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Insurance pricing using H∞-control
Applied Mathematics and Computation 232 (2014) 685–697
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Insurance pricing using H1-control Alexandros A. Zimbidis Department of Statistics, Athens University of Economics and Business, 76 Patision St., Athens 104 34, Greece
a r t i c l e
i n f o
Keywords: Insurance pricing H1-control Robust stochastic stabilization Linear matrix inequalities Time varying delayed systems
a b s t r a c t The paper considers a typical insurance system ‘‘suffering’’ from the three standard ‘‘curses’’: (a) the stochastic nature of claims, (b) the inherent delays in claims settlement and reserving process and (c) the uncertainty concept that endows many of its parameters and especially the investment process. We construct a general multidimensional model for pricing simultaneously one, two or more different insurance products. The responsible decision maker uses the incomplete information of claims and aims to balance the system by the means of a feedback mechanism. The robust stabilization controller of the system is obtained by the means of H1-control using typical linear matrix inequalities. Finally, a numerical application is fully investigated providing further insight into the practical problem of pricing assuming the simplest case of a portfolio with a single product. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction The pricing process is one of the basic concerns of an insurance company. The traditional static point of view has already been replaced by the dynamic view using the tools of stochastic control theory. Actually, typical insurance systems have been described by the means of stochastic differential or difference equations while the pricing process has been established as an optimization problem under a certain objective function. In general the ‘‘insurance pricing puzzle’’ is quite complicated and difficult. There are several issues and modelling technicalities such as: the stochastic nature of the parameters, the inherent delay of the available information, the uncertainty of the economic environment and many others. An insurance company starting with an initial capital it continuously receives premiums while pays claims at specific time points and always targets to a positive (but not huge) balance in any future time. Past research papers have investigated the pricing problem of an insurance system allowing for stochastic variables or delays or other technicalities. Here, we mention only a few of these research efforts as an indicative guide to the huge problem of pricing. Martin-Löf [6] was amongst the pioneers for using the tools of control theory in the insurance pricing context. Vandebroek and Dhaene [10] apply the control theory techniques in the non-life insurance business. Norberg [7], using the standard Brownian motion as the basic tool for modelling, formulates the problem of pricing and reserving within a continuous time framework. His results coincide with the typical results of the traditional approach from risk theory. Zimbidis and Haberman [13] examine the combined effect of delay and feedback on the pricing process. Emms et al. [4] explore the optimal premium strategies within a competitive insurance market using the tools of a standard Brownian motion. Young [12] investigates the pricing of life contracts when mortality is modelled stochastically via the instantaneous Sharpe ratio. Zimbidis [14] further develops the concept of competitive insurance markets using the fractional Brownian motion as the modelling tool for the claim process. Finally, we provide an interesting reference for the modeling of uncertainty in the financial setting. Denis and Martini [3] introduce the uncertainty concept in the financial pricing of a contingent claim. They E-mail address: [email protected] http://dx.doi.org/10.1016/j.amc.2014.01.105 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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A.A. Zimbidis / Applied Mathematics and Computation 232 (2014) 685–697
