
This paper presents a hybrid-triggered catastrophe bond (CAT bond) pricing model. We take earthquake CAT bonds as an example for model construction and numerical analysis. According to the characteristics of earthquake disasters, we choose direct economic loss and magnitude as trigger indicators. The marginal distributions of the two trigger indicators are depicted using extreme value theory, and the joint distribution is established by using a copula function. Furthermore, we derive a multi-year hybrid-triggered CAT bond pricing formula under stochastic interest rates. The numerical experiments show that the bond price is negatively correlated with maturity, market interest rate and dependence of trigger indicators, and positively correlated with trigger level and coupon rate. This study can be used as a reference for formulating reasonable CAT bond pricing strategies.
Citation: Longfei Wei, Lu Liu, Jialong Hou. Pricing hybrid-triggered catastrophe bonds based on copula-EVT model[J]. Quantitative Finance and Economics, 2022, 6(2): 223-243. doi: 10.3934/QFE.2022010
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This paper presents a hybrid-triggered catastrophe bond (CAT bond) pricing model. We take earthquake CAT bonds as an example for model construction and numerical analysis. According to the characteristics of earthquake disasters, we choose direct economic loss and magnitude as trigger indicators. The marginal distributions of the two trigger indicators are depicted using extreme value theory, and the joint distribution is established by using a copula function. Furthermore, we derive a multi-year hybrid-triggered CAT bond pricing formula under stochastic interest rates. The numerical experiments show that the bond price is negatively correlated with maturity, market interest rate and dependence of trigger indicators, and positively correlated with trigger level and coupon rate. This study can be used as a reference for formulating reasonable CAT bond pricing strategies.
Stochastic nonlinear control is a hot topic because of its wide application in economic and engineering fields. The pioneer work is [1,2,3,4,5,6,7,8]. Specifically, [1,2,3] propose designs with quadratic Lyapunov functions coupled with weighting functions and [4,5,6,7,8] develop designs with quartic Lyapunov function, which are further developed by [9,10,11,12]. Recently, a class of stochastic systems (SSs) whose Jacobian linearizations may have unstable modes, has received much attention. Such systems are also called stochastic high-order nonlinear systems (SHONSs), which include a class of stochastic benchmark mechanical systems [13] as a special case. In this direction, [14] studies the state-feedback control with stochastic inverse dynamics; [15] develops a stochastic homogeneous domination method, which completely relaxes the order restriction required in [13], and investigates the output-feedback stabilization for SHONSs with unmeasurable states; [16] investigates the output-feedback tracking problem; [17] studies the adaptive state-feedback design for state-constrained systems. It should be emphasized that [13,14,15,16,17], only achieve stabilization in asymptotic sense (as time goes to infinity). However, many real applications appeals for prescribed-time stabilization, which permits the worker to prescribe the convergence times in advance.
For the prescribed-time control, [18,19] design time-varying feedback to solve the regulation problems of normal-form nonlinear systems; [20] considers networked multi-agent systems; [21,22] investigates the prescribed-time design for linear systems in controllable canonical form; [23] designs the output-feedback controller for uncertain nonlinear strict-feedback-like systems. It should be noted that the results in [18,19,20,21,22,23] don't consider stochastic noise. For SSs, [24] is the first paper to address the stochastic nonlinear inverse optimality and prescribed-time stabilization problems; The control effort is further reduced in [25]; Recently, [26] studies the prescribed-time output-feedback for SSs without/with sensor uncertainty. It should be noted that [24,25,26] don't consider SHONSs. From practical applications, it is important to solves the prescribed-time control problem of SHONSs since it permits the worker to set the convergence times in advance.
Motivated by the above discussions, this paper studies the prescribed-time design for SSs with high-order structure. The contributions include:
1) This paper proposes new prescribed-time design for SHONSs. Since Jacobian linearizations of such a system possibly have unstable modes, all the prescribed-time designs in [24,25,26] are invalid. New design and analysis tools should be developed.
2) This paper develops a more practical design than those in [13,14,15,16,17]. Different from the designs in [13,14,15,16,17] where only asymptotic stabilization can be achieved, the design in this paper can guarantee that the closed-loop system is prescribed-time mean-square stable, which is superior to those in [13,14,15,16,17] since it permits the worker to prescribe the convergence times in advance without considering the initial conditions.
The remainder of this paper is organized as follows. In Section 2, the problem is formulated. In Section 3, the controller is designed and the stability analysis is given. Section 4 uses an example to explain the validity of the prescribed-time design. The conclusions are collected in Section 6.
