Citation: Zhenghui Li, Zhenzhen Wang, Zhehao Huang. Modeling Business Cycle with Financial Shocks Basing on Kaldor-Kalecki Model[J]. Quantitative Finance and Economics, 2017, 1(1): 44-66. doi: 10.3934/QFE.2017.1.44
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Recently, [1] introduced the process SH,K={SH,Kt,t≥0} on the probability space (Ω,F,P) with indices H∈(0,1) and K∈(0,1], named the sub-bifractional Brownian motion (sbfBm) and defined as follows:
SH,Kt=12(2−K)/2 (BH,Kt+BH,K−t), |
where {BH,Kt,t∈R} is a bifractional Brownian motion (bfBm) with indices H∈(0,1) and K∈(0,1], namely, {BH,Kt,t∈R} is a centered Gaussian process, starting from zero, with covariance
E[BH,KtBH,Ks]=12K[(|t|2H+|s|2H)K−|t−s|2HK], |
with H∈(0,1) and K∈(0,1].
Clearly, the sbfBm is a centered Gaussian process such that SH,K0=0, with probability 1, and Var(SH,Kt)=(2K−22HK−1)t2HK. Since (2H−1)K−1<K−1≤0, it follows that 2HK−1<K. We can easily verify that SH,K is self-similar with index HK. When K=1, SH,1 is the sub-fractional Brownian motion (sfBm). For more on sub-fractional Brownian motion, we can see [2,3,4,5] and so on. The following computations show that for all s,t≥0,
RH,K(t,s)=E(SH,KtSH,Ks)=(t2H+s2H)K−12(t+s)2HK−12|t−s|2HK | (1.1) |
and
C1|t−s|2HK≤E[(SH,Kt−SH,Ks)2]≤C2|t−s|2HK, | (1.2) |
where
C1=min{2K−1,2K−22HK−1}, C2=max{1,2−22HK−1}. | (1.3) |
(See [1]). [6] investigated the collision local time of two independent sub-bifractional Brownian motions. [7] obtained Berry-Esséen bounds and proved the almost sure central limit theorem for the quadratic variation of the sub-bifractional Brownian motion. For more on sbfBm, we can see [8,9,10].
Reference [11] studied the limits of bifractional Brownian noises. [12] obtained limit results of sub-fractional Brownian and weighted fractional Brownian noises. Motivated by all these studies, in this paper, we will study the increment process {SH,Kh+t−SH,Kh,t≥0} of SH,K and the noise generated by SH,K and see how close this process is to a process with stationary increments. In principle, since the sub-bifractional Brownian motion is not a process with stationary increments, its increment process depends on h.
We have organized our paper as follows: In Section 2 we prove our main result that the increment process of SH,K converges to the fractional Brownian motion BHK. Section 3 is devoted to a different view of this main result and we analyze the noise generated by the sub-bifractional Brownian motion and study its asymptotic behavior. In Section 4 we prove limit theorems to the sub-bifractional Brownian motion from a correlated non-stationary Gaussian sequence. Finally, Section 5 describes the behavior of the tangent process of sbfBm.
In this section, we prove the following main result which says that the increment process of the sub-bifractional Brownian motion SH,K converges to the fractional Brownian motion with Hurst index HK.
Theorem 2.1. Let K∈(0,1). Then, as h→∞,
{SH,Kh+t−SH,Kh,t≥0}d⇒{BHKt,t≥0}, |
where d⇒ means convergence of all finite dimensional distributions and BHK is the fractional Brownian motion with Hurst index HK.
In order to prove Theorem 2.1, we first show a decomposition of the sub-bifractional Brownian motion with parameters H and K into the sum of a sub-fractional Brownian motion with Hurst parameter HK plus a stochastic process with absolutely continuous trajectories. Some similar results were obtained in [13] for the bifractional Brownian motion and in [14] for the sub-fractional Brownian motion. Such a decomposition is useful in order to derive easier proofs for different properties of sbfBm (like variation, strong variation and Chung's LIL).
