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Long-term behavior of stochastic interest rate models with jumps and memory
Insurance: Mathematics and Economics 53 (2013) 266–272
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Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime
Long-term behavior of stochastic interest rate models with jumps and memory Jianhai Bao a , Chenggui Yuan b,∗ a
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410075, China
b
Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, UK
highlights • We show the nonnegative property of solutions for a class of stochastic equations. • We investigate the long-term return for stochastic interest rate models. • An application to a two-factor CIR model is presented.
article
abstract
info
Article history: Received November 2011 Received in revised form February 2013 Accepted 18 May 2013
The long-term interest rates, for example, determine when homeowners refinance their mortgages in mortgage pricing, play a dominant role in life insurance, decide when one should exchange a long bond to a short bond t in pricing an option. In this paper, for a one-factor model, we reveal that the long-term return t −µ 0 X (s)ds for some µ ≥ 1, in which X (t ) follows an extension of the Cox–Ingersoll–Ross model with jumps and memory, converges almost surely to a reversion level which is random itself. Such a convergence can be applied in the determination of models of participation in the benefit or of saving products with a guaranteed minimum return. As an immediate application of the result obtained for the one-factor model, for a class t of two-factor model, we also investigate the almost sure convergence of the long-term return t −µ 0 Y (s)ds for some µ ≥ 1, where Y (t ) follows an extended Cox–Ingersoll–Ross model with stochastic reversion level −X (t )/(2β) in which X (t ) follows an extension of the square root process. This result can be applied to, e.g., how the percentage of interest should be determined when insurance companies promise a certain fixed percentage of interest on their insurance products such as bonds, life-insurance and so on. © 2013 Elsevier B.V. All rights reserved.
MSC: 60H10 60H30 Keywords: Interest rate Cox–Ingersoll–Ross model Jump Memory One-factor model Two-factor model Long-term return
1. Introduction
models of the short-term interest rate in the framework
Cox et al. (1985) proposed the short-term interest rate dynamics as dS (t ) = κ(γ − S (t ))dt + σ
S (t )dW (t )
for positive constants κ, γ and σ and standard Brownian motion {W (t ) : t ≥ 0}. This model is known as the Cox–Ingersoll–Ross (CIR) model and has some empirically relevant properties, e.g., the randomly moving interest rate is elastically pulled towards the long-term constant value γ . In order to better capture the properties of empirical data, Chan et al. (1992) nested a wide range of
∗
Corresponding author. E-mail address: [email protected] (C. Yuan).
0167-6687/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.insmatheco.2013.05.006
dS (t ) = κ(γ − S (t ))dt + σ S (t )α dW (t )
(1)
for α ≥ 1/2 with appropriate restrictions on the parameters κ, γ , α . Here γ , toward which rates drift, stands for the long-term mean of the process, κ means the speed of the drift, σ measures the volatility, and 2α denotes the variance elasticity. In particular, by the X2 test to the one-month US. Treasury bill yields, Chan et al. (1992) compared the ability of each model with different α to capture the volatility of the term structure, found that the value of α is the most important feature differentiating interest rate models, and revealed that the most successful models in capturing the dynamics of the short-term interest rate are those that allow the volatility of interest rate changes to be highly sensitive to the level of the riskless rate (e.g., α ≥ 1). Note that the long-term mean γ , the speed of drift κ and the volatility σ are not constants either and there is strong evidence to indicate that they are Markov jump pro-
J. Bao, C. Yuan / Insurance: Mathematics and Economics 53 (2013) 266–272
cesses. Another generalization of the CIR model is to use regimeswitching such as in Ang and Bekaert (2002) and Gray (1996), to name a few. On the other hand, from the economic point of view, there is some evidence indicating that certain events happening before the trading periods influence the current and future asset price, and therefore many scholars introduce delays to the financial models. For example, Arriojas et al. (2007) took delay into consideration for the price process of underlying assets and developed a Black–Scholes type formula. Benhabib (2004) considered a linear, flexible price model, where nominal interest rates are measured by a flexible distributed delay. Stoica (2004) computed the logarithmic utility of an insider when the financial market is modeled by a stochastic delay equation and offered an alternative to the anticipating delayed Black–Scholes formula. However, the meanreverting square root process cannot explain some empirical phenomena, such as stochastic volatility. To explain these phenomena, jump processes are also used in the financial models, e.g., Bardhan and Chao (1993), Chan (1999), Henderson and Hobson (2003) and Merculio and Runggaldier (1993). There is extensive literature on quantitative and qualitative properties of the generalized CIR-type models. For instance, different convergence results and the corresponding applications of the long-term returns are found in Deelstra and Delbaen (1995, 1997) and Zhao (2009). Strong convergence of the Monte Carlo simulations are studied in Deelstra and Delbaen (1998) and Wu et al. (2008, 2009), and the representations of solutions are presented in Arriojas et al. (2007) and Stoica (2004). Deelstra and Delbaen (1995, 1997) investigated the long-term returns of the CIR model, and Zhao (2009) extended those results to the jump models. Noting that the reversion level γ in (1) is a constant, in order to better reflect the time dependence caused by the cyclical nature of the economy or by expectations concerning the future impact of monetary, as in Deelstra and Delbaen (1995, 1997), we can assume that the short-term interest rate model has a stochastic reversion level. As described above, there is a natural motivation for considering the stochastic interest rate model where all three features, delay, jumps and time dependence of reversion level, are presented. In this paper, we consider the stochastic interest rate model with jumps and memory in the form
dX (t ) = {2 β X (t ) + δ(t )}dt + σ X γ (t − τ ) |X (t )|dW (t ) + g (X (t −), u)N˜ (dt , du), U X0 = ξ ∈ D ,
(2)
where X (t −) := lims↑t X (s) and D denotes all real bounded càdlàg functions defined on [−τ , 0] for some τ > 0. The integral g ( X ( t −), u)N˜ (dt , du) depends on the Poisson measure and is U regarded as a jump. The diffusion term is dependent on the past through X γ (t − τ ) and so is called delay or memory. Precise assumptions on the data of the problem (2) are given in Section 2 below. The long-term interest rates play an important role in finance and insurance. For instance, the long-term interest rates determine when homeowners refinance their mortgages in mortgage pricing, play a dominant role in life insurance, decide when one should exchange a long bond to a short bond in pricing an option. In this light, for the instantaneous interest rate model (2), it is interesting t to investigate the long-term return t −µ 0 X (s)ds for some µ ≥ 1.
t
We shall reveal that the long-term return t −µ 0 X (s)ds converges almost surely to a stochastic reversion level, which will be stated in Theorem 1 below. As stated in Deelstra and Delbaen (2000), t the limit of long-term return t −µ 0 X (s)ds can be applied in the determination of models of participation in the benefit or of saving products with a guaranteed minimum return.
267
As we know, one-factor models imply that the instantaneous returns on bonds of all maturities are perfectly correlated, which is clearly inconsistent with reality, e.g., Longstaff and Schwartz (1992). However empirical research, e.g., Brigo and Mercurio (2006), Cassola and Barros Luis (2001) and Longstaff and Schwartz (1992), has suggested that two-factor models, including the shortterm interest rate and the instantaneous variance of changes in the short-term interest rate, are better than one-factor models to capture the behavior of the term structure in the real world. This is because the two-factor models allow contingent claim values to reflect both the current level of interest rates as well as the current level of interest rate volatility. Cox et al. (1985, p. 399) introduced a model by two independent factors, r1 and r2 , and the instantaneous interest rate is the sum of two factors, that is,
r (t ) = r1 (t ) + r2 (t )
dri (t ) = κi (θi − ri (t ))dt + σi (t ) ri (t )dBi (t ),
i = 1, 2,
where B1 (t ) and B2 (t ) are independent Brownian motions, θi is the long-term mean factor ri reverts to, and κi and σi are constants. In another example of a multi-factor model, the domestic short-rate rd and the European short-rate re satisfy the following stochastic differential equation (SDE) drd = [a + b(re − rd )]dt + σd dBd (t ), dre = c (d − re )dt + σe dBe (t ), Cov(dBd (t ), dBe (t )) = ρ dt ,
where Bd (t ) and Be (t ) are Brownian motions with instantaneous correlation ρ , and a, b, c , d, σd , σe are positive constants, Corzo and Schwartz (2000) investigated how to price the European bond and several other interest rate derivatives. As an immediate application of Theorem 1, we consider the long-term return of the two-factor model in the form
dX (t ) = {2β1 X (t )+ δ(t )}dt + σ1 X γ1 (t − τ ) |X (t )|dW1 (t ) +ϑ1 X (t ) uN˜ 1 (dt , du), U (3) dY (t ) = {2β2 Y (t )+ X (t )}dt + σ2 Y γ2 (t − τ ) |Y (t )|dW2 (t ) +ϑ2 Y (t ) uN˜ 2 (dt , du) U
with the initial data (X (t ), Y (t )) = (ξ (t ), η(t )), t ∈ [−τ , 0]. Here W1 (t ), W2 (t ) are Brownian motions, N1 (dt , du), N2 (dt , du) represent Poisson counting measures, defined on (Ω , F , {Ft }t ≥0 , P), with characteristic measures λ1 (·) and λ2 (·) respectively, and ξ , η ∈ D . More details on the parameters of model (3) are to be presented in Section 4. In model (3), the short interest rate Y (t ) follows an extended CIR model with stochastic reversion level −X (t )/(2β), where X (t ) follows an extension of the square root process. For model (3), we are interested t in the almost sure convergence of the long-term return t −µ 0 Y (s)ds for some µ ≥ 1. Such a convergence can be applied to the finance and insurance markets. For example, the customer wants a return as high as possible, and insurance companies wonder how the percentage of interest should be determined when they promise a certain fixed percentage of interest on their insurance products such as bonds, life-insurance and so on. The rest of the paper is organized as follows. In Section 2 we introduce some preliminaries, show the nonnegative property of X (t ) as nominal instantaneous interest rate determined by (2), and give an auxiliary lemma of Theorem 1. Section 3 is devoted to t the almost sure convergence of long-term return t −µ 0 X (s)ds for some µ ≥ 1 with X (t ) a generalized CIR model determined by (2), in which the corresponding results in Deelstra and Delbaen (1995, 1997) and Zhao (2009) are extended. In the final section, as an application of Theorem 1,we consider the almost sure convergence t of long-term return t −µ 0 Y (s)ds for some µ ≥ 1 determined by (3) with stochastic reversion level −X (t )/(2β), where the results of Zhao (2009, Theorem 2) are developed.