Consider the following class of SHONSs
dx1=xp2dt+φT1(x)dω, | (2.1) |
dx2=udt+φT2(x)dω, | (2.2) |
where x=(x1,x2)T∈R2 and u∈R are the system state and control input. p≥1 is an odd integral number. The functions φi:R2→Rm are smooth in x with φi(0)=0, i=1,2. ω is an m-dimensional independent standard Wiener process.
The assumptions we need are as follows.
Assumption 1. There is a constant c>0 such that
|φ1(x)|≤c|x1|p+12, | (2.3) |
|φ2(x)|≤c(|x1|p+12+|x2|p+12). | (2.4) |
We introduce the function:
μ(t)=(Tt0+T−t)m,t∈[t0,t0+T), | (2.5) |
where m≥2 is an integral number. Obviously, the function μ(t) is monotonically increasing on [t0,t0+T) with μ(t0)=1 and limt→t0+Tμ(t)=+∞.
Our control goal is to design a prescribed-time state-feedback controller, which guarantees that the closed-loop system has an almost surely unique strong solution and is prescribed-time mean-square stable.
In the following, we design a time-varying controller for system (2.1)–(2.2) step by step.
Step 1. In this step, our goal is to design the virtual controller x∗2.
Define V1=14ξ41, ξ1=x1, from (2.1), (2.3) and (2.5) we have
LV1(ξ1)=ξ31xp2+32ξ21|φ1|2≤ξ31xp2+32c2ξp+31≤ξ31(xp2−x∗p2)+ξ31x∗p2+32c2μpξp+31. | (3.1) |
Choosing
x∗2=−μ(2+32c2)1/pξ1≜−μα1ξ1, | (3.2) |
which substitutes into (3.1) yields
LV1(ξ1)≤−2μpξp+31+ξ31(xp2−x∗p2), | (3.3) |
where α1=(2+32c2)1/p.
Step 2. In this step, our goal is to design the actual controller u.
Define ξ2=xk−x∗2, from (3.2) we get
ξ2=x2+μα1ξ1. | (3.4) |
By using (2.1)–(2.2) and (3.4) we get
dξ2=(u+mTμ1+1/mα1ξ1+μα1xp2)dt+(φT2+μα1φT1)dω. | (3.5) |
We choose the following Lyapunov function
V2(ξ1,ξ2)=V1(ξ1)+14ξ42. | (3.6) |
It follows from (3.3), (3.5)–(3.6) that
LV2≤−2μpξp+31+ξ31(xp2−x∗p2)+ξ32(u+mTμ1+1/mα1ξ1+μα1xp2)+32ξ22|φ2+μα1φ1|2. | (3.7) |
By using (3.4) we have
ξ31(xp2−x∗p2)≤|ξ1|3|ξ2|(xp−12+x∗p−12)≤|ξ1|3|ξ2|((β1+1)μp−1αp−11ξp−11+β1ξp−12)=(β1+1)μp−1αp−11|ξ1|p+2|ξ2|+β1|ξ1|3|ξ2|p, | (3.8) |
where
β1=max{1,2p−2}. | (3.9) |
By using Young's inequality we get
(β1+1)μp−1αp−11|ξ1|p+2|ξ2|≤13μpξp+31+13+p(3+p3(2+p))−(2+p)((β1+1)αp−11)p+3μ−3ξp+32 | (3.10) |
and
β1|ξ1|3|ξ2|p≤13μpξ3+p1+pp+3(3+p9)−3/pβ(3+p)/p1μ−3ξp+32. | (3.11) |
Substituting (3.10)–(3.11) into (3.8) yields
ξ31(xp2−x∗p2)≤23μpξp+31+(pp+3(p+39)−3/pβ(p+3)/p1+1p+3(p+33(p+2))−(p+2)((β1+1)αp−11)p+3)μ−3ξp+32. | (3.12) |
From (2.3), (2.4), (3.2) and (3.4) we have
|φ2+μα1φ1|2≤2|φ2|2+2μ2α21|φ1|2≤4c2(|x1|1+p+|x2|1+p)+2c2α21μ2|x1|1+p≤β2μ1+p|ξ1|p+1+4c22p|ξ2|1+p, | (3.13) |
where
β2=4c2(1+2pαp+11)+2c2α21. | (3.14) |
By using (3.13) we get
32ξ22|φ2+μα1φ1|2≤13μpξ3+p1+(23+p(p+33(1+p))−(1+p)/2(32β2)(3+p)/2+6c22p)μ3(1+p)/2ξp+32. | (3.15) |
It can be inferred from (3.7), (3.12) and (3.15) that
LV2≤−μpξp+31+ξ32(u+mTμ1+1/mα1ξ1+μα1xp2)+β3μ3(p+1)/2ξp+32, | (3.16) |
where
β3=p3+p(3+p9)−3/pβ(3+p)/p1+13+p(p+33(p+2))−(p+2)((β1+1)αp−11)3+p+23+p(p+33(1+p))−(1+p)/2(32β2)(3+p)/2+6c22p. | (3.17) |
We choose the controller
u=−mTμ1+1/mα1ξ1−μα1xp2−(1+β3)μ3(p+1)/2ξp2, | (3.18) |
then we get
LV2≤−μpξp+31−μ3(p+1)/2ξp+32. | (3.19) |
Now, we describe the main stability analysis results for system (2.1)–(2.2).