We consider the following decomposition of the covariance function of the sub-bifractional Brownian motion:
RH,K(t,s)=E(SH,KtSH,Ks)=(t2H+s2H)K−12(t+s)2HK−12|t−s|2HK |
=[(t2H+s2H)K−t2HK−s2HK] |
+[t2HK+s2HK−12(t+s)2HK−12|t−s|2HK]. | (2.1) |
The second summand in (2.1) is the covariance of a sub-fractional Brownian motion with Hurst parameter HK. The first summand turns out to be a non-positive definite and with a change of sign it will be the covariance of a Gaussian process. Let {Wt,t≥0} a standard Brownian motion, for any 0<K<1, define the process XK={XKt,t≥0} by
XKt=∫∞0(1−e−θt)θ−1+K2dWθ. | (2.2) |
Then, XK is a centered Gaussian process with covariance:
E(XKtXKs)=∫∞0(1−e−θt)(1−e−θs)θ−1−Kdθ |
=∫∞0(1−e−θt)θ−1−Kdθ−∫∞0(1−e−θt)e−θsθ−1−Kdθ |
=∫∞0(∫t0θe−θudu)θ−1−Kdθ−∫∞0(∫t0θe−θudu)e−θsθ−1−Kdθ |
=∫t0(∫∞0θ−Ke−θudθ)du−∫t0(∫∞0θ−Ke−θ(u+s)dθ)du |
=Γ(1−K)K[tK+sK−(t+s)K], | (2.3) |
where Γ(α)=∫∞0xα−1e−xdx.
Therefore we obtain the following result:
Lemma 2.1. Let SH,K be a sub-bifractional Brownian motion, K∈(0,1) and assume that {Wt,t≥0} is a standard Brownian motion independent of SH,K. Let XK be the process defined by (2.2). Then the processes {√KΓ(1−K)XKt2H+SH,Kt,t≥0} and {SHKt,t≥0} have the same distribution, where {SHKt,t≥0} is a sub-fractional Brownian motion with Hurst parameter HK.
Proof. Let Yt=√KΓ(1−K)XKt2H+SH,Kt. Then, from (2.1) and (2.3), we have, for s,t≥0,
E(YsYt)=KΓ(1−K)E(XKs2HXKt2H)+E(SH,KsSH,Kt) |
=t2HK+s2HK−(t2H+s2H)K |
+(t2H+s2H)K−12(t+s)2HK−12|t−s|2HK |
=t2HK+s2HK−12(t+s)2HK−12|t−s|2HK, |
which completes the proof.
Lemma 2.1 implies that
{SH,Kt,t≥0}d={SHKt−√KΓ(1−K)XKt2H,t≥0} | (2.4) |
where d= means equality of all finite-dimensional distributions.
By Theorem 2 in [13], the process XK has a version with trajectories that are infinitely differentiable trajectories on (0,∞) and absolutely continuous on [0,∞).
Reference [15] presented a decomposition of the sub-fractional Brownian motion into the sum of a fractional Brownian motion plus a stochastic process with absolutely continuous trajectories. Namely, we have the following lemma.
Lemma 2.2. Let BH be a fractional Brownian motion with Hurst parameter H, SH be a sub-fractional Brownian motion with Hurst parameter H and B={Bt,t≥0} is a standard Brownian motion. Let
YHt=∫∞0(1−e−θt)θ−1+2H2dBθ. | (2.5) |
(1) If 0<H<12 and suppose that BH and B are independent, then the processes
{√HΓ(1−2H)YHt+BHt,t≥0} and {SHt,t≥0} have the same distribution.
(2) If 12<H<1 and suppose that SH and B are independent, then the processes
{√H(2H−1)Γ(2−2H)YHt+SHt,t≥0} and {BHt,t≥0} have the same distribution.
Proof. See the proof of Theorem 2.2 in [15] or the proof of Theorem 3.5 in [14].
By (2.4) and Lemma 2.2, we get, as 0<HK<12,
{SH,Kt,t≥0}d={BHKt+√HKΓ(1−2HK)YHKt−√KΓ(1−K)XKt2H,t≥0} | (2.6) |
and as 12<HK<1,
{SH,Kt,t≥0}d={BHKt−√HK(2HK−1)Γ(2−2HK)YHKt−√KΓ(1−K)XKt2H,t≥0}. | (2.7) |
The following Lemma 2.3 comes from Proposition 2.2 in [11].