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Applying the Itô formula yields that for any t ∈ (0, T ]
2. Preliminaries Throughout this paper, let (Ω , F , {Ft }t ≥0 , P) be a complete probability space with the filtration {Ft }t ≥0 satisfying the usual conditions (i.e. it is right continuous and F0 contains all P-null sets). Let W (t ) be a scalar Brownian motion defined on the probability space (Ω , F , {Ft }t ≥0 , P). Let B (R+ ) be the Borel σ -algebra on R+ , and λ(dx) a σ -finite measure defined on B (R+ ). Let Dp ⊂ R+ be a countable set and p = (p(t )), t ∈ Dp , a stationary Ft -Poisson point process on R+ with characteristic measure λ(·). Denote N (dt , du) by the Poisson counting measure associated with p, i.e., N (t , U ) = s∈Dp ,s≤t IU (p(s)) for U ∈ B (R+ ). We assume
λ(U ) < ∞ and let N˜ (dt , du) := N (dt , du) − dt λ(du) be the compensated Poisson measure associated with N (dt , du). For the sake of convenience, we will denote C > 0 by a generic constant whose values may change from line to line. For model (2), we make the following assumptions:
Eφk (X (t )) = Eφk (ξ (0)) + E
lim
1
t →∞
tµ
+
2
t
E
φk′′ (X (s))X 2γ (s − τ )X (s)ds
0
t 0
{φk (X (s) + g (X (s), u)) U
− φk (X (s)) − φk′ (X (s))g (X (s), u)}λ(du)ds. By (i)–(iii), Taylor’s expansion and (A4), it then follows from (4) that for t ∈ (0, T ]
Eφk (X (t )) ≤
C k
1
t
{(φk′ (θ g (X (s), u) + X (s))
+E 0
0
U
− φk′ (X (s)))g (X (s), u)}dθ λ(du)ds t 1 C ≤ +E {(φk′ (θ g (X (s), u) + X (s)) k
0
0
U
− φk′ (X (s)))g (X (s), u)}1{X (s)≤0} dθ λ(du)ds t C 1 1 ≤ + 2K 2 λ 2 (U )E X − (s)1{X (s)≤0} ds
t
δ(s)ds = ν a.s. 0
|g (x, u) − g (y, u)|2 λ(du) ≤ K |x − y|2 ,
σ2
+E
(A3) g : Ω × R × U → R with g (0, u) = 0 and there exists K > 0 such that
φk′ (X (s)){2β X (s) + δ(s)}ds 0
(A1) β < 0, σ > 0 and γ ∈ [0, 12 ). (A2) δ : Ω × R+ → R+ , and there exist constants µ ≥ 1 and ν ≥ 0 such that
t
≤
x, y ∈ R.
U
k C
0 1 2
1 2
+ 2K λ (U )Tak−1 t 1 1 Eφk (X (s))ds. + 2K 2 λ 2 (U ) k
0
(A4) For any θ ∈ [0, 1], x + θ g (x, u) ≥ 0 whenever x > 0.
This, together with the Gronwall inequality, gives that
Remark 1. There are some examples, where (A4) holds, e.g., for x ∈ R and u ∈ U , g (x, u) ≥ 0 or −g (x, u) ≤ x whenever g (x, u) ≤ 0.
EX (t ) − ak−1 ≤ Eφk (X (t )) ≤ C −
1 k
+ ak − 1 ,
t ∈ (0, T ].
Since (2) is mainly used to model stochastic volatility or interest rate or an asset price, it is critical that the solution {X (t )}t ≥−τ will never become negative, which is guaranteed by the following lemma.
Thus, EX − (t ) = 0 as k → ∞ and therefore X (t ) ≥ 0 a.s. for any t ∈ (0, T ]. Hence the nonnegative property of the solution {X (t )}t ≥0 follows from the arbitrariness of T > 0.
Lemma 1. Under (A1)–(A4), (2) admits a unique nonnegative solution {X (t )}t ≥0 for any ξ ∈ D .
Remark 2. Wu et al. (2008) studied the strong convergence of Monte Carlo simulations of mean-reverting square root process with jumps in the form
Proof. An application of Zhao (2010, Theorems 2.1 and 2.2) gives that (2) has a unique strong solution X (t ) on [0, τ ]. Repeating this procedure we see that (2) also admits a unique strong solution X (t ) on [τ , 2τ ]. Hence (2) has a unique strong solution X (t ) on the horizon t ≥ 0. Moreover, carrying out a similar argument to that of Wu et al. (2009, Theorem 2.1), we deduce that there exists C > 0 such that for any T > 0, q > 0
E|X (t )|q ≤ C ,
t ∈ [0, T ].
(4)
To end the proof, it is sufficient to show the nonnegative property of the solution {X (t )}t ∈[0,T ] . We adopt the method of Yamada and Watanabe (1971). Let a a0 1 = 1 and ak = exp(−k(k + 1)/2), k = 1, 2 · · · . Then a k−1 kx dx = 1 and there is a continuous k nonnegative function ψk (x), x ∈ R+ , which possesses the support (ak , ak−1 ), has the integral 1 and satisfies ψk (x) ≤ kx2 . Define an auxiliary function φk (x) = 0 for x ≥ 0 and
φk (x) :=
−x
y
ψk (u)du,
dy 0
x < 0.
0
Then φk ∈ C 2 (R; R+ ) has the following properties: (i) −1 ≤ φk′ (x) ≤ 0 for −ak−1 < x < −ak , otherwise φk′ (x) = 0; (ii) |φk′′ (x)| ≤ k|2x| for −ak−1 < x < −ak , otherwise φk′′ (x) = 0;
(iii) x− − ak−1 ≤ φk (x) ≤ x− , x ∈ R.
dS (t ) = α[µ − S (t )]dt + σ
|S (t )|dW (t ) + δ S (t −)dN˜ (t )
for α, µ, σ > 0, and investigated the nonnegative property of S (t ). Our work is a generalization of this model. Zhao (2009) also showed the nonnegative property of the solution X (t ) to (2) with γ = 0, g (x, u) = 0 for x < 0 and U g 2 (x, u)λ(du) ≤ K |x| for some constant K > 0. The following auxiliary lemma gives an estimate of X (t ), determined by the one-factor model (2), and plays an important role in analyzing the asymptotic behavior of long-term return t t −µ 0 X (s)ds for some µ ≥ 1. Lemma 2. Let (A1)–(A4) hold and assume further that 4β + K < 0. Then there exist κ > 0 and C > 0 such that
E(e−κβρ X 2 (ρ)) ≤ C + C E
ρ
e−κβ s (δ 2 (s) + 1)ds,
0
where ρ > 0 is a bounded stopping time. Proof. We first recall the Young inequality: for any a, b > 0 and α ∈ (0, 1) aα b1−α ≤ α a + (1 − α)b.
(5)
J. Bao, C. Yuan / Insurance: Mathematics and Economics 53 (2013) 266–272
Let κ > 0 and ϵ > 0 be arbitrary. By the Itô formula, (A3) and the inequality (5), we obtain that d(e
−κβ t
=e
X (t )) = −κβ e 2
−κβ t
−κβ t
X (t )dt + e 2
−κβ t
dX (t ) 2
Thus, substituting this into (8) one has tµ
2
+
U
≤e {((4 − κ)β + ϵ + K )X 2 (t ) + C1 (ϵ)X 4γ (t − τ ) + C1 (ϵ)δ 2 (t )}dt + M1 (t ) + M2 (t ) ≤ e−κβ t {((4 − κ)β + ϵ + K )X 2 (t ) + ϵ eκβτ X 2 (t − τ ) + C1 (ϵ)δ 2 (t ) + C2 (ϵ)}dt + M1 (t ) + M2 (t ) for some constants C1 (ϵ) > 0 and C2 (ϵ) > 0 dependent on ϵ , 3
where M1 (t ) := 2σ e−κβ t X 2 (t )X γ (t − τ )dW (t ) and M2 (t ) := e−κβ t U {g 2 (X (t ), u)+2X (t )g (X (t ), u)}N˜ (dt , du). Integrating from 0 to ρ and taking expectations on both sides, we arrive at
E(e−κβρ X 2 (ρ))
ρ ≤ C ∥ξ ∥2 + ((4 − κ)β + 2ϵ + K )E e−κβ s X 2 (s)ds 0 ρ 2 + (C1 (ϵ) ∨ C2 (ϵ))E (δ (s) + 1)ds. 0
Due to 4β + K < 0, we choose κ > 0 and ϵ > 0 such that (4 − κ)β + 2ϵ + K = 0. 3. Almost sure convergence of long-term returns Our first main result in this paper is stated as follows. Theorem 1. Let (A1)–(A4) hold and 4β + K < 0. Assume further that there exist λ > 0 and θ ∈ [1, 2µ] such that (6)
∞
√
X γ (t − τ ) |X (t )|
(1 + t )µ
0 0
Then lim
t →∞
1
δ(s) X (s) + ds = 0 a.s. 2β
0
(7)
0
δ(s) X (t ) − ξ (0) ds = X (s) + 2β 2β t σ − X γ (s − τ ) |X (s)|dW (s) 2β 0 t 1 − g (X (s−), u)N˜ (ds, du). 2β 0 U
X (t ) = e
ξ (0) +
t
e−2β s δ(s)ds
0 t +σ e−2β s X γ (s − τ ) |X (s)|dW (s) 0 t −2 β s ˜ + e g (X (s−), u)N (ds, du) .