incorporate uncertainty via a family of martingale measures and calculate the superreplication price of a European contingent claim. In this paper, we investigate the pricing problem considering all the three complications mentioned before: Stochastic nature, delay and uncertainty. The new modelling concept is the uncertainty. We solve the complications by using the tools of H1-control. Additionally, we make an allowance for pricing a mixed portfolio of different insurance products. We start from the very basics, where we consider an insurance company with an initial reserve, an incoming premium flow plus incoming investment proceeds while also an additional outcome of claim payments. The pricing rule (premium determination) is based on past claim experience and over an initial balancing feedback mechanism evaluating the total accumulated surplus (refunding a certain portion of surplus back to the policyholders by reducing the initial level of premium). Now, although the typical description mentioned above indicates premium as input and claims as output variables, in the typical system design we should consider (the vice versa situation) claims as input variable while premiums as output variable. Then, our target is to simultaneously stabilize the premium level charged to the policyholders while also stabilize the state variable: the accumulated surplus fund of the insurance company. The stabilization may be obtained by the attachment of an additional feedback mechanism. That may be described with equations and Fig. 1 below. [Surplus Fund at time t] = [Initial Fund of the Company at time 0] + [Investment Proceeds up to time t] + [Premiums received up to time t] [Claims paid up to time t]. [Premium received at time t] = [Average of paid claims per time-unit up to time t] [% of the Surplus Fund at time t]. The paper is organised as follows: Section 2 contains a short introductory guide to (H infinity) H1-control and linear matrix inequalities (LMI’s). Section 3 describes the basic structure for the pricing process and the set of required assumptions. Furthermore, we analyze the problem in the multidimensional case allowing for pricing two or more different insurance products. Section 4 presents the theoretical solution of the model for the general multidimensional case while provides a detailed numerical application and practical considerations as regards the insurance pricing for the simplest case of a portfolio with a single product. Finally, Section 5 concludes the paper. 2. H‘-control and linear matrix inequalities Control theory and especially optimal control theory has played an important role in many scientific areas and practical problems over the past decades. The last two or three decades, control theory has been fully applied to actuarial problems. A new direction of research for control theory the very last years is the H1-control. Actually, H1-control is optimal control design when considering the worst exogenous input for a closed loop system. So, H1-control offers an ideal framework to investigate problems under uncertain (but somehow bounded) parameters and conditions. Below, we provide a short note on the relevant theory as regards the H1-control (see [2] for more details) and linear matrix inequalities (LMIs) (see [1] for more details) that is the powerful tool for solving the respective stability problems. Before going further, we formalize the notation for the matrices i.e. Let A represents a matrix, then A ð P; 6; Þ 0 denotes that A is symmetric positive definite (symmetric positive semi-definite, symmetric negative definite, symmetric negative semi-definite). 2.1. H1-control (or H infinity control) We assume an uncertain linear stochastic delayed and controlled differential system,
dxðtÞ ¼ ðAðtÞxðtÞ þ Ad ðtÞxðt sðtÞÞ þ BðtÞuðtÞ þ Bt tðtÞÞdt þ ðEðtÞxðtÞ þ Ed ðtÞxðt sðtÞÞ þ Et tðtÞÞdWðtÞ t P 0;
ð1Þ
zðtÞ ¼ CxðtÞ þ C d xðt sðtÞÞ þ DuðtÞ t P 0;
ð2Þ
zðtÞ ¼ /ðtÞ t 2 ½l; 0;
ð3Þ
where xðtÞ 2 Rn is the state variable,
Fig. 1. Control system design of an insurance system.
