in mean square, as
t → ∞. Then 1
t
∫ t
0
Xsds → δ̄
−2β
in mean square, as t → ∞.
STOCHASTIC INTEREST RATE MODELS 259
Proof. Similarly to the proof of theorem 2 we obtain
sup
u≤t
EX2
u ≤ C1 + sup
u≤t
Ee4βu
(∫ u
0
δse
−2βs ds
)2
+ C3
(
sup
u≤t
EX2
u
)1/2
.
Since
sup
u≤t
Ee4βu
(∫ u
0
δse
−2βs ds
)2
≤ sup
u≤t
Ee4βu
∫ u
0
δ2
s ds
∫ u
0
e−4βs ds ≤
≤ sup
u≤t
Ee4βu e−4βu
−4β
∫ u
0
δ2
s ds ≤ 1
−4β
∫ t
0
Eδ2
s ds ≤ C4(t + 1),
then
sup
u≤t
EX2
u − C3(sup
u≤t
EX2
u)1/2 ≤ C1 + C4(t + 1),
(
(sup
u≤t
EX2
u)1/2 − C3/2
)2
≤ C1 + C4(t + 1) + C2
3/4,
whence it follows that EX2
t ≤ C(t + 1), where constant C is independent
on t. Therefore
E
(Xt − X0
t
)2
≤ 2
(EX2
t
t2
+
EX2
0
t2
)
→ 0, t → ∞.
1
t2
E
(∫ t
0
g(Xs) dws
)2
≤ 1
t2
∫ t
0
b(EX2
s )1/2 ds ≤
≤ C
t2
∫ t
0
(s + 1)1/2 ds = C
(t + 1)3/2 − 1
t2
→ 0, t → ∞.
1
t2
E
(∫ t
0
∫
R
q(Xs, y)ν̃(ds, dy)
)2
≤ 1
t2
∫ t
0
b(EX2
s )1/2 ds → 0, t → ∞.
Therefore, taking into account (14), we obtain 1
t
∫ t
0
Xs ds → δ̄
−2β
in mean
square. �
4. A two-factor stochastic interest rate model
Let us consider an application of theorem 2 and theorem 3. We study
the two-factor model
drt = k(γt − rt) + σ
√
|rt| dwt +
∫
R
q(rt, y)ν̃(dt, dy),
dγt = k̃(γ∗ − γt) dt + σ̃
√
γt dwt,
k, k̃ > 0; γ∗, σ and σ̃ are positive constants; (wt)t≥0 and (w̃t)t≥0 are two
Wiener processes; ν̃(dt, dy) = ν(dt, dy) − Π(dy)dt, ν(dt, dy) is the Poisson
260 VLADIMIR ZUBCHENKO
measure independent on wt, Eν(dt, dy) = Π(dy)dt; Π(·) is a sigma-finite
measure on the σ-algebra of Borel sets in R.
Suppose that the first equation of the model has a solution and that
∫
R
|q(x1, y)−q(x2, y)|2 Π(dy) ≤ b|x1−x2| ∀x1, x2 ∈ R, q(0, y) = 0 ∀ y ∈ R.
We are interested in the convergence of the long-term return 1
t
∫ t
0
rs ds.
Applying theorem 2 to the second equation of the model we have, that
1
t
∫ t
0
γs ds → y∗ in mean square. Really, if we define Yt = 4γt/σ̃
2, then Yt
satisfies stochastic differential equation of the kind
dYt = (δt + 2β̃Yt)dt + g̃(Yt)dw̃t,
with β̃ = −k̃/2, δt = 4k̃γ∗/σ̃2 ∀ t ∈ R+, g̃(Yt) = 2
√
Yt.
We can note, that [11] there exists a solution of this equation, it is unique
and non-negative. Since conditions of theorem 2 are fulfilled, then
1
t
∫ t
0
Ys ds → 4γ∗/σ̃2 in mean square and accordingly 1
t
∫ t
0
γs ds → γ∗ in
mean square. Further, taking into consideration remark to the theorem 2,
we have Eγ2
t ≤ C, where constant C is independent on t.
Now we consider first equation of the model. If we define Xt = 4rt/σ
2,
then Xt satisfies, in the notations of theorem 3, the equation of the following
kind
dXt = (2βXt + δt)dt + g(Xt)dwt +
∫
R
q1(Xt, y)ν̃(dt, dy),
with β = −k/2, δt = 4kγt/σ
2, g(Xt) = 2
√|Xt|, q1(Xt, y) = 4
σ2 q(
σ2
4
Xt, y).
Since conditions of theorem 3 are fulfilled and 1
t
∫ t
0
δs ds = (1
t
∫ t
0
γs ds) 4k
σ2 →
γ∗ 4k
σ2 in mean square, then 1
t
∫ t
0
Xs ds → 4γ∗
σ2 in mean square and finally
1
t
∫ t
0
rs ds → γ∗ in mean square.
We conclude that the long-term return converges in mean square to the
long-term constant value γ∗.
References
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nian motion with drift, Ann. I.H.P., 41 (3), (2005), 335–347.
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perpetual integral functionals of diffusions, Elect. Comm. in Probab., 11,
(2006), 108–117.
3. Borysenko O., Malyshev I., The limit behaviour of integral functional of the
solution of stochastic differential equation depending on small parameter,
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STOCHASTIC INTEREST RATE MODELS 261
5. Deelstra, G. and Delbaen, F., Long-term returns in stochastic interest rate
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Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: zubchenko@univ.kiev.ua
|
| id | nasplib_isofts_kiev_ua-123456789-4528 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T17:08:03Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Zubchenko, V. 2009-11-24T15:40:59Z 2009-11-24T15:40:59Z 2007 Long-term returns in stochastic interest rate models / V. Zubchenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 247–261. — Бібліогр.: 11 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4528 We consider the behavior of integral functional of the solution of stochastic differential equation with coefficients contained small parameter. The dependence on the order of small parameter in every term of equation with Wiener process and Poisson measure term is studied. We observe the convergence of the long-term return, using an extension of the Cox-Ingersoll-Ross stochastic model of the short interest rate. Obtained results are applied for studying of two-factor stochastic interest rate model. en Інститут математики НАН України Long-term returns in stochastic interest rate models Article published earlier |
| spellingShingle | Long-term returns in stochastic interest rate models Zubchenko, V. |
| title | Long-term returns in stochastic interest rate models |
| title_full | Long-term returns in stochastic interest rate models |
| title_fullStr | Long-term returns in stochastic interest rate models |
| title_full_unstemmed | Long-term returns in stochastic interest rate models |
| title_short | Long-term returns in stochastic interest rate models |
| title_sort | long-term returns in stochastic interest rate models |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4528 |
| work_keys_str_mv | AT zubchenkov longtermreturnsinstochasticinterestratemodels |