0
U
τn := inf t ≥ 0
t
t 0
δ 2 (s) ds ≥ n . (1 + s)2µ
δ 2 (s)ds ≤ L(1 + t )θ a.s.
0
This, together with θ ∈ [1, 2µ], leads to
(8)
On the other hand, an application of Itô formula to e−2β t X (t ) yields that 2β t
(9)
In light of (6) there exists an L > 0 such that
Proof. From (2)
t
dW (t ) exists a.s.
For each n > ∥ξ ∥ define a stopping time
t
tµ
e2β(t −s) δ(s)ds
0
To derive the assertion (7), it is sufficient to verify that Ii (t ) → 0 a.s., i = 1, . . . , 6, as t → ∞. Since β < 0 and µ ≥ 1, I1 (t ) → 0 as t → ∞. Following a similar argument to that of Deelstra and Delbaen and noting that limt →∞ [(1 + t )µ − (1 + √ µ(1995, p. 168), µ t − t ) ]/(1 + t ) = 0, by (A2) we deduce that I2 (t ) → 0 a.s. for t → ∞. Next, in order to show I3 (t ) → 0 a.s. and I4 (t ) → 0 a.s. whenever t → ∞, by Deelstra and Delbaen (1995, Kronecker’s lemma), it suffices to check that
t
δ 2 (s)ds ≤ λ a.s.
2β t µ
+
−κβ t
t
1
δ(s) (e2β t − 1)ξ (0) ds = 2β 2β t µ
µ 1 σ 1 1+ − 2 β t 2β t e (1 + t )µ t e−2β s δ(s)X γ (s − τ ) |X (s)|dW (s) × 0 µ t σ 1 1 X γ (s − τ ) |X (s)|dW (s) − 1+ µ 2β t (1 + t ) 0 µ t 1 1 1 g (X (s−), u)N˜ (ds, du) − 1+ 2β t (1 + t )µ 0 U µ 1 1 1 + 1+ 2β t e−2β t (1 + t )µ t × e−2β s g (X (s−), u)N˜ (ds, du) 0 U µ µ 1 1 σ 1 =: I1 (t ) + I2 (t ) + 1+ I3 (t ) − 1+ I4 ( t ) 2β t 2β t µ µ 1 1 1 1 − 1+ I5 (t ) + 1+ I6 (t ). 2β t 2β t
g 2 (X (t ), u)λ(du) dt + M1 (t ) + M2 (t )
1 lim sup θ t t →∞
X (s) +
0
2γ
+
t
1
(4 − κ)β X (t ) + σ X (t )X (t − τ ) + 2δ(t )X (t ) 2
269
s 2 δ (u)du δ 2 (s) ds = lim 0 2µ s →∞ ( 1 + s ) ( 1 + s)2µ 0 ∞ s ds + 2µ δ 2 (u)du ( 1 + s)2µ+1 0 0 ∞ L 1 ≤ lim + 2µL ds s→∞ (1 + s)2µ−θ (1 + s)2µ+1−θ 0 < ∞ a.s. ∞
Hence {τn = ∞} ↑ Ω and then it is sufficient to verify (9) on {τn = ∞}. Furthermore, observing that J (t ) :=
t
0
√
X γ (s − τ ) |X (s)|
(1 + s)µ
1{s≤τn } dW (s)
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J. Bao, C. Yuan / Insurance: Mathematics and Economics 53 (2013) 266–272
is a local martingale, we only need to check that J (t ) is an L2 -bounded martingale. By the Itô isometry and the inequality (5), we obtain that
E |J (t )| =
E{X 2γ (s − τ )X (s)}1{s≤τn }
t
2
(1 + s)2µ
0
E{X (s)1{s≤τn } }
t
ds
2(1 + s)2µ
0
ds
E{X 4γ (s − τ )1{s≤τn } }
t
+
2(1 + s)2µ
0
ds
t E{X 2 (s)1{s≤τn } } (1 − 2γ ) ds + ds 2 µ 2(1 + s)2µ 0 2(1 + s) 0 t E{X 2 (s − τ )1{s≤τn } } +γ ds (1 + s)2µ 0 t κβ s −κβ(s∧τn ) 2 e E{e X (s ∧ τn )} 1 − 2γ + ds ≤ 2µ 2(2µ − 1) 2 ( 1 + s ) 0 t κβ(s−τ ) −κβ(s∧τn −τ ) 2 e E{e X (s ∧ τn − τ )} +γ ds 2 (1 + s) µ 0
≤
J1 ( t ) ≤ C
t
1 + s + eκβ s E
0
e−κβ r δ 2 (r )dr
4. An application to the two-factor CIR model
1 + s + eκβ s
t
s
≤C
0
e−κβ r E(δ 2 (r )1{r ≤τn } )dr
2(1 + s)2µ
0
eκβ s
t
≤ C +C 0
2(1 + s)
τn
ds
s
2µ
ds
2(1 + s)2µ
0
s∧τn
e−κβ r E(δ 2 (r )1{r ≤τn } )drds 0
δ 2 (r ) dr ≤ C 1 + n . 2 µ (1 + r )
≤ C + CE 0
(10)
Noting that J2 (t ) ≤ C + γ ∥ξ ∥2 t
e
×
κβ(s−τ )
t
τ
(E{e
eκβ(s−τ ) 2(1 + s)2µ
ds + γ
−κβ(s∧τn −τ ) 2
X (s ∧ τn − τ )1{s∧τn >τ } })
2(1 + s)2µ
τ
g (X (s−), u)
t 0
U
( 1 + s) µ
g (X (s−), u)
t
2
=E 0
1{s≤τn } N˜ (ds, du)
U
(1 + s)2µ
One-factor models imply that the instantaneous returns on bonds of all maturities are perfectly correlated, which is clearly inconsistent with reality, e.g. Longstaff and Schwartz (1992). On the other hand, empirical research, e.g., Brigo and Mercurio (2006), Cassola and Barros Luis (2001) and Longstaff and Schwartz (1992), has suggested that two-factor models, including the short-term interest rate and the instantaneous variance of changes in the short-term interest rate, are better than one-factor models to capture the behavior of the term structure in the real world. This is because the two-factor models allow contingent claim values to reflect both the current level of interest rates as well as the current level of interest rate volatility. As an immediate application of Theorem 1, for two-factor model (3), in this section t we consider the almost sure convergence of the long-term t −µ 0 Y (s)ds for some µ ≥ 1, where the short interest rate Y (t ) follows an extended CIR model with stochastic reversion level −X (t )/(2β), and X (t ) follows an extension of the square root process. For two-factor model (3), we assume that (A5) β1 < 0, σ1 > 0, γ1 ∈ [0, 12 ), ϑ1 > 0 and δ(t ) satisfies (A2).
ds,
and carrying out a similar argument to that of (10), we conclude that there exists C (n, µ, α) > 0 such that J2 (t ) ≤ C (n, µ, α). Finally, I5 (t ) → 0 a.s. and I6 (t ) → 0 a.s. whenever t → ∞ follow by observing that
E
(11)
The linear case g (x, u) = C |u|x, for some C > 0, does not satisfy (11). However, Theorem 1 is available for such a fundamental case. This proof is different from that of Zhao (2009) by incorporation of the memory, since it shows how to deal with the delay term when showing that J (t ) is an L2 -bounded martingale.
By Lemma 2 with κ > 0 it follows that
g 2 (x, u)λ(du) ≤ K |x| for some constant K > 0. U
=: (1 − 2γ )/(4µ − 2) + J1 (t ) + J2 (t ).
1
dX (t ) = {2β X (t ) + δ(t )}dt + σ |X (t )| 2 +l dW (t ), X (0) = x > 0,
whereas Deelstra and Delbaen (1995) investigated the long-term returns of such a model only when l = 0. Zhao (2009) discussed the long-time behavior of (2) with γ = 0, g (x, u) = 0 for x < 0, u ∈ U, and
t
Remark 4. For β < 0, σ > 0 and l ∈ [0, 12 ), Theorem 1 applies to the generalized mean-reverting model
2
≤
pricing, play a dominant role in life insurance, decide when one should exchange a long bond to a short bond in pricing an option.
2
1{s≤τn } λ(du)ds.
Remark 3. By Theorem 1, under some appropriate conditions on the parameters, we deduce that the long-term return converges almost surely to a reversion level which is random itself. As stated in Deelstra and Delbaen (1995, 1997), the long-term return determines when homeowners refinance their mortgages in mortgage
(A6) β2 < 0, σ2 > 0, γ2 ∈ [0, 12 ), ϑ2 > 0 and ϑ22 −4β2 .
(A7) For θ ∈ [1, 2µ] (where µ is defined in (A2)), ∞ a.s.
U
∞ 0
u2 λ2 (du) < δ 4 (t ) (1+t )2θ
dt <
The following auxiliary lemma gives an estimate of the generalized square root process X (t ), determined by the first equation in (3), and plays a key role in revealing the long-term behavior of two-factor model (3). Lemma 3. Let (A5) and (A6) hold and assume that
Γ (ϑ1 , λ1 ) := ϑ12
u2 (6 + 4ϑ1 u + ϑ12 u2 )λ1 (du) < −8β1 .
(12)
U
Then (3) admits a unique nonnegative solution (X (t ), Y (t ))t ≥0 , and there exist κ > 0 and C > 0 such that
E(e−κβ1 ρ X 4 (ρ)) ≤ C + C E
ρ
e−κβ1 s (δ 4 (s) + 1)ds,
0
where ρ > 0 is a bounded stopping time.