A.A. Zimbidis / Applied Mathematics and Computation 232 (2014) 685–697
687
uðtÞ 2 Rm is the controlled input variable, mðtÞ 2 Rp is the disturbance input variable (zero or positive valued & square integrable), zðtÞ 2 Rq is the controlled output variable, s(t) is time-varying bounded delay time satisfying the following conditions
d sðtÞ 6 h < 1 dt
0 6 sðtÞ 6 l
/(t) is any given initial data in the Rn -valued family F0 measurable stochastic process f(s) with supl6s60 EjfðsÞj2 < 1, where E[. . .] stands for the expectation operator with respect to the given probability measure P. W = {W(t); t P 0} is a scalar Brownian motion defined on a complete probability space (X, F, P) with a natural filtration fFt gtP0 . Bt, Et C, Cd, D are known constant matrices A(t), Ad(t), B(t), E(t), Ed(t) are matrix-valued functions with time-varying uncertainties as
AðtÞ ¼ A þ DAðtÞ; Ad ðtÞ ¼ Ad þ DAd ðtÞ; BðtÞ ¼ B þ DBðtÞ; EðtÞ ¼ E þ DEðtÞ; Ed ðtÞ ¼ Ed þ DEd ðtÞ where A, Ad, B, E and Ed are known real constant matrices while DAðtÞ; DAd ðtÞ; DBðtÞ; DEðtÞ and DEd ðtÞ are unknown matrices representing time-varying parameters uncertainties. These uncertainties are norm-bounded and may be described as follows
½DAðtÞ DAd ðtÞ DBðtÞ DEðtÞ DEd ðtÞ ¼ HFðtÞ½Na Nad Nb Ne Ned where H; N a ; N ad ; N b ; N e and N ed are known real constant matrices and F(t) is an unknown matrix function with Lebesgue measurable elements, satisfying the condition
F s ðtÞ FðtÞ I; where F s ðtÞ stands for the transpose matrix of FðtÞ. Furthermore, we provide the formal definitions for robust stability and robust performance for the system (1)-(3). Definition 1. The system (1)–(3) with u(t) = 0, t(t) = 0 is said to be robust stochastically stable if there exists a positive constant q such that
lim E
Z
T!1
T
xs ðtÞ xðtÞ dt < q sup EjfðsÞj2
ð4Þ
l6s60
0
for all admissible uncertainties DAðtÞ; DAd ðtÞ; DEðtÞ and DEd ðtÞ Definition 2. Given a scalar c > 0, the unforced stochastic system (1)–(3) with u(t) = 0 is said to be robust stochastically stable with disturbance attenuation c if it is robust stochastically stable in the sense of definition 1 and under zero initial conditions,
E½
Z
1
zs ðtÞ zðtÞ dt < c2
0
Z
1
W s ðtÞ WðtÞ dt
ð5Þ
0
Below, we provide two basic theorems that present the solutions to the robust stabilization problem for system (1)-(3). The first theorem corresponds to the special case of a zero disturbance input variable, t(t) = 0. The second theorem solves the general format of the problem using the validity of Theorem 1. Both detailed proofs (for Theorems 1 and 2) are described in Chen et al. [2]. So, we have the following Theorems 1 and 2: Theorem 1. Consider the system (1)-(3) with t(t) = 0. Then for given scalars l > 0, h < 1, this system is robust stochastically d stabilizable for any time-delay s(t) satisfying 0 6 sðtÞ 6 l; dt sðtÞ 6 h if for some prescribed scalar d, there exist matrices X 0; Y; Z; W 1 ; W 2 ; W 3 ; K; Q 0; S 0; R 0 and scalars e1 > 0; e2 > 0; e3 > 0 such that the following linear matrix inequalities (LMIs) (6) and (7) hold:
2
X1 6 Xs 6 2
6 6 0 6 6 s 6X 6 6 lZ 6 6 s 4 L11 s
L21
X2 X3
L11
0
lZ s lY s
0
X
ð1 dÞAd S
0
ð1 dÞSs As
ð1 hÞS
0
0
L12
0 lY
0 0
S 0
0 lQ
0 0
0
Ls12
0
0
U1
0
Ls22
0
0
0
d
L21
3
0 7 7 7 L22 7 7 7 0 7 0 7 0 7 7 7 0 5
U2
ð6Þ
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A.A. Zimbidis / Applied Mathematics and Computation 232 (2014) 685–697
2
W1
W2
6 s 4 W2
7 dAd Q 5 0
W3 d Q s As
0
3
0
ð7Þ
Q
d
(Note: some of the ZEROS in (6) and (7) correspond to blocks of zeros),
where
X1 ¼ Z þ Z s þ lW 1 ; X2 ¼ XðAs þ dAsd Þ þ K s Bs þ Y Z s þ lW 2 ; X3 ¼ Y Y s þ e1 HHs þ d2 Ad R Asd þ lW 3 ; n
U1 ¼ diag e1 I; e2 I;
o
l 1h
n
U2 ¼ diag X e2 HHs ;
e3 I ;
l 1h
o ðR e3 HHs Þ ;
h
L11 ¼ XN a s þ KNb s XN e s h
l
L12 ¼ SNad s SNed s h
L21 ¼ XEs
l 1h
h
L22 ¼ SEsd
l 1h
1h
l 1h
i XN e s ;
i SNed s ;
i XEs ; SEsd
i
and the stabilizing control law is described as
uðtÞ ¼ K X 1 xðtÞ:
ð8Þ