Lemma 2.3. Let XKt be defined by (2.2). Then, as h→∞,
E[(XK(h+t)2H−XKh2H)2]=Γ(1−K)K2KH2K(1−K)t2h2(HK−1)(1+o(1)). |
Therefore, as h→∞,
{XK(h+t)2H−XKh2H,t≥0}d⇒{Xt≡0,t≥0}. |
Lemma 2.4. Let YHt be defined by (2.5). Then, as h→∞,
E[(YHKh+t−YHKh)2]=22HK−2Γ(2−2HK)t2h2(HK−1)(1+o(1)). |
Therefore, as h→∞,
{YHKh+t−YHKh,t≥0}d⇒{Yt≡0,t≥0}. |
Proof. By Proposition 2.1 in [15], we have
E(YHtYHs)={Γ(1−2H)2H[t2H+s2H−(t+s)2H],if 0<H<12;Γ(2−2H)2H(2H−1)[(t+s)2H−t2H−s2H],if 12<H<1. |
When 0<HK<12, we get
E(YHKtYHKs)=Γ(1−2HK)2HK[t2HK+s2HK−(t+s)2HK]. |
In particular, for every t≥0,
E[(YHKt)2]=Γ(1−2HK)2HK(2−22HK)t2HK. |
Hence, we obtain
E[(YHKh+t−YHKh)2]=−Γ(1−2HK)2HK22HK[(h+t)2HK+h2HK]+Γ(1−2HK)2HK2(2h+t)2HK. |
Then, for every large h>0, by using Taylor's expansion, we have
I:=2HKΓ(1−2HK)E[(YHKh+t−YHKh)2] |
=−22HK[(h+t)2HK+h2HK]+2(2h+t)2HK |
=−22HKh2HK[(1+th−1)2HK+1]+2h2HK(2+th−1)2HK |
=−22HKh2HK[2+2HKth−1+HK(2HK−1)t2h−2(1+o(1))] |
+2h2HK[22HK+22HK−12HKth−1+22HK−2HK(2HK−1)t2h−2(1+o(1))] |
=22HK−1HK(1−2HK)t2h2(HK−1)(1+o(1)). |
Thus,
E[(YHKh+t−YHKh)2]=22HK−2(1−2HK)Γ(1−2HK)t2h2(HK−1)(1+o(1)) |
=22HK−2Γ(2−2HK)t2h2(HK−1)(1+o(1)). |
Similarly, we can prove the case 12<HK<1. Therefore we finished the proof of Lemma 2.4.
Proof of Theorem 2.1. It is obvious that Theorem 2.1 is the consequence of (2.6), (2.7), Lemma 2.3 and Lemma 2.4.
In this section, we can understand Theorem 2.1 by considering the sub-bifractional Brownian noise, which is increments of sub-bifractional Brownian motion. For every integer n≥0, the sub-bifractional Brownian noise is defined by
Yn:=SH,Kn+1−SH,Kn. |
Denote
R(a,a+n):=E(YaYa+n)=E[(SH,Ka+1−SH,Ka)(SH,Ka+n+1−SH,Ka+n)]. | (3.1) |
We obtain
R(a,a+n)=fa(n)+g(n)−g(2a+n+1), | (3.2) |
where
fa(n)=[(a+1)2H+(a+n+1)2H]K−[(a+1)2H+(a+n)2H]K |
−[a2H+(a+n+1)2H]K+[a2H+(a+n)2H]K |
and
g(n)=12[(n+1)2HK+(n−1)2HK−2n2HK]. |
We know that the function g is the covariance function of the fractional Brownian noise with Hurst index HK. Thus we need to analyze the function fa to understand "how far" the sub-bifractional Brownian noise is from the fractional Brownian noise. In other words, how far is the sub-bifractional Brownian motion from a process with stationary increments?
The sub-bifractional Brownian noise is not stationary. However, the meaning of the following theorem is that it converges to a stationary sequence.