(13)
J. Bao, C. Yuan / Insurance: Mathematics and Economics 53 (2013) 266–272
Proof. By Lemma 1, under (A5) and (A6), (3) admits a unique nonnegative solution (X (t ), Y (t ))t ≥0 . Also, by the Itô formula and the inequality (5) we derive that d(e−κβ1 t X 4 (t )) = −κβ1 e−κβ1 t X 4 (t )dt + e−κβ1 t dX 4 (t )
for sufficiently small ϵ > 0 and some constant C (ϵ) > 0. Integrating from 0 to t on both sides leads to X 2 (t ) − ξ 2 (0) ≤ ϵ∥ξ ∥2 τ + (4β1 + 2ϵ + m(ϑ1 , λ1 ))
t
X 2 (s)ds + C (ϵ)
×
= e−κβ1 t (8 − κ)β1 X 4 (t ) + 4δ(t )X 3 (t )
˜ 1 (t ) + M ˜ 2 (t ) +M
0
+ ϑ1
(14)
Remark 5. In fact, (3) admits a unique nonnegative solution (X (t ), Y (t ))t ≥0 under u2 λ1 (du) < −4β1 ,
1 tθ
X (s)ds ≤ 2
t →∞
t
tµ
Y (s)ds = 0
ν , 4β1 β2
a.s.
lim
t →∞
1
t
tµ
X (s)ds = − 0
ν 2β1
,
a.s.
(16)
On the other hand, for θ ∈ [1, 2µ] such that (6), if there exists C > 0 such that 1 lim sup θ t →∞ t
t
X 2 (s)ds ≤ C ,
a.s.,
(17)
dX 2 (t ) =
U
(2u + ϑ1 u2 )X 2 (s)N˜ 1 (du, ds). U
(1 + s)θ
dW1 (s)
∞
X 2 (s)
( 1 + s) θ
U
N˜ 1 (du, ds)
δ 4 (s) ds ≥ n . (1 + s)2θ
t 0
By (A5) we see that {ρn = ∞} ↑ Ω . Following the argument of Theorem 1, we only need to show that M (t ) :=
3
X 2 (s)X γ1 (s − τ )
t
(1 + s)θ
0
1{s≤ρn } dW1 (s)
is L2 -bounded. By the Itô isometry and the inequality (5), it follows that
E |M (t )| = E 2
t
X 3 (s)X 2γ1 (s − τ )
1{s≤ρ } ds
n (1 + s)2θ t X 4 (s) ≤ C +E 1 ds 2θ {s≤ρn } 0 (1 + s) t 4 X (s − τ ) +E 1 ds 2θ {s≤ρn } 0 (1 + s) =: C + J˜1 (t ) + J˜2 (t ).
0
3
3
0
exist a.s. For each n > ∥ξ ∥, define a stopping time
(4β1 + m(ϑ1 , λ1 ))X 2 (t ) + 2δ(t )X (t ) + σ12 X (t )X 2γ1 (t − τ ) dt
+ 2σ X 2 (t )X γ1 (t − τ )dW1 (t ) + ϑ1 (2u + ϑ1 u2 )X 2 (t )N˜ 1 (du, dt )
0
t
and
+ 2σ1 X 2 (t )X γ1 (t − τ )dW1 (t ) + ϑ1 (2u + ϑ1 u2 )X 2 (t )N˜ 1 (du, dt ) U ≤ (4β1 + ϵ + m(ϑ1 , λ1 ))X 2 (t ) + ϵ X 2 (t − τ ) + C (ϵ)(1 + δ 2 (t )) dt
3
3
0
then, together with (16) and Theorem 1, (15) holds. Therefore, we only need to verify that. Again, by the Itô formula and the inequality (5), it follows from (3) that
δ 2 (s)ds 0
X 2 (s)X γ1 (s − τ )dW1 (s)
X 2 (s)X γ1 (s − τ )
ρn := inf t ≥ 0
Proof. By (A5) and Theorem 1, we deduce that
∞
0
(15)
κ˜ t θ
t
Note from θ ∈ [1, 2µ] and (6) that the first two terms on the right hand side are finite almost surely. In order to prove (17), by Kronecker’s lemma it is sufficient to show that
J2 (∞) :=
Theorem 2. Under (A5)–(A7), (6) and (12),
t
κ˜ t θ
ϑ1 + θ κ˜ t
0
Our second main result in this paper is presented as follows.
2σ
+
C
+
tθ
0
rather than (12), which is imposed just to guarantee (13).
1
C (1 + t )
t
J1 (∞) :=
U
lim
(2u + ϑ1 u2 )X 2 (s)N˜ 1 (du, ds). U
By virtue of (12), we choose ϵ > 0 such that κ˜ := 4β1 + 2ϵ + m(ϑ1 , λ1 ) < 0. Thus, for θ ∈ [1, 2µ], by (A5)
˜ 1 (t ) and M ˜ 2 (t ) for any κ > 0 and sufficiently small ϵ > 0, where M are two local martingales. Then (13) follows by integrating from 0 to ρ , taking expectations on both sides of (14) and, in particular, choosing κ > 0 and ϵ > 0 such that (8−κ)β1 +2ϵ+Γ (ϑ1 , λ1 ) = 0 due to (12).
3
X 2 (s)X γ1 (s − τ )dW1 (s)
t
U
m(ϑ1 , λ1 ) := ϑ12
t
0
≤ e−κβ1 t {((8 − κ)β1 + ϵ + Γ (ϑ1 , λ1 ))X 4 (t ) + ϵ eκβ1 τ X 4 (t − τ ) + C (ϵ)(δ 4 (t ) + 1)}dt
(1 + δ 2 (s))ds 0
+ 2σ
˜ 1 (t ) + M ˜ 2 (t ) + 6σ (t )X (t − τ ) + M + ((1 + ϑ1 u)4 − 1 − 4ϑ1 u)λ1 (du)X 4 (t )
t
0
2γ1
2 3 1X
271
For κ > 0, we obtain from Lemma 3 that J˜1 (t ) ≤
t
eκβ1 s E{e−pβ1 (s∧τn ) X 4 (s ∧ τn )}
0
t
eκβ1 s
≤C
(1 + s)2θ
0 t
≤ C +C 0
ds
(1 + s)2θ s −κβ r 4 1 1+s+ 0 e E(δ (r )1{r ≤τn } )dr eκβ1 s
(1 + s)2θ
ds
s
e−κβ1 r E(δ 4 (r )1{r ≤τn } )drds 0
272
J. Bao, C. Yuan / Insurance: Mathematics and Economics 53 (2013) 266–272 t
e−κβ1 r E(δ 4 (r )1{r ≤τn } )
= C +C 0
≤ C (1 + n),
t
r
eκβ1 s
(1 + s)2θ
and, similarly, that J˜2 (t ) ≤ C (1 + n), so J1 (∞) exists.
dsdr
Remark 6. From Theorem 2, for the two-factor model (3), we cont clude that the long-term return t −µ 0 Y (s)ds converges almost surely to a random variable which generally depends on the economic environment. Such a convergence can be used in the finance and insurance markets. For example, the customer wants a return as high as possible, and insurance companies wonder how the percentage of interest should be determined when they promise a certain fixed percentage of interest on their insurance products such as bonds, life-insurance and so on. Remark 7. Theorem 2 holds for the two-factor CIR-type mode (3) with delay τ = 0 whenever γi ∈ [0, 21 ), i = 1, 2. This situation is not covered by Zhao (2009, Theorem 2) due to the fact that our jump–diffusion coefficient is only Lipschitz continuous, and not Hölder continuous with exponent 21 . Acknowledgment We are indebted to the referee for his/her helpful comments which has greatly improved our earlier version. References Ang, A., Bekaert, G., 2002. Regime switching in interest rates. Journal of Business & Economic Statistics. Arriojas, M., Hu, Y., Mohammed, S.-E., Pap, G., 2007. A delayed Black and Scholes formula. Stochastic Analysis and Applications 25, 471–492. Bardhan, I., Chao, X., 1993. Pricing options on securities with discontinuous returns. Stochastic Processes and their Applications 48, 123–137. Benhabib, J., 2004. Interest rate policy in continuous time with discrete delays. Journal of Money, Credit and Banking 36, 1–15.
Brigo, D., Mercurio, F., 2006. Interest Rate Models—Theory and Practice: With Smile, Inflation, and Credit. Springer, New-York. Cassola, N., Barros Luis, J., 2001. A two-factor model of the german term structure of interest rates. In: European Central Bank Working Paper No. 46. Available at SSRN: http://ssrn.com/abstract=356020. Chan, T., 1999. Pricing contingent claims on stocks driven by Lévy processes. The Annals of Applied Probability 9, 504–528. Chan, K., Karolyi, A., Longstaff, F.A., Sanders, A.B., 1992. An empirical comparison of alternative models of the short-term interest rate. Journal of Finance 47, 1209–1227. Corzo, T., Schwartz, E., 2000. Convergence within the European Union: evidence from interest rates. Economic Notes 29, 243–268. Cox, J.C., Ingersoll, J.E., Ross, S.A., 1985. A theory of the term structure of interest rates. Econometrica 53, 385–407. Deelstra, G., Delbaen, F., 1995. Long-term returns in stochastic interest rate models. Insurance: Mathematics & Economics 17, 163–169. Deelstra, G., Delbaen, F., 1997. Long-term returns in stochastic interest rate models: different convergence results. Applied Stochastic Models and Data Analysis 13, 401–407. Deelstra, G., Delbaen, F., 1998. Convergence of discretized stochastic (interest rate) processes with stochastic drift term. Applied Stochastic Models and Data Analysis 14, 77–84. Deelstra, G., Delbaen, F., 2000. Long-term returns in stochastic interest rate models: application. Astin Bulletin 30, 123–140. Gray, S.F., 1996. Modeling the conditional distribution of interest rates as a regimeswitching process. Journal of Financial Economics 42, 27–76. Henderson, V., Hobson, D., 2003. Coupling and option price comparisons in a jumpdiffusion model. Stochastics and Stochastics Reports 75, 79–101. Longstaff, F.A., Schwartz, E.A., 1992. Interest rate volatility and the term structure: a two-factor general equilibrium model. The Journal of Finance 47, 1259–1282. Merculio, F., Runggaldier, J.W., 1993. Option pricing for jump diffusions: approximations and their interpretation. Applied Mathematical Finance 3, 191–200. Stoica, G., 2004. A stochastic delay financial model. Proceedings of the American Mathematical Society 133, 1837–1841. Wu, F., Mao, X., Chen, K., 2008. Strong convergence of Monte Carlo simulations of the mean-reverting square root process with jump. Applied Mathematics and Computation 206, 494–505. Wu, F., Mao, X., Chen, K., 2009. The Cox–Ingersoll–Ross model with delay and strong convergence of its Euler–Maruyama approximate solutions. Applied Numerical Mathematics 59, 2641–2658. Yamada, T., Watanabe, S., 1971. On the uniqueness of solutions of stochastic differential equations. Journal of Mathematics of Kyoto University 11, 155–167. Zhao, J., 2009. Long time behaviour of stochastic interest rate models. Insurance: Mathematics & Economics 44, 459–463. Zhao, J., 2010. Strong solution of a class of SDEs with jumps. Stochastic Analysis and Applications 5, 735–746.