Theorem 2. Consider the system (1)-(3). Then for given scalars l > 0, h < 1, this system is robust stochastically stabilizable with disturbance attenuation c > 0 for any time-delay s(t) satisfying 0 6 sðtÞ 6 l; dtd sðtÞ 6 h if for some prescribed scalar d, there exist matrices X 0; Y; Z; W 1 ; W 2 ; W 3 ; K; Q 0; S 0; R 0 and scalars e1 > 0; e2 > 0; e3 > 0 such that LMI (7) and the following LMI holds:
2
X1 Xs2
6 6 6 6 0 6 6 6 0 6 6 s 6 X C þ DK s 6 6 Ls11 6 6 6 Ls21 6 6 Xs 4
lZ
X2 X3
0
0
XC s þ KDs
L11
L21
X
ð1 dÞAd S
Bt
0
0
0
0
L12
L22
0
3
lZ s 7 lY s 7 7
ð1 dÞSs As
ð1 hÞS
0
Bst
0
c2 I
0
0
L23
0
0
0
I
0
0
0
0
d
0
Cd
Ss
s
SC d
0
0
Ls12
0
0
U1
0
0
0
0
Ls22
Ls23
0
0
U2
0
0
0
0
0
0
0
0
S
0
lY s
0
0
0
0
0
0
lQ
7 7 7 7 7 7 7 0 7 7 7 7 7 7 7 5
ð9Þ
(some of the ZEROS in (9) correspond to blocks of zeros), where X1, X2, X3, U1, U2, L11, L12, L21, L22, are defined in previous l s Theorem 1 and L23 ¼ Est 1h Et . Then, the stabilizing control law is described as
uðtÞ ¼ K X 1 xðtÞ: 2.2. Linear matrix inequalities (LMI’s) The general format of a typical Linear Matrix Inequality is the following
ð10Þ
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A.A. Zimbidis / Applied Mathematics and Computation 232 (2014) 685–697 m X xi F i 0;
FðxÞ ¼ F 0 þ
ð11Þ
i¼1
where x ¼ ðx1 ; x2 ; x3 ; . . . ; xm Þ 2 Rm is the variable to be determined, while F i ¼ F si 2 Rnxn i ¼ 0; 1; 2; . . . ; m, additionally, we can state that the LMI above may be easily transformed to a set of n inequalities in x. The relationship of LMIs and dynamic systems has been early recognised by Lyapunov [5]. He demonstrated that the stability of the basic differential equation
d xðtÞ ¼ A xðtÞ dt is obtained when there is a matrix P such that the following conditions hold
P 0 and As P þ PA 0: Now, we may easily transform the conditions above into the standard format of a LMI using a matrix of matrices as below
P
0
0 ðAs P þ PAÞ
0:
Furthermore, Yakubovich [11] was the first who formally established the importance of the LMI’s in the solution of the control problems. The solution for LMI’s has been initially based on the ellipsoid algorithm and on the interior-point methods. These two basic approaches have been further exploited over the last years and also inspire other similar ones. In this paper, we use a quite recent algorithm proposed by Orsi et al. [8] and Rami et al. [9] that is based on the classical alternating projection method. Actually, they use this algorithm to solve ‘‘convex feasibility problems where the constraints are given by the intersection of two convex cones in a Hilbert space’’. Then, as an application, they derive the solution of an LMI problem calculating a sequence of matrix eigenvalue–eigenvector decompositions. Below, we provide a short description of the relevant algorithm that is fully described in the papers mentioned above (and especially in Orsi et al. [8, p. 4982]), aiming to find x 2 Rm satisfying a strict LMI (11) (or the non-strict version where the symbol ‘‘’’ is replaced by ‘‘’’). Before provide the formal description of the algorithm, we establish two special functionals ‘‘_ ’’ and ‘‘^ ’’ from Sn the set of real symmetric matrices to Rp and vice versa, where p ¼ nðnþ1Þ , as below: 2
0
s11
B B s12 B S¼B B s13 B @ s1n
s12
s13
s22
s23
s23
s33
s2n
s3n
s1n
1
C s2n C _ C s s3n C C ! S ¼ ðs11 ; s12 ; s13 ; ; s1n ; s22 ; s23 ; ; s2n ; s33 ; ; s3n ; :; snn Þ C A snn 0
s11 B B s12 ^ B S ¼ ðs11 ; s12 ; s13 ; . . . ; s1n ; s22 ; s23 ; . . . ; s2n ; s33 ; . . . ; s3n ; . . . ; snn Þs ! S ¼ B B s13 B @
s12
s13
s22
s23
s23
s33
s1n
s2n
s3n
s1n
C s2n C C s3n C C C A snn
2.3. Algorithm for solving the strict LMI (11) Data: A set of m + 1 symmetric matrices F i ¼ F si 2 Rnxn i ¼ 0; 1; 2; . . . ; m Initialization: Choose initial conditions and calculate the basic matrix G
-1
, as below
Choose values for the parameters q, t such that q > 0 and t 2 (0, 2) (e.g. q = 1 & t = 1,99). Choose any x0 2 R and x 2 Rm . Choose any real symmetric matrix S 2 Rnxn (that is a slack variable). Define matrix Q as Q :¼ ½F^0 ; F^1 ; . . . ; F^m 2 Rp x ðmþ1Þ ; p ¼ nðnþ1Þ . 2 Calculate matrix G :¼ Q Q s þ 12 diagð^I þ 1Þ, where I is the n n identity matrix and 1 2 Rp while 1 ¼ ð1; 1; . . . ; 1Þs . (vi) Calculate the inverse matrix G1.