Theorem 3.1. For each n, as a→∞, we have
fa(n)=2H2K(K−1)a2(HK−1)(1+o(1)) | (3.3) |
and
g(2a+n+1)=22HK−2HK(2HK−1)a2(HK−1)(1+o(1)). | (3.4) |
Therefore lima→∞fa(n)=0 and lima→∞g(2a+n+1)=0 for each n.
Proof. (3.3) is obtained by Theorem 3.3 in Maejima and Tudor. For (3.4), we have
g(2a+n+1)=12[(2a+n+2)2HK+(2a+n)2HK−2(2a+n+1)2HK] |
=22HK−1a2HK[(1+n+22a−1)2HK+(1+n2a−1)2HK−2(1+n+12a−1)2HK] |
=22HK−1a2HK[1+2HKn+22a−1+HK(2HK−1)(n+22)2a−2(1+o(1)) |
+1+2HKn2a−1+HK(2HK−1)(n2)2a−2(1+o(1)) |
−2(1+2HKn+12a−1+HK(2HK−1)(n+12)2a−2(1+o(1)))] |
=22HK−2HK(2HK−1)a2(HK−1)(1+o(1)). |
Hence the proof of Theorem 3.1 is completed.
We are now interested in the behavior of the sub-bifractional Brownian noise (3.1) with respect to n (as n→∞). We have the following result.
Theorem 3.2. For integers a,n≥0, let R(a,a+n) be given by (3.1). Then for large n,
R(a,a+n)=HK(K−1)[(a+1)2H−a2H]n2(HK−1)+(1−2H)+o(n2(HK−1)+(1−2H)). |
Proof. By (3.2), we have
R(a,a+n)=fa(n)+g(n)−g(2a+n+1). |
By the proof of Theorem 4.1 in [11], we get, for large n, the term fa(n) behaves as
HK(K−1)[(a+1)2H−a2H]n2(HK−1)+(1−2H)+o(n2(HK−1)+(1−2H)). |
We know that the term g(n) behaves as HK(2HK−1)n2(HK−1) for large n. For g(2a+n+1), it is similar to the computation for Theorem 3.1, we can obtain g(2a+n+1) also behaves as HK(2HK−1)n2(HK−1) for large n. Hence we have finished the proof of Theorem 3.2.
It is easy to obtain the following corollary.
Corollary 3.1. For integers a≥1 and n≥0, let R(a,a+n) be given by (3.1). Then, for every a∈N, we have
∑n≥0R(a,a+n)<∞. |
Proof. By Theorem 3.2, we get that the main term of R(a,a+n) is n2HK−2H−1, and since 2HK−2H−1<−1, the series is convergent.
In this section, we prove two limit theorems to the sub-bifractional Brownian motion. Define a function g(t,s),t≥0,s≥0 by
g(t,s)=∂2RH,K(t,s)∂t∂s=4H2K(K−1)(t2H+s2H)K−2(ts)2H−1+HK(2HK−1)|t−s|2HK−2 |
−HK(2HK−1)(t+s)2HK−2 |
=:g1(t,s)+g2(t,s)−g3(t,s), | (4.1) |
for (t,s) with t≠s, t≠0, s≠0 and t+s≠0.
Theorem 4.1. Assume that 2HK>1 and let {ξj,j=1,2,⋯} be a sequence of standard normal random variables. g(t,s) is defined by (4.1). Suppose that E(ξiξj)=g(i,j). Then, as n→∞,
{n−HK[nt]∑j=1ξj,t≥0}d⇒{SH,Kt,t≥0}. |
Remark 1. Theorem 4.1 and 4.2 (below) are similar to the central limit theorem and can be used as a basis for many subsequent studies.
In order to prove Theorem 4.1, we need the following lemma.
Lemma 4.1. When 2HK>1, we have
∫t0∫s0g(u,v)dudv=(t2H+s2H)K−12(t+s)2HK−12|t−s|2HK. |
Proof. It follows from the fact that g(t,s)=∂2RH,K(t,s)∂t∂s for every t≥0,s≥0 and by using that 2HK>1.