Contents lists available at SciVerse ScienceDirect
Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime
Long-term behavior of stochastic interest rate models with jumps and memory Jianhai Bao a , Chenggui Yuan b,∗ a
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410075, China
b
Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, UK
highlights • We show the nonnegative property of solutions for a class of stochastic equations. • We investigate the long-term return for stochastic interest rate models. • An application to a two-factor CIR model is presented.
article
abstract
info
Article history: Received November 2011 Received in revised form February 2013 Accepted 18 May 2013
The long-term interest rates, for example, determine when homeowners refinance their mortgages in mortgage pricing, play a dominant role in life insurance, decide when one should exchange a long bond to a short bond t in pricing an option. In this paper, for a one-factor model, we reveal that the long-term return t −µ 0 X (s)ds for some µ ≥ 1, in which X (t ) follows an extension of the Cox–Ingersoll–Ross model with jumps and memory, converges almost surely to a reversion level which is random itself. Such a convergence can be applied in the determination of models of participation in the benefit or of saving products with a guaranteed minimum return. As an immediate application of the result obtained for the one-factor model, for a class t of two-factor model, we also investigate the almost sure convergence of the long-term return t −µ 0 Y (s)ds for some µ ≥ 1, where Y (t ) follows an extended Cox–Ingersoll–Ross model with stochastic reversion level −X (t )/(2β) in which X (t ) follows an extension of the square root process. This result can be applied to, e.g., how the percentage of interest should be determined when insurance companies promise a certain fixed percentage of interest on their insurance products such as bonds, life-insurance and so on. © 2013 Elsevier B.V. All rights reserved.
MSC: 60H10 60H30 Keywords: Interest rate Cox–Ingersoll–Ross model Jump Memory One-factor model Two-factor model Long-term return
1. Introduction
models of the short-term interest rate in the framework
Cox et al. (1985) proposed the short-term interest rate dynamics as dS (t ) = κ(γ − S (t ))dt + σ
S (t )dW (t )
for positive constants κ, γ and σ and standard Brownian motion {W (t ) : t ≥ 0}. This model is known as the Cox–Ingersoll–Ross (CIR) model and has some empirically relevant properties, e.g., the randomly moving interest rate is elastically pulled towards the long-term constant value γ . In order to better capture the properties of empirical data, Chan et al. (1992) nested a wide range of
∗
Corresponding author. E-mail address: [email protected] (C. Yuan).
0167-6687/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.insmatheco.2013.05.006
dS (t ) = κ(γ − S (t ))dt + σ S (t )α dW (t )
(1)
for α ≥ 1/2 with appropriate restrictions on the parameters κ, γ , α . Here γ , toward which rates drift, stands for the long-term mean of the process, κ means the speed of the drift, σ measures the volatility, and 2α denotes the variance elasticity. In particular, by the X2 test to the one-month US. Treasury bill yields, Chan et al. (1992) compared the ability of each model with different α to capture the volatility of the term structure, found that the value of α is the most important feature differentiating interest rate models, and revealed that the most successful models in capturing the dynamics of the short-term interest rate are those that allow the volatility of interest rate changes to be highly sensitive to the level of the riskless rate (e.g., α ≥ 1). Note that the long-term mean γ , the speed of drift κ and the volatility σ are not constants either and there is strong evidence to indicate that they are Markov jump pro-
J. Bao, C. Yuan / Insurance: Mathematics and Economics 53 (2013) 266–272
cesses. Another generalization of the CIR model is to use regimeswitching such as in Ang and Bekaert (2002) and Gray (1996), to name a few. On the other hand, from the economic point of view, there is some evidence indicating that certain events happening before the trading periods influence the current and future asset price, and therefore many scholars introduce delays to the financial models. For example, Arriojas et al. (2007) took delay into consideration for the price process of underlying assets and developed a Black–Scholes type formula. Benhabib (2004) considered a linear, flexible price model, where nominal interest rates are measured by a flexible distributed delay. Stoica (2004) computed the logarithmic utility of an insider when the financial market is modeled by a stochastic delay equation and offered an alternative to the anticipating delayed Black–Scholes formula. However, the meanreverting square root process cannot explain some empirical phenomena, such as stochastic volatility. To explain these phenomena, jump processes are also used in the financial models, e.g., Bardhan and Chao (1993), Chan (1999), Henderson and Hobson (2003) and Merculio and Runggaldier (1993). There is extensive literature on quantitative and qualitative properties of the generalized CIR-type models. For instance, different convergence results and the corresponding applications of the long-term returns are found in Deelstra and Delbaen (1995, 1997) and Zhao (2009). Strong convergence of the Monte Carlo simulations are studied in Deelstra and Delbaen (1998) and Wu et al. (2008, 2009), and the representations of solutions are presented in Arriojas et al. (2007) and Stoica (2004). Deelstra and Delbaen (1995, 1997) investigated the long-term returns of the CIR model, and Zhao (2009) extended those results to the jump models. Noting that the reversion level γ in (1) is a constant, in order to better reflect the time dependence caused by the cyclical nature of the economy or by expectations concerning the future impact of monetary, as in Deelstra and Delbaen (1995, 1997), we can assume that the short-term interest rate model has a stochastic reversion level. As described above, there is a natural motivation for considering the stochastic interest rate model where all three features, delay, jumps and time dependence of reversion level, are presented. In this paper, we consider the stochastic interest rate model with jumps and memory in the form
dX (t ) = {2 β X (t ) + δ(t )}dt + σ X γ (t − τ ) |X (t )|dW (t ) + g (X (t −), u)N˜ (dt , du), U X0 = ξ ∈ D ,
(2)
where X (t −) := lims↑t X (s) and D denotes all real bounded càdlàg functions defined on [−τ , 0] for some τ > 0. The integral g ( X ( t −), u)N˜ (dt , du) depends on the Poisson measure and is U regarded as a jump. The diffusion term is dependent on the past through X γ (t − τ ) and so is called delay or memory. Precise assumptions on the data of the problem (2) are given in Section 2 below. The long-term interest rates play an important role in finance and insurance. For instance, the long-term interest rates determine when homeowners refinance their mortgages in mortgage pricing, play a dominant role in life insurance, decide when one should exchange a long bond to a short bond in pricing an option. In this light, for the instantaneous interest rate model (2), it is interesting t to investigate the long-term return t −µ 0 X (s)ds for some µ ≥ 1.
t
We shall reveal that the long-term return t −µ 0 X (s)ds converges almost surely to a stochastic reversion level, which will be stated in Theorem 1 below. As stated in Deelstra and Delbaen (2000), t the limit of long-term return t −µ 0 X (s)ds can be applied in the determination of models of participation in the benefit or of saving products with a guaranteed minimum return.