(i) (ii) (iii) (iv) (v)
1
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A.A. Zimbidis / Applied Mathematics and Computation 232 (2014) 685–697
Then we may start the algorithm described with the following steps Step 1: Calculate and replace via the formula x0 :¼ ð1 tÞ x0 þ t maxfq; x0 g. Step 2: Find an eigenvalue–eigenvector decomposition of S. Let S ¼ V D V s with D ¼ diagðd1 ; d2 ; . . . ; dn Þ the respective matrix of eigenvalues while V the respective matrix of eigenvectors. Step 3: Define a new matrix D ¼ diagðmaxfq; d1 g ; maxfq; d2 g ; . . . ; maxfq; dn g Þ Step 4: Calculate and replace matrix S via the formula S ¼ V ðð1 tÞD þ tDÞ V s . Step 5: Calculate a :¼ G1 ðQ ½x0 ; xs s ^ SÞ. Step 6: Calculate and replace x0 and x via the formula ½x0 ; xs s :¼ ½x0 ; xs s Q s a. ^ Step 7: Define T :¼ 0; 5 a . Step 8: Redefine matrix T by duplicating all the elements of the first diagonal. Step 9: Replace matrix S via the formula S :¼ S T Step 10: If x0 > 0 and the minimum eigenvalue of S is greater than zero then the solution is x=x0 otherwise we return to step 1 and continue the algorithm. 3. The pricing process 3.1. Portfolio with a single product – one dimension We consider a typical insurance system with an initial reserve, incoming variable premiums determined by a feedback mechanism according to past experience and claims driven by a drifted Brownian motion. The reserve is invested in a safe asset with a variable yet uncertain force of interest. Then, the reserve of the company obeys the following stochastic differential equation.
dRðtÞ ¼ ½Reserve at time ðt þ dtÞ ½Reserve at time ðtÞ ¼ ½Investment income earned in ðt; t þ dtÞ þ ½Premiums in ðt; t þ dtÞ ½Claims in ðt; t þ dtÞ ⁄ (Claims also include the element of administration expenses) or using standard notation the reserve obeys the following relationship
dRðtÞ ¼ rðtÞRðtÞdt þ p ðtÞdt ½mðtÞdt þ rðtÞdWðtÞ;
ð12Þ
while the premium is established via the following feedback mechanism
p ðtÞ ¼ pðtÞ f Rðt lðtÞÞ; where R(t): is the reserve at time t, rðtÞ : is the force of interest at time t, p ðtÞ: actual premium rate at time t, p(t): standard premium rate at time t, f: profit sharing factor, lðtÞ: time delay for claims settlement at time t, mðtÞ: average claim rate at time t, rðtÞ: volatility claim rate at time t, further assume that WðtÞ: standard Brownian motion.
ð13Þ
rðtÞ ¼ k mðtÞ
Combining equations (12) and (13) and after some algebra we finally obtain the system
dRðtÞ ¼ ½rðtÞRðtÞ f Rðt lðtÞÞ þ pðtÞ mðtÞdt þ ½kmðtÞdWðtÞ;
ð14Þ
p ðtÞ ¼ pðtÞ f Rðt lðtÞÞ:
ð15Þ
Following the reasoning of Zimbidis and Haberman [13], parameter (f) – the feedback factor – normally lies in the interval of [0, 1]. The practical justification is more or less obvious. An insurance company should be prepared to refund a smaller or larger portion of the accumulated surplus but certainly reluctant to exceed the 100%. Furthermore, a negative feedback factor corresponds to unreasonable and unnecessary premium loading. So, f normally lies in [0, 1]. Additionally, supporting Eq. (13) we also assume that there is no distribution of surplus to shareholders or anyone else (e.g. the tax authorities) except to the policyholders.