Proof of Theorem 4.1. It is enough to show that, as n→∞,
In:=E[(n−HK[nt]∑i=1ξi)(n−HK[ns]∑j=1ξj)]→E(SH,KtSH,Ks). |
In fact, we have
In=n−2HK[nt]∑i=1[ns]∑j=1E(ξiξj)=n−2HK[nt]∑i=1[ns]∑j=1g(i,j). |
Note that
g(in,jn)=4H2K(K−1)[(in)2H+(jn)2H]K−2(ijn2)2H−1 |
+HK(2HK−1)|in−jn|2HK−2−HK(2HK−1)(in+jn)2HK−2 |
=n2(1−HK)g(i,j). | (4.2) |
Thus, as n→∞,
In=n−2HK[nt]∑i=1[ns]∑j=1n2HK−2g(in,jn) |
=n−2[nt]∑i=1[ns]∑j=1g(in,jn) |
→∫t0∫s0g(u,v)dudv |
=(t2H+s2H)K−12(t+s)2HK−12|t−s|2HK |
=E(SH,KtSH,Ks). |
Hence, we finished the proof of Theorem 4.1.
We now consider more general sequence of nonlinear functional of standard normal random variables. Let f be a real valued function such that f(x) does not vanish on a set of positive measure, E[f(ξ1)]=0 and E[(f(ξ1))2]<∞. Let Hk denote the k-th Hermite polynomial with highest coefficient 1. We have
f(x)=∞∑k=1ckHk(x), |
where ∑∞k=1c2kk!<∞ and ck=E[f(ξj)Hk(ξj)] (see e.g. [16]). Assume that c1≠0. Let ηj=f(ξj),j=1,2,⋯, where {ξj,j=1,2,⋯} is the same sequence of standard normal random variables as before.
Theorem 4.2. Assume that 2HK>32 and let {ξj,j=1,2,⋯} be a sequence of standard normal random variables. g(t,s) is defined by (4.1). Suppose that E(ξiξj)=g(i,j). Then, as n→∞,
{n−HK[nt]∑j=1ηj,t≥0}d⇒{c1SH,Kt,t≥0}. |
Proof. Note that ηj=f(ξj)=c1ξj+∑∞k=2ckHk(ξj). We obtain
n−HK[nt]∑j=1ηj=c1n−HK[nt]∑j=1ξj+n−HK[nt]∑j=1∞∑k=2ckHk(ξj). |
Using Theorem 4.1, it is enough to show that, as n→∞,
E[(n−HK[nt]∑j=1∞∑k=2ckHk(ξj))2]→0. | (4.3) |
In fact, we get
Jn:=E[(n−HK[nt]∑j=1∞∑k=2ckHk(ξj))2] |
=n−2HK[nt]∑i=1[nt]∑j=1∞∑k=2∞∑l=2ckclE[Hk(ξi)Hl(ξj)]. |
We know that, if ξ and η are two random variables with joint Gaussian distribution such that E(ξ)=E(η)=0, E(ξ2)=E(η2)=1 and E(ξη)=r, then
E[Hk(ξ)Hl(η)]=δk,lrkk!, |
where
δk,l={1,if k=l;0,if k≠l. |
Thus,
Jn=n−2HK[nt]∑i=1[nt]∑j=1∞∑k=2c2k(E(ξiξj))kk! |
=n−2HK[nt]∞∑k=2c2kk!+n−2HK[nt]∑i,j=1;i≠j∞∑k=2c2kk![g(i,j)]k. |
Since |g(i,j)|≤(E(ξ2i))12(E(ξ2j))12=1, we get, by (4.2),
Jn≤n−2HK[nt]∞∑k=2c2kk!+n−2HK[nt]∑i,j=1;i≠j∞∑k=2c2kk![g(i,j)]2 |
=n−2HK[nt]∞∑k=2c2kk!+n−2HK∞∑k=2c2kk![nt]∑i,j=1;i≠j[g(i,j)]2 |
≤tn1−2HK∞∑k=2c2kk!+n2(HK−1)(∞∑k=2c2kk!)n−2[nt]∑i,j=1;i≠j[g(in,jn)]2. | (4.4) |
On one hand, by ∑∞k=2c2kk!<∞ and 2HK>32>1, we get, as n→∞,
tn1−2HK∞∑k=2c2kk!→0. | (4.5) |
On the other hand, we have
n−2[nt]∑i,j=1;i≠j[g(in,jn)]2=n−2[nt]∑i,j=1;i≠j[g1(in,jn)+g2(in,jn)−g3(in,jn)]2 |
≤3n−2[nt]∑i,j=1;i≠j{[g1(in,jn)]2+[g2(in,jn)]2+[g3(in,jn)]2}. |
Since |g1(u,v)|≤C(uv)HK−1 and 2HK>32>1, we obtain