267
As we know, one-factor models imply that the instantaneous returns on bonds of all maturities are perfectly correlated, which is clearly inconsistent with reality, e.g., Longstaff and Schwartz (1992). However empirical research, e.g., Brigo and Mercurio (2006), Cassola and Barros Luis (2001) and Longstaff and Schwartz (1992), has suggested that two-factor models, including the shortterm interest rate and the instantaneous variance of changes in the short-term interest rate, are better than one-factor models to capture the behavior of the term structure in the real world. This is because the two-factor models allow contingent claim values to reflect both the current level of interest rates as well as the current level of interest rate volatility. Cox et al. (1985, p. 399) introduced a model by two independent factors, r1 and r2 , and the instantaneous interest rate is the sum of two factors, that is,
r (t ) = r1 (t ) + r2 (t )
dri (t ) = κi (θi − ri (t ))dt + σi (t ) ri (t )dBi (t ),
i = 1, 2,
where B1 (t ) and B2 (t ) are independent Brownian motions, θi is the long-term mean factor ri reverts to, and κi and σi are constants. In another example of a multi-factor model, the domestic short-rate rd and the European short-rate re satisfy the following stochastic differential equation (SDE) drd = [a + b(re − rd )]dt + σd dBd (t ), dre = c (d − re )dt + σe dBe (t ), Cov(dBd (t ), dBe (t )) = ρ dt ,
where Bd (t ) and Be (t ) are Brownian motions with instantaneous correlation ρ , and a, b, c , d, σd , σe are positive constants, Corzo and Schwartz (2000) investigated how to price the European bond and several other interest rate derivatives. As an immediate application of Theorem 1, we consider the long-term return of the two-factor model in the form
dX (t ) = {2β1 X (t )+ δ(t )}dt + σ1 X γ1 (t − τ ) |X (t )|dW1 (t ) +ϑ1 X (t ) uN˜ 1 (dt , du), U (3) dY (t ) = {2β2 Y (t )+ X (t )}dt + σ2 Y γ2 (t − τ ) |Y (t )|dW2 (t ) +ϑ2 Y (t ) uN˜ 2 (dt , du) U
with the initial data (X (t ), Y (t )) = (ξ (t ), η(t )), t ∈ [−τ , 0]. Here W1 (t ), W2 (t ) are Brownian motions, N1 (dt , du), N2 (dt , du) represent Poisson counting measures, defined on (Ω , F , {Ft }t ≥0 , P), with characteristic measures λ1 (·) and λ2 (·) respectively, and ξ , η ∈ D . More details on the parameters of model (3) are to be presented in Section 4. In model (3), the short interest rate Y (t ) follows an extended CIR model with stochastic reversion level −X (t )/(2β), where X (t ) follows an extension of the square root process. For model (3), we are interested t in the almost sure convergence of the long-term return t −µ 0 Y (s)ds for some µ ≥ 1. Such a convergence can be applied to the finance and insurance markets. For example, the customer wants a return as high as possible, and insurance companies wonder how the percentage of interest should be determined when they promise a certain fixed percentage of interest on their insurance products such as bonds, life-insurance and so on. The rest of the paper is organized as follows. In Section 2 we introduce some preliminaries, show the nonnegative property of X (t ) as nominal instantaneous interest rate determined by (2), and give an auxiliary lemma of Theorem 1. Section 3 is devoted to t the almost sure convergence of long-term return t −µ 0 X (s)ds for some µ ≥ 1 with X (t ) a generalized CIR model determined by (2), in which the corresponding results in Deelstra and Delbaen (1995, 1997) and Zhao (2009) are extended. In the final section, as an application of Theorem 1,we consider the almost sure convergence t of long-term return t −µ 0 Y (s)ds for some µ ≥ 1 determined by (3) with stochastic reversion level −X (t )/(2β), where the results of Zhao (2009, Theorem 2) are developed.
268
J. Bao, C. Yuan / Insurance: Mathematics and Economics 53 (2013) 266–272
Applying the Itô formula yields that for any t ∈ (0, T ]
2. Preliminaries Throughout this paper, let (Ω , F , {Ft }t ≥0 , P) be a complete probability space with the filtration {Ft }t ≥0 satisfying the usual conditions (i.e. it is right continuous and F0 contains all P-null sets). Let W (t ) be a scalar Brownian motion defined on the probability space (Ω , F , {Ft }t ≥0 , P). Let B (R+ ) be the Borel σ -algebra on R+ , and λ(dx) a σ -finite measure defined on B (R+ ). Let Dp ⊂ R+ be a countable set and p = (p(t )), t ∈ Dp , a stationary Ft -Poisson point process on R+ with characteristic measure λ(·). Denote N (dt , du) by the Poisson counting measure associated with p, i.e., N (t , U ) = s∈Dp ,s≤t IU (p(s)) for U ∈ B (R+ ). We assume
λ(U ) < ∞ and let N˜ (dt , du) := N (dt , du) − dt λ(du) be the compensated Poisson measure associated with N (dt , du). For the sake of convenience, we will denote C > 0 by a generic constant whose values may change from line to line. For model (2), we make the following assumptions:
Eφk (X (t )) = Eφk (ξ (0)) + E
lim
1
t →∞
tµ
+
2
t
E
φk′′ (X (s))X 2γ (s − τ )X (s)ds
0
t 0
{φk (X (s) + g (X (s), u)) U
− φk (X (s)) − φk′ (X (s))g (X (s), u)}λ(du)ds. By (i)–(iii), Taylor’s expansion and (A4), it then follows from (4) that for t ∈ (0, T ]
Eφk (X (t )) ≤
C k
1
t
{(φk′ (θ g (X (s), u) + X (s))
+E 0
0
U
− φk′ (X (s)))g (X (s), u)}dθ λ(du)ds t 1 C ≤ +E {(φk′ (θ g (X (s), u) + X (s)) k
0
0
U
− φk′ (X (s)))g (X (s), u)}1{X (s)≤0} dθ λ(du)ds t C 1 1 ≤ + 2K 2 λ 2 (U )E X − (s)1{X (s)≤0} ds
t
δ(s)ds = ν a.s. 0
|g (x, u) − g (y, u)|2 λ(du) ≤ K |x − y|2 ,
σ2
+E
(A3) g : Ω × R × U → R with g (0, u) = 0 and there exists K > 0 such that
φk′ (X (s)){2β X (s) + δ(s)}ds 0
(A1) β < 0, σ > 0 and γ ∈ [0, 12 ). (A2) δ : Ω × R+ → R+ , and there exist constants µ ≥ 1 and ν ≥ 0 such that
t
≤
x, y ∈ R.
U
k C
0 1 2
1 2
+ 2K λ (U )Tak−1 t 1 1 Eφk (X (s))ds. + 2K 2 λ 2 (U ) k
0
(A4) For any θ ∈ [0, 1], x + θ g (x, u) ≥ 0 whenever x > 0.
This, together with the Gronwall inequality, gives that
Remark 1. There are some examples, where (A4) holds, e.g., for x ∈ R and u ∈ U , g (x, u) ≥ 0 or −g (x, u) ≤ x whenever g (x, u) ≤ 0.
EX (t ) − ak−1 ≤ Eφk (X (t )) ≤ C −
1 k
+ ak − 1 ,
t ∈ (0, T ].
Since (2) is mainly used to model stochastic volatility or interest rate or an asset price, it is critical that the solution {X (t )}t ≥−τ will never become negative, which is guaranteed by the following lemma.
Thus, EX − (t ) = 0 as k → ∞ and therefore X (t ) ≥ 0 a.s. for any t ∈ (0, T ]. Hence the nonnegative property of the solution {X (t )}t ≥0 follows from the arbitrariness of T > 0.
Lemma 1. Under (A1)–(A4), (2) admits a unique nonnegative solution {X (t )}t ≥0 for any ξ ∈ D .
Remark 2. Wu et al. (2008) studied the strong convergence of Monte Carlo simulations of mean-reverting square root process with jumps in the form
Proof. An application of Zhao (2010, Theorems 2.1 and 2.2) gives that (2) has a unique strong solution X (t ) on [0, τ ]. Repeating this procedure we see that (2) also admits a unique strong solution X (t ) on [τ , 2τ ]. Hence (2) has a unique strong solution X (t ) on the horizon t ≥ 0. Moreover, carrying out a similar argument to that of Wu et al. (2009, Theorem 2.1), we deduce that there exists C > 0 such that for any T > 0, q > 0
E|X (t )|q ≤ C ,
t ∈ [0, T ].
(4)
To end the proof, it is sufficient to show the nonnegative property of the solution {X (t )}t ∈[0,T ] . We adopt the method of Yamada and Watanabe (1971). Let a a0 1 = 1 and ak = exp(−k(k + 1)/2), k = 1, 2 · · · . Then a k−1 kx dx = 1 and there is a continuous k nonnegative function ψk (x), x ∈ R+ , which possesses the support (ak , ak−1 ), has the integral 1 and satisfies ψk (x) ≤ kx2 . Define an auxiliary function φk (x) = 0 for x ≥ 0 and
φk (x) :=
−x
y
ψk (u)du,
dy 0
x < 0.
0
Then φk ∈ C 2 (R; R+ ) has the following properties: (i) −1 ≤ φk′ (x) ≤ 0 for −ak−1 < x < −ak , otherwise φk′ (x) = 0; (ii) |φk′′ (x)| ≤ k|2x| for −ak−1 < x < −ak , otherwise φk′′ (x) = 0;
(iii) x− − ak−1 ≤ φk (x) ≤ x− , x ∈ R.
dS (t ) = α[µ − S (t )]dt + σ
|S (t )|dW (t ) + δ S (t −)dN˜ (t )
for α, µ, σ > 0, and investigated the nonnegative property of S (t ). Our work is a generalization of this model. Zhao (2009) also showed the nonnegative property of the solution X (t ) to (2) with γ = 0, g (x, u) = 0 for x < 0 and U g 2 (x, u)λ(du) ≤ K |x| for some constant K > 0. The following auxiliary lemma gives an estimate of X (t ), determined by the one-factor model (2), and plays an important role in analyzing the asymptotic behavior of long-term return t t −µ 0 X (s)ds for some µ ≥ 1. Lemma 2. Let (A1)–(A4) hold and assume further that 4β + K < 0. Then there exist κ > 0 and C > 0 such that
E(e−κβρ X 2 (ρ)) ≤ C + C E
ρ
e−κβ s (δ 2 (s) + 1)ds,
0
where ρ > 0 is a bounded stopping time. Proof. We first recall the Young inequality: for any a, b > 0 and α ∈ (0, 1) aα b1−α ≤ α a + (1 − α)b.