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3.2. Portfolio with two products – two dimensions Now, we consider a typical insurance system with two products. The extra complication is the interaction (incoming or outgoing reserve portion) between the two reserves. Keeping unchanged the other characteristics of the model, we may identify that the individual reserve of each company obeys the following stochastic differential equation dR(t) = [Reserve at time (t + dt)] [Reserve at time (t)] = [profit sharing coming from the other product in (t, t + dt)] [profit sharing given to the other product in (t, t + dt)] + [Investment income earned in (t,t + dt)] + [Premiums in (t, t + dt)] [Claims in (t, t + dt)] or extending the previous standard notation we obtain
dR1 ðtÞ ¼ ½r1 ðtÞR1 ðtÞ þ q21 ðtÞR2 ðt l2 ðtÞÞ q12 ðtÞR1 ðt l1 ðtÞÞ f1 R1 ðt l1 ðtÞÞ þ p1 ðtÞ m1 ðtÞdt þ ½k1 ðtÞm1 ðtÞdWðtÞ; p 1 ðtÞ ¼ p1 ðtÞ f1 R1 ðt l1 ðtÞÞ dR2 ðtÞ ¼ ½r2 ðtÞR2 ðtÞ þ q12 ðtÞR1 ðt l1 ðtÞÞ q21 ðtÞR2 ðt l2 ðtÞÞ f2 R2 ðt l2 ðtÞÞ þ p2 ðtÞ m2 ðtÞdt þ ½k2 ðtÞm2 ðtÞdWðtÞ; p 2 ðtÞ ¼ p2 ðtÞ f2 R2 ðt l2 ðtÞÞ; where the subscript 1 and 2 correspond to the first and second product respectively while q12 ðtÞ: proportion of reserve transferred from the first to the second product at time t, q21 ðtÞ: proportion of reserve transferred from the second to the first product at time t, Using the vector representation we obtain the following vector equations
" dR ðtÞ ¼ " þ
p ðtÞ ¼
#
dR1 ðtÞ
¼
dR2 ðtÞ 1
0
0
1
p 1 ðtÞ p 2 ðtÞ
("
#"
0
0 #)
r 2 ðtÞ ("
m1 ðtÞ
dt þ
m2 ðtÞ
¼
#"
r 1 ðtÞ
1 0 0 1
p1 ðtÞ p2 ðtÞ
#
" þ
R2 ðtÞ k1
0
0
k2
þ
R1 ðtÞ
f1 0
#"
0 f2
#"
q12 ðtÞ f1
q21 ðtÞ
q12 ðtÞ
q21 ðtÞ f2
m1 ðtÞ m2 ðtÞ
#)
R1 ðt l1 ðtÞÞ
#
R2 ðt l2 ðtÞÞ
" þ
1 0 0 1
#"
p1 ðtÞ
#
p2 ðtÞ
dWðtÞ;
R1 ðt l1 ðtÞÞ : R2 ðt l2 ðtÞÞ
3.3. Portfolio with n-products – multidimensional case Now, we may easily extend the model to a mixed portfolio with n products – dimensions. The respective vector representation of the general multidimensional model is as follows
80 3 2 r 1 ðtÞ 0 ... 0 1 dR1 ðtÞ > > > 7 >B 7 6 C6 > 6 dR ðtÞ 7 > 6 dR ðtÞ 7 r 2 ðtÞ . . . 0 C 7 6 2 7 B 7 C B 6 ... 7 > ... . . . . . . . . . A6 . . . 7 > @ 5 > 5 4 4 > > : 0 0 . . . r n ðtÞ dRn ðtÞ dRn ðtÞ 12 3 0 q11 ðtÞ f1 q21 ðtÞ ... qn1 ðtÞ R1 ðt l1 ðtÞÞ C6 7 B 7 B q ðtÞ 6 q22 ðtÞ f2 . . . qn2 ðtÞ C C6 R2 ðt l2 ðtÞÞ 7 B 12 C6 7 þB C6 7 B C6 7 B ... ... ... ... ... A4 5 @ q1n ðtÞ q2n ðtÞ . . . qnn ðtÞ fn Rn ðt ln ðtÞÞ 12 39 0 0 1 0 . . . 0 k1 m1 ðtÞ > 0 > > > C6 7> B B B 0 1 . . . 0 C6 m2 ðtÞ 7= B 0 k2 C6 7 B B þB þ fB C6 7 dt B . . . . . . . . . . . . C6 :::::::::::: 7> B ... ... > A4 5> @ @ > > ; 0 0 . . . 1 mn ðtÞ 0 0 2
dR1 ðtÞ
3
0 1 0 . . . 0 12 p1 ðtÞ 3 C6 7 B 7 B 0 6 1 ... 0 C C6 p2 ðtÞ 7 B þB C6 7 B . . . . . . . . . . . . C6 . . . 7 A4 5 @ ... ... ...