n−2[nt]∑i,j=1;i≠j[g1(in,jn)]2→∫t0∫t0g21(u,v)dudv≤C∫t0∫t0(uv)2HK−2dudv<∞. | (4.6) |
We know that
n−2[nt]∑i,j=1;i≠j[g3(in,jn)]2→∫t0∫t0g23(u,v)dudv |
=H2K2(2HK−1)2∫t0∫t0(u+v)4HK−4dudv |
<∞, | (4.7) |
since 2HK>32>1.
We have also
n−2[nt]∑i,j=1;i≠j[g2(in,jn)]2→∫t0∫t0g22(u,v)dudv |
=H2K2(2HK−1)2∫t0∫t0(u−v)4HK−4dudv |
<∞, | (4.8) |
since 2HK>32. Thus (4.3) holds from (4.4)–(4.8) and 2HK>32. The proof is completed.
Remark 2. [11] pointed out, when 2HK>1, the convergence of
n2(HK−1)n−2[nt]∑i,j=1;i≠j[g2(in,jn)]2 |
had been already proved in [16]. But we can not find the details in [16]. Here we only give the proof when 2HK>32, because the holding condition for (4.8) is 2HK>32.
In this section, we study an approximation in law of the fractional Brownian motion via the tangent process generated by the sbfBm SH,K.
Theorem 5.1. Let H∈(0,1) and K∈(0,1). For every t0>0, as ϵ→0, we have, the tangent process
{SH,Kt0+ϵu−SH,Kt0ϵHK,u≥0}d⇒{BHKu,u≥0}, | (5.1) |
where BHKu is the fractional Brownian motion with Hurst index HK.
Proof. As 0<HK<12, by (2.6), we get
{SH,Kt,t≥0}d={BHKt+√HKΓ(1−2HK)YHKt−√KΓ(1−K)XKt2H,t≥0}. |
By (2.5) in [12], there exists a constant C(H,K)>0 such that
E[(XK(t0+ϵu)2H−XK(t0)2HϵHK)2]=C(H,K)t2(HK−1)0u2ϵ2(1−HK)(1+o(1)), |
which tends to zero, as ϵ→0, since 1−HK>0.
On the other hand, similar to the proof of Lemma 2.4, we obtain
E[(YHKt0+ϵu−YHKt0ϵHK)2]=22HK−2Γ(2−2HK)t2(HK−1)0u2ϵ2(1−HK)(1+o(1)), |
which also tends to zero, as ϵ→0. Therefore (5.1) holds. Similarly, (5.1) also holds for the case 12<HK<1. We finished the proof.
In this paper, we prove that the increment process generated by the sub-bifractional Brownian motion converges to the fractional Brownian motion. Moreover, we study the behavior of the noise associated to the sbfBm and the behavior of the tangent process of the sbfBm. In the future, we will investigate limits of Gaussian noises.
Nenghui Kuang was supported by the Natural Science Foundation of Hunan Province under Grant 2021JJ30233. The author wishes to thank anonymous referees for careful reading of the previous version of this paper and also their comments which improved the paper.
The author declares there is no conflict of interest.
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1. | Nenghui Kuang, Huantian Xie, Least squares type estimators for the drift parameters in the sub-bifractional Vasicek processes, 2023, 26, 0219-0257, 10.1142/S0219025723500042 |