Theorem 1. For the plant (2.1)–(2.2), if Assumption 1 holds, with the controller (3.18), the following conclusions are held.
1) The plant has an almost surely unique strong solution on [t0,t0+T);
2) The equilibrium at the origin of the plant is prescribed-time mean-square stable with limt→t0+TE|x|2=0.
Proof. By using Young's inequality we get
14μξ41≤μpξp+31+β4μ−3, | (3.20) |
14μξ42≤μ3(p+1)/2ξp+32+β4μ−3, | (3.21) |
where
β4=p−1p+3(3+p4)−4/(p−1)(14)(3+p)/(p−1). | (3.22) |
It can be inferred from (3.19)–(3.21) that
LV2≤−14μξ41−14μξ42+2β4μ−3=−μV2+2β4μ−3. | (3.23) |
From (2.1), (2.2) and (3.18), the local Lipschitz condition is satisfied by the plant. By (3.23) and using Lemma 1 in [24], the plant has an almost surely unique strong solution on [t0,t0+T), which shows that conclusion 1) is true.
Next, we verify conclusion 2).
For each positive integer k, the first exit time is defined as
ρk=inf{t:t≥t0,|x(t)|≥k}. | (3.24) |
Choosing
V=e∫tt0μ(s)dsV2. | (3.25) |
From (3.23) and (3.25) we have
LV=e∫tt0μ(s)ds(LV2+μV2)≤2β4μ−3e∫tt0μ(s)ds. | (3.26) |
By (3.26) and using Dynkin's formula we get
EV(ρk∧t,x(ρk∧t))=Vn(t0,x0)+E{∫ρk∧tt0LV(x(τ),τ)dτ}≤Vn(t0,x0)+2β4∫tt0μ−3e∫τt0μ(s)dsdτ. | (3.27) |
Using Fatou Lemma, from (3.27) we have
EV(t,x)≤Vn(t0,x0)+2β4∫tt0μ−3e∫τt0μ(s)dsdτ,t∈[t0,t0+T). | (3.28) |
By using (3.25) and (3.28) we get
EV2≤e−∫tt0μ(s)ds(Vn(t0,x0)+2β4∫tt0μ−3e∫τt0μ(s)dsdτ),t∈[t0,t0+T). | (3.29) |
By using (3.6) and (3.29) we obtain
limt→t0+TE|x|2=0. | (3.30) |
Consider the prescribed-time stabilization for the following system
dx1=x32dt+0.1x21dω, | (4.1) |
dx2=udt+0.2x1x2dω, | (4.2) |
where p1=3, p2=1. Noting 0.2x1x2≤0.1(x21+x22), Assumption 1 is satisfied.
Choosing
μ(t)=(11−t)2,t∈[0,1), | (4.3) |
According to the design method in Section 3, we have
u=−3μ1.5x1−1.5μx32−57μ6(x2+1.5μx1)3 | (4.4) |
In the practical simulation, we select the initial conditions as x1(0)=−6, x2(0)=5. Figure 1 shows the responses of (4.1)–(4.4), from which we can obtain that limt→t0+TE|x1|2=limt→t0+TE|x2|2=0, which means that the controller we designed is effective.
This paper proposes a new design method of prescribed-time state-feedback for SHONSs. the controller we designed can guarantee that the closed-loop system has an almost surely unique strong solution and the equilibrium at the origin of the closed-loop system is prescribed-time mean-square stable. The results in this paper are more practical than those in [13,14,15,16,17] since the design in this paper permits the worker to prescribe the convergence times in advance without considering the initial conditions.
There are some related problems to investigate, e.g., how to extend the results to multi-agent systems [27], impulsive systems [28,29,30] or more general high-order systems [31,32,33,34].
This work is funded by Shandong Provincial Natural Science Foundation for Distinguished Young Scholars, China (No. ZR2019JQ22), and Shandong Province Higher Educational Excellent Youth Innovation team, China (No. 2019KJN017).
The authors declare there is no conflict of interest.
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