0 0 ... 1 12 3 0 m1 ðtÞ C6 7 7 6 0 C C6 m2 ðtÞ 7 C6 7dWðtÞ 7 6 ... C A4 . . . 5
. . . kn
mn ðtÞ
pn ðtÞ
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A.A. Zimbidis / Applied Mathematics and Computation 232 (2014) 685–697
qii ðtÞ ¼ 1 ½qi1 ðtÞ þ qi2 ðtÞ þ þ qi;i1 ðtÞ þ qi;iþ1 ðtÞ þ þ qi;n ðtÞ; 2
p 1 ðtÞ
3
0
1
0
...
0
12
p1 ðtÞ
3
0
7 B C6 7 B 6 6 p ðtÞ 7 B 0 6 p ðtÞ 7 B 0 1 ... 0 C 7 C 7 B 6 B 6 2 2 7¼B C6 7þB p ðtÞ ¼ 6 7 B C6 7 B 6 6 . . . 7 B . . . . . . . . . . . . C6 . . . 7 B . . . 5 @ A4 5 @ 4 p n ðtÞ
0
0
...
1
0
...
f2
...
...
...
0
. . . fn
f1
pn ðtÞ
0
0
12
R1 ðt l1 ðtÞÞ
3
C6 7 7 6 0 C C6 R2 ðt l2 ðtÞÞ 7 C6 7 C6 7 7 6 ... C . . . A4 5 Rn ðt ln ðtÞÞ
4. The general theoretical solution and numerical application 4.1. The theoretical solution As regards the theoretical solution, it is straight-forward when applying Theorem 2.We must only match the notation and symbols accordingly. So, if we assume the following
3 dR1 ðtÞ 7 6 6 dR ðtÞ 7 6 2 7 7; xðtÞ ¼ 6 7 6 6 ... 7 5 4
3 p1 ðtÞ 7 6 6 p ðtÞ 7 6 2 7 7; uðtÞ ¼ 6 7 6 6 ... 7 5 4
2
3 m1 ðtÞ 7 6 6 m ðtÞ 7 6 2 7 7; tðtÞ ¼ 6 7 6 6 ... 7 5 4
2
2
pn ðtÞ
dRn ðtÞ
mn ðtÞ
1
0
C B C B B 0 r2 . . . 0 C C B A¼B C; B... ... ... ...C C B A @ 0 0 . . . rn
B B B B Ad ðtÞ ¼ B B B @
0
r1
0
0
1
0
...
0
...
0
1
C B C B B 0 1 ... 0 C C B BðtÞ ¼ B C; B... ... ... ...C C B A @ 0 0 ... 1 3 p 1 ðtÞ 7 6 6 7 6 p2 ðtÞ 7 7 6 zðtÞ6 7; 6 ... 7 7 6 5 4
pn ðtÞ 2
0
1 0 B B B 0 1 B D¼B B... ... B @ 0 0
0
q11 ðtÞ f1 q12 ðtÞ
0
q21 ðtÞ
...
q1n ðtÞ
q2n ðtÞ
1
1
C C ... 0 C C C; ... ...C C A ... 1
0
...
0
0
C C C A
. . . qnn ðtÞ fn
1
0
k1
C C 0 C C C; ... ... C C A . . . 1
B B B 0 B Et ¼ B B ... B @ 0
0
...
f1 B B B 0 B Cd ¼ B B ... B @ 0
C
C qn2 ðtÞ C C
...
1 . . . ...
1
qn1 ðtÞ
...
q22 ðtÞ f2 . . .
...
B B B 0 B Bt ¼ B B ... B @ 0 ...
AðtÞ ¼ A þ DAðtÞ
0 f2 ... 0
0
0 k2 ... 0
...
0
1
C C 0 C C C ... ... C C A . . . kn ...