(5)
J. Bao, C. Yuan / Insurance: Mathematics and Economics 53 (2013) 266–272
Let κ > 0 and ϵ > 0 be arbitrary. By the Itô formula, (A3) and the inequality (5), we obtain that d(e
−κβ t
=e
X (t )) = −κβ e 2
−κβ t
−κβ t
X (t )dt + e 2
−κβ t
dX (t ) 2
Thus, substituting this into (8) one has tµ
2
+
U
≤e {((4 − κ)β + ϵ + K )X 2 (t ) + C1 (ϵ)X 4γ (t − τ ) + C1 (ϵ)δ 2 (t )}dt + M1 (t ) + M2 (t ) ≤ e−κβ t {((4 − κ)β + ϵ + K )X 2 (t ) + ϵ eκβτ X 2 (t − τ ) + C1 (ϵ)δ 2 (t ) + C2 (ϵ)}dt + M1 (t ) + M2 (t ) for some constants C1 (ϵ) > 0 and C2 (ϵ) > 0 dependent on ϵ , 3
where M1 (t ) := 2σ e−κβ t X 2 (t )X γ (t − τ )dW (t ) and M2 (t ) := e−κβ t U {g 2 (X (t ), u)+2X (t )g (X (t ), u)}N˜ (dt , du). Integrating from 0 to ρ and taking expectations on both sides, we arrive at
E(e−κβρ X 2 (ρ))
ρ ≤ C ∥ξ ∥2 + ((4 − κ)β + 2ϵ + K )E e−κβ s X 2 (s)ds 0 ρ 2 + (C1 (ϵ) ∨ C2 (ϵ))E (δ (s) + 1)ds. 0
Due to 4β + K < 0, we choose κ > 0 and ϵ > 0 such that (4 − κ)β + 2ϵ + K = 0. 3. Almost sure convergence of long-term returns Our first main result in this paper is stated as follows. Theorem 1. Let (A1)–(A4) hold and 4β + K < 0. Assume further that there exist λ > 0 and θ ∈ [1, 2µ] such that (6)
∞
√
X γ (t − τ ) |X (t )|
(1 + t )µ
0 0
Then lim
t →∞
1
δ(s) X (s) + ds = 0 a.s. 2β
0
(7)
0
δ(s) X (t ) − ξ (0) ds = X (s) + 2β 2β t σ − X γ (s − τ ) |X (s)|dW (s) 2β 0 t 1 − g (X (s−), u)N˜ (ds, du). 2β 0 U
X (t ) = e
ξ (0) +
t
e−2β s δ(s)ds
0 t +σ e−2β s X γ (s − τ ) |X (s)|dW (s) 0 t −2 β s ˜ + e g (X (s−), u)N (ds, du) .
0
U
τn := inf t ≥ 0
t
t 0
δ 2 (s) ds ≥ n . (1 + s)2µ
δ 2 (s)ds ≤ L(1 + t )θ a.s.
0
This, together with θ ∈ [1, 2µ], leads to
(8)
On the other hand, an application of Itô formula to e−2β t X (t ) yields that 2β t
(9)
In light of (6) there exists an L > 0 such that
Proof. From (2)
t
dW (t ) exists a.s.
For each n > ∥ξ ∥ define a stopping time
t
tµ
e2β(t −s) δ(s)ds
0
To derive the assertion (7), it is sufficient to verify that Ii (t ) → 0 a.s., i = 1, . . . , 6, as t → ∞. Since β < 0 and µ ≥ 1, I1 (t ) → 0 as t → ∞. Following a similar argument to that of Deelstra and Delbaen and noting that limt →∞ [(1 + t )µ − (1 + √ µ(1995, p. 168), µ t − t ) ]/(1 + t ) = 0, by (A2) we deduce that I2 (t ) → 0 a.s. for t → ∞. Next, in order to show I3 (t ) → 0 a.s. and I4 (t ) → 0 a.s. whenever t → ∞, by Deelstra and Delbaen (1995, Kronecker’s lemma), it suffices to check that
t
δ 2 (s)ds ≤ λ a.s.
2β t µ
+
−κβ t
t
1
δ(s) (e2β t − 1)ξ (0) ds = 2β 2β t µ
µ 1 σ 1 1+ − 2 β t 2β t e (1 + t )µ t e−2β s δ(s)X γ (s − τ ) |X (s)|dW (s) × 0 µ t σ 1 1 X γ (s − τ ) |X (s)|dW (s) − 1+ µ 2β t (1 + t ) 0 µ t 1 1 1 g (X (s−), u)N˜ (ds, du) − 1+ 2β t (1 + t )µ 0 U µ 1 1 1 + 1+ 2β t e−2β t (1 + t )µ t × e−2β s g (X (s−), u)N˜ (ds, du) 0 U µ µ 1 1 σ 1 =: I1 (t ) + I2 (t ) + 1+ I3 (t ) − 1+ I4 ( t ) 2β t 2β t µ µ 1 1 1 1 − 1+ I5 (t ) + 1+ I6 (t ). 2β t 2β t
g 2 (X (t ), u)λ(du) dt + M1 (t ) + M2 (t )
1 lim sup θ t t →∞
X (s) +
0
2γ
+
t
1
(4 − κ)β X (t ) + σ X (t )X (t − τ ) + 2δ(t )X (t ) 2
269
s 2 δ (u)du δ 2 (s) ds = lim 0 2µ s →∞ ( 1 + s ) ( 1 + s)2µ 0 ∞ s ds + 2µ δ 2 (u)du ( 1 + s)2µ+1 0 0 ∞ L 1 ≤ lim + 2µL ds s→∞ (1 + s)2µ−θ (1 + s)2µ+1−θ 0 < ∞ a.s. ∞
Hence {τn = ∞} ↑ Ω and then it is sufficient to verify (9) on {τn = ∞}. Furthermore, observing that J (t ) :=
t
0
√
X γ (s − τ ) |X (s)|
(1 + s)µ
1{s≤τn } dW (s)
270
J. Bao, C. Yuan / Insurance: Mathematics and Economics 53 (2013) 266–272
is a local martingale, we only need to check that J (t ) is an L2 -bounded martingale. By the Itô isometry and the inequality (5), we obtain that
E |J (t )| =
E{X 2γ (s − τ )X (s)}1{s≤τn }
t
2
(1 + s)2µ
0
E{X (s)1{s≤τn } }
t
ds
2(1 + s)2µ
0
ds
E{X 4γ (s − τ )1{s≤τn } }
t
+
2(1 + s)2µ
0
ds
t E{X 2 (s)1{s≤τn } } (1 − 2γ ) ds + ds 2 µ 2(1 + s)2µ 0 2(1 + s) 0 t E{X 2 (s − τ )1{s≤τn } } +γ ds (1 + s)2µ 0 t κβ s −κβ(s∧τn ) 2 e E{e X (s ∧ τn )} 1 − 2γ + ds ≤ 2µ 2(2µ − 1) 2 ( 1 + s ) 0 t κβ(s−τ ) −κβ(s∧τn −τ ) 2 e E{e X (s ∧ τn − τ )} +γ ds 2 (1 + s) µ 0
≤
J1 ( t ) ≤ C
t
1 + s + eκβ s E
0
e−κβ r δ 2 (r )dr
4. An application to the two-factor CIR model
1 + s + eκβ s
t
s
≤C
0
e−κβ r E(δ 2 (r )1{r ≤τn } )dr
2(1 + s)2µ
0
eκβ s
t
≤ C +C 0
2(1 + s)
τn
ds
s
2µ
ds
2(1 + s)2µ
0
s∧τn
e−κβ r E(δ 2 (r )1{r ≤τn } )drds 0
δ 2 (r ) dr ≤ C 1 + n . 2 µ (1 + r )
≤ C + CE 0
(10)
Noting that J2 (t ) ≤ C + γ ∥ξ ∥2 t
e
×
κβ(s−τ )
t
τ
(E{e
eκβ(s−τ ) 2(1 + s)2µ
ds + γ
−κβ(s∧τn −τ ) 2
X (s ∧ τn − τ )1{s∧τn >τ } })
2(1 + s)2µ
τ
g (X (s−), u)
t 0
U
( 1 + s) µ
g (X (s−), u)
t
2
=E 0
1{s≤τn } N˜ (ds, du)
U
(1 + s)2µ
One-factor models imply that the instantaneous returns on bonds of all maturities are perfectly correlated, which is clearly inconsistent with reality, e.g. Longstaff and Schwartz (1992). On the other hand, empirical research, e.g., Brigo and Mercurio (2006), Cassola and Barros Luis (2001) and Longstaff and Schwartz (1992), has suggested that two-factor models, including the short-term interest rate and the instantaneous variance of changes in the short-term interest rate, are better than one-factor models to capture the behavior of the term structure in the real world. This is because the two-factor models allow contingent claim values to reflect both the current level of interest rates as well as the current level of interest rate volatility. As an immediate application of Theorem 1, for two-factor model (3), in this section t we consider the almost sure convergence of the long-term t −µ 0 Y (s)ds for some µ ≥ 1, where the short interest rate Y (t ) follows an extended CIR model with stochastic reversion level −X (t )/(2β), and X (t ) follows an extension of the square root process. For two-factor model (3), we assume that (A5) β1 < 0, σ1 > 0, γ1 ∈ [0, 12 ), ϑ1 > 0 and δ(t ) satisfies (A2).
ds,
and carrying out a similar argument to that of (10), we conclude that there exists C (n, µ, α) > 0 such that J2 (t ) ≤ C (n, µ, α). Finally, I5 (t ) → 0 a.s. and I6 (t ) → 0 a.s. whenever t → ∞ follow by observing that
E
(11)
The linear case g (x, u) = C |u|x, for some C > 0, does not satisfy (11). However, Theorem 1 is available for such a fundamental case. This proof is different from that of Zhao (2009) by incorporation of the memory, since it shows how to deal with the delay term when showing that J (t ) is an L2 -bounded martingale.
By Lemma 2 with κ > 0 it follows that
g 2 (x, u)λ(du) ≤ K |x| for some constant K > 0. U
=: (1 − 2γ )/(4µ − 2) + J1 (t ) + J2 (t ).
1
dX (t ) = {2β X (t ) + δ(t )}dt + σ |X (t )| 2 +l dW (t ), X (0) = x > 0,
whereas Deelstra and Delbaen (1995) investigated the long-term returns of such a model only when l = 0. Zhao (2009) discussed the long-time behavior of (2) with γ = 0, g (x, u) = 0 for x < 0, u ∈ U, and
t
Remark 4. For β < 0, σ > 0 and l ∈ [0, 12 ), Theorem 1 applies to the generalized mean-reverting model
2
≤
pricing, play a dominant role in life insurance, decide when one should exchange a long bond to a short bond in pricing an option.
