6.2.1 Assumptions
Here, the necessary notation and basic assumptions for our model are described. Some assumptions are almost the same with that in Zimbidis and Haberman (2001) [82], Pantelous and Papageorgiou (2013) [51] and Pantelous and Yang (2014, 2015) [52, 53]. so only a brief explanation for the different assumptions is provided here.
Assumption 6.1: Same with Assumption 3.1 in Chapter 3.
Assumption 6.2: Same with Assumption 3.2 and Assumption 3.3 in Chapter 3. Assumption 6.3: Same with Assumption 3.4 in Chapter 3.
Assumption 6.4: Let {σt;t ≥ 0} be a arbitrary switching signal with state space
S ={1,2· · ·N}. σt is a piecewise constant function of time and the transition proba-
bility is is unknown or not existed. We assume that the switching signalσtis unknown
a prior, but its instantaneous value is available in real time.
Assumption 6.5: Positive integerτi represents the time delay when the system oper-
ates in the regime i. Then we denote
τmax= max{τi, i∈ S},
τmin= min{τi, i∈ S}.
We consider a mode-dependent time-varying delay,τt, which is upper and lower bounded,
i.e. τmin≤τt≤τmaxwithτmin, τmax∈N. So, considering a specific time-delay interval, at the end of each year [t, t+ 1], we have the exact information up to the end of the year t−τt. As indicated in previous chapters’ assumption, the value forτi can be estimated
using past experience and statistical data. Moreover, the national and international regulatory policy might be also applied for defining the upper bound of this interval. Assumption 6.6: Same with Assumption 3.6 in Chapter 3.
Assumption 6.7: Same with Assumption 3.7 in Chapter 3..
6.2.2 Model Formulation
In the present chapter, the P-R process is described by a arbitrary regime switching system with time-varying delays which extend the model used in Chapter 5. Assume Rt = (R1,tR2,t· · ·Rm,t)T be the vector expression of the accumulated reserves, where
Ri,t is the accumulated reserves ofithproduct at timet. As in Chapter 5, the premium
process is formulated as follow:
Pt+1 = ˆCt+1−[Eσt + ∆Eσt,t]Rt−τt −[Zσt+ ∆Zσt,t]Ut. (6.2.1)
Ut∈Rm is the control input. Here, we develop the model into the arbitrary switched
system. Let Rt = (R1,tR2,t· · ·Rm,t)T be the vector expression of the reserves, where
Same with those in Chapter 5, Jσt is the investment return matrices in timet for the risk-free asset. It is possible to include also risky assets but we leave it for a future work. Switching signalσtis a piecewise constant function of time which takes valueiin
the finite setS = [1 2· · ·N]. We assume that the switching signal σt is governed by
a Arbitrary jump process (see Assumption 6.4). The premiums are assumed to be the
earned premiums and claims areincurred claims as well. From the equations (6.2.1) and (6.2.2), we get
Rt+1 = [Jσt + ∆Jσt,t]Rt+e{Cˆt+1−[Eσt+ ∆Eσt,t]Rt−τt −[Zσt + ∆Zσt,t]Ut} −Ct+1
= [Jσt + ∆Jσt,t]Rt−e[Eσt+ ∆Eσt,t]Rt−τt−e[Zσt + ∆Zσt,t]Ut+wt+1.
The parametersJi,EiandZiare real constant base matrices. ∆Ji,t, ∆Ei,tand ∆Zi,t
are the respective parameter uncertainties. For the purpose of the modelling process, Ji and Ei respectively could be a risk-free interest rate and a constant-base return to
the policyholders. Then, Zi is a parameter of the control input. Finally, ∆Ji,t, ∆Ei,t
and ∆Zi,t are unknown matrices representing time-varying parameter uncertainties,
and they are assumed to be of the form:
[∆Ji,t −e∆Ei,t −e∆Zi,t] =MiFt[N1i N2i N3i], (6.2.3)
Mi, N1i, N2i, N3i are known real constant matrices and Ft :N→ Rs×j is an unknown time-varying matrix function satisfying
FtTFt≤I, ∀t∈N, (6.2.4)
∆Ji,t , ∆Ei,t and ∆Zi,t are said to be admissible if they satisfy both (6.2.3) and (6.2.4).
Thus we have the following discrete time arbitrary regime switching linear P-R system:
Θ4:
Rt+1= [Jσt + ∆Jσt,t]Rt−e[Eσt+ ∆Eσt,t]Rt−τt−e[Zσt + ∆Zσt,t]Ut+wt+1
Rt=ϕt fort∈[−τmax,0].
The system hasN system regimes. We denote system Θ4without controller element
Ut and disturbance wt+1 as Θ41. System Θ4 without disturbance wt+1 is denoted as
Lemma 6.1. (Gu et al. 2003 [30]) Assume that τt : Z+ → 1,2, ... and τt < τmax,
where τmax is a positive integer, then for any positive-definite matrix P ∈ Rn×n and
vector functionRt, we have
τmax t−1 X m=t−τmax YTmQYm> t−1 X m=t−τσt YTm} Q t−1 X m=t−τσt Ym
This chapter is concerned with the robust stability analysis and design problems for the arbitrary switched P-R system and it is complementary research of markovian switched system in Chapter 5. Our objective is to present an approach to investigate and manipulate the stability of arbitrary switched P-R system.