2
1{s≤τn } λ(du)ds.
Remark 3. By Theorem 1, under some appropriate conditions on the parameters, we deduce that the long-term return converges almost surely to a reversion level which is random itself. As stated in Deelstra and Delbaen (1995, 1997), the long-term return determines when homeowners refinance their mortgages in mortgage
(A6) β2 < 0, σ2 > 0, γ2 ∈ [0, 12 ), ϑ2 > 0 and ϑ22 −4β2 .
(A7) For θ ∈ [1, 2µ] (where µ is defined in (A2)), ∞ a.s.
U
∞ 0
u2 λ2 (du) < δ 4 (t ) (1+t )2θ
dt <
The following auxiliary lemma gives an estimate of the generalized square root process X (t ), determined by the first equation in (3), and plays a key role in revealing the long-term behavior of two-factor model (3). Lemma 3. Let (A5) and (A6) hold and assume that
Γ (ϑ1 , λ1 ) := ϑ12
u2 (6 + 4ϑ1 u + ϑ12 u2 )λ1 (du) < −8β1 .
(12)
U
Then (3) admits a unique nonnegative solution (X (t ), Y (t ))t ≥0 , and there exist κ > 0 and C > 0 such that
E(e−κβ1 ρ X 4 (ρ)) ≤ C + C E
ρ
e−κβ1 s (δ 4 (s) + 1)ds,
0
where ρ > 0 is a bounded stopping time.
(13)
J. Bao, C. Yuan / Insurance: Mathematics and Economics 53 (2013) 266–272
Proof. By Lemma 1, under (A5) and (A6), (3) admits a unique nonnegative solution (X (t ), Y (t ))t ≥0 . Also, by the Itô formula and the inequality (5) we derive that d(e−κβ1 t X 4 (t )) = −κβ1 e−κβ1 t X 4 (t )dt + e−κβ1 t dX 4 (t )
for sufficiently small ϵ > 0 and some constant C (ϵ) > 0. Integrating from 0 to t on both sides leads to X 2 (t ) − ξ 2 (0) ≤ ϵ∥ξ ∥2 τ + (4β1 + 2ϵ + m(ϑ1 , λ1 ))
t
X 2 (s)ds + C (ϵ)
×
= e−κβ1 t (8 − κ)β1 X 4 (t ) + 4δ(t )X 3 (t )
˜ 1 (t ) + M ˜ 2 (t ) +M
0
+ ϑ1
(14)
Remark 5. In fact, (3) admits a unique nonnegative solution (X (t ), Y (t ))t ≥0 under u2 λ1 (du) < −4β1 ,
1 tθ
X (s)ds ≤ 2
t →∞
t
tµ
Y (s)ds = 0
ν , 4β1 β2
a.s.
lim
t →∞
1
t
tµ
X (s)ds = − 0
ν 2β1
,
a.s.
(16)
On the other hand, for θ ∈ [1, 2µ] such that (6), if there exists C > 0 such that 1 lim sup θ t →∞ t
t
X 2 (s)ds ≤ C ,
a.s.,
(17)
dX 2 (t ) =
U
(2u + ϑ1 u2 )X 2 (s)N˜ 1 (du, ds). U
(1 + s)θ
dW1 (s)
∞
X 2 (s)
( 1 + s) θ
U
N˜ 1 (du, ds)
δ 4 (s) ds ≥ n . (1 + s)2θ
t 0
By (A5) we see that {ρn = ∞} ↑ Ω . Following the argument of Theorem 1, we only need to show that M (t ) :=
3
X 2 (s)X γ1 (s − τ )
t
(1 + s)θ
0
1{s≤ρn } dW1 (s)
is L2 -bounded. By the Itô isometry and the inequality (5), it follows that
E |M (t )| = E 2
t
X 3 (s)X 2γ1 (s − τ )
1{s≤ρ } ds
n (1 + s)2θ t X 4 (s) ≤ C +E 1 ds 2θ {s≤ρn } 0 (1 + s) t 4 X (s − τ ) +E 1 ds 2θ {s≤ρn } 0 (1 + s) =: C + J˜1 (t ) + J˜2 (t ).
0
3
3
0
exist a.s. For each n > ∥ξ ∥, define a stopping time
(4β1 + m(ϑ1 , λ1 ))X 2 (t ) + 2δ(t )X (t ) + σ12 X (t )X 2γ1 (t − τ ) dt
+ 2σ X 2 (t )X γ1 (t − τ )dW1 (t ) + ϑ1 (2u + ϑ1 u2 )X 2 (t )N˜ 1 (du, dt )
0
t
and
+ 2σ1 X 2 (t )X γ1 (t − τ )dW1 (t ) + ϑ1 (2u + ϑ1 u2 )X 2 (t )N˜ 1 (du, dt ) U ≤ (4β1 + ϵ + m(ϑ1 , λ1 ))X 2 (t ) + ϵ X 2 (t − τ ) + C (ϵ)(1 + δ 2 (t )) dt
3
3
0
then, together with (16) and Theorem 1, (15) holds. Therefore, we only need to verify that. Again, by the Itô formula and the inequality (5), it follows from (3) that
δ 2 (s)ds 0
X 2 (s)X γ1 (s − τ )dW1 (s)
X 2 (s)X γ1 (s − τ )
ρn := inf t ≥ 0
Proof. By (A5) and Theorem 1, we deduce that
∞
0
(15)
κ˜ t θ
t
Note from θ ∈ [1, 2µ] and (6) that the first two terms on the right hand side are finite almost surely. In order to prove (17), by Kronecker’s lemma it is sufficient to show that
J2 (∞) :=
Theorem 2. Under (A5)–(A7), (6) and (12),
t
κ˜ t θ
ϑ1 + θ κ˜ t
0
Our second main result in this paper is presented as follows.
2σ
+
C
+
tθ
0
rather than (12), which is imposed just to guarantee (13).
1
C (1 + t )
t
J1 (∞) :=
U
lim
(2u + ϑ1 u2 )X 2 (s)N˜ 1 (du, ds). U
By virtue of (12), we choose ϵ > 0 such that κ˜ := 4β1 + 2ϵ + m(ϑ1 , λ1 ) < 0. Thus, for θ ∈ [1, 2µ], by (A5)
˜ 1 (t ) and M ˜ 2 (t ) for any κ > 0 and sufficiently small ϵ > 0, where M are two local martingales. Then (13) follows by integrating from 0 to ρ , taking expectations on both sides of (14) and, in particular, choosing κ > 0 and ϵ > 0 such that (8−κ)β1 +2ϵ+Γ (ϑ1 , λ1 ) = 0 due to (12).
3
X 2 (s)X γ1 (s − τ )dW1 (s)
t
U
m(ϑ1 , λ1 ) := ϑ12
t
0
≤ e−κβ1 t {((8 − κ)β1 + ϵ + Γ (ϑ1 , λ1 ))X 4 (t ) + ϵ eκβ1 τ X 4 (t − τ ) + C (ϵ)(δ 4 (t ) + 1)}dt
(1 + δ 2 (s))ds 0
+ 2σ
˜ 1 (t ) + M ˜ 2 (t ) + 6σ (t )X (t − τ ) + M + ((1 + ϑ1 u)4 − 1 − 4ϑ1 u)λ1 (du)X 4 (t )
t
0
2γ1
2 3 1X
271
For κ > 0, we obtain from Lemma 3 that J˜1 (t ) ≤
t
eκβ1 s E{e−pβ1 (s∧τn ) X 4 (s ∧ τn )}
0
t
eκβ1 s
≤C
(1 + s)2θ
0 t
≤ C +C 0
ds
(1 + s)2θ s −κβ r 4 1 1+s+ 0 e E(δ (r )1{r ≤τn } )dr eκβ1 s
(1 + s)2θ
ds
s
e−κβ1 r E(δ 4 (r )1{r ≤τn } )drds 0
272
J. Bao, C. Yuan / Insurance: Mathematics and Economics 53 (2013) 266–272 t
e−κβ1 r E(δ 4 (r )1{r ≤τn } )
= C +C 0
≤ C (1 + n),
t
r
eκβ1 s
(1 + s)2θ
and, similarly, that J˜2 (t ) ≤ C (1 + n), so J1 (∞) exists.
dsdr
Remark 6. From Theorem 2, for the two-factor model (3), we cont clude that the long-term return t −µ 0 Y (s)ds converges almost surely to a random variable which generally depends on the economic environment. Such a convergence can be used in the finance and insurance markets. For example, the customer wants a return as high as possible, and insurance companies wonder how the percentage of interest should be determined when they promise a certain fixed percentage of interest on their insurance products such as bonds, life-insurance and so on. Remark 7. Theorem 2 holds for the two-factor CIR-type mode (3) with delay τ = 0 whenever γi ∈ [0, 21 ), i = 1, 2. This situation is not covered by Zhao (2009, Theorem 2) due to the fact that our jump–diffusion coefficient is only Lipschitz continuous, and not Hölder continuous with exponent 21 . Acknowledgment We are indebted to the referee for his/her helpful comments which has greatly improved our earlier version. References Ang, A., Bekaert, G., 2002. Regime switching in interest rates. Journal of Business & Economic Statistics. Arriojas, M., Hu, Y., Mohammed, S.-E., Pap, G., 2007. A delayed Black and Scholes formula. Stochastic Analysis and Applications 25, 471–492. Bardhan, I., Chao, X., 1993. Pricing options on securities with discontinuous returns. Stochastic Processes and their Applications 48, 123–137. Benhabib, J., 2004. Interest rate policy in continuous time with discrete delays. Journal of Money, Credit and Banking 36, 1–15.
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