1
C C 0 C C C ... ... C C A . . . fn ...
EðtÞ ¼ 0 ; Ed ðtÞ ¼ 0 ; C ¼ 0 ; Nad ¼ 0 ; Nb ¼ 0 ; Ne ¼ 0 ; Ned ¼ 0
DAd ðtÞ ¼ 0;
DBðtÞ ¼ 0;
DEðtÞ ¼ 0;
DEd ðtÞ ¼ 0
DAðtÞ ¼ HFðtÞN a where H; N a are known real constant matrices and F(t) is an unknown matrix function with Lebesgue measurable elements, satisfying the condition F sðtÞ FðtÞ I. The optimal solution is obtained via the application of LMIs (9) and (7). As regards the limitations imposed by the hypotheses described in Theorem 2, we argue the following: (1) The delay occurring in an insurance system is certainly a bounded variable, so there is no problem to accept the inequality, 0 6 sðtÞ 6 l. (2) But the second inequality dtd sðtÞ < h ¼ 1 imposes a certain restriction. If we consider the discrete version of the specific restriction, we may rewrite it as Ds(t) < Dt. That means, the delay should not be enhanced faster than the time elapsed. Otherwise, the system becomes unstable. The detailed solution is calculated in the following subsection for the single dimension.
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4.2. Numerical application for a single dimension (product) Now, we apply the theoretical solution for the single product case and solve our problem, assuming the following xðtÞ ¼ RðtÞ; uðtÞ ¼ pðtÞ; tðtÞ ¼ mðtÞ
AðtÞ ¼ rðtÞ ¼ r þ DrðtÞ; EðtÞ ¼ 0;
Ed ðtÞ ¼ 0;
Nad ¼ 0;
Nb ¼ 0;
Ad ðtÞ ¼ f ; Et ¼ k;
Ne ¼ 0;
BðtÞ ¼ 1;
C ¼ 0;
Bt ðtÞ ¼ 1;
C d ¼ f ;
D ¼ 1;
Ned ¼ 0:
Substituting in matrices (9) and (7) and since our problem has only one dimension (there is no meaning for the symbol of transpose ()s), we obtain, 3 2 2Z þ lW 1 g0 0 0 K XN a 0 0 0 0 X lZ 7 6 6 g0 g ð1 dÞf S 1 0 0 0 0 0 0 0 lY 7 7 6 6 0 ð1 dÞf S ð1 hÞS 0 f S 0 0 0 0 0 0 0 7 7 6 7 6 l 2 7 6 0 1 0 c 0 0 0 0 k k 0 0 1h 7 6 7 6 6 K 0 f S 0 1 0 0 0 0 0 0 0 7 7 6 6 XN a 0 0 0 0 e1 0 0 0 0 0 0 7 7 6 70 6 6 0 0 0 0 0 0 e2 0 0 0 0 0 7 7 6 l 6 0 0 0 0 0 0 0 1h e3 0 0 0 0 7 7 6 7 6 6 0 0 0 k 0 0 0 0 X þ e2 H2 0 0 0 7 7 6 7 6 l l 6 0 0 0 k 1h 0 0 0 0 0 1h ðR e3 H2 Þ 0 0 7 7 6 7 6 0 5 X 0 0 0 0 0 0 0 0 0 S 4 lZ lY 0 0 0 0 0 0 0 0 0 lQ
g ¼ 2! þ e1 H2 þ d2 f 2 R þ lW 3 ; 2
W1
6 4 W2 0
W2 W3
g 0 ¼ Xðr df Þ þ K þ Y Z þ lW 2 ;
3
0
7 df Q 5 0
df Q
Q
Furthermore assuming
A ¼ r;
Et ¼ k;
C d ¼ f ;
Na ¼ w;
l ¼ 1; h ¼ 0:5; c ¼ 1; H ¼ 1; FðtÞ ¼ 1; d ¼ 1: We obtain
2 6 6 6 6 6 6 6 6 6 6 6 6 N¼6 6 6 6 6 6 6 6 6 6 6 6 4
2Z þ W 1
g0
0
0
K
wX
0
0
0
g0
g
0
1
0
0
0
0
0
0
0
0; 5S
0
f S
0
0
0
0
0
1
0
1
0
0
0
0
k 0
K
0
f S
0
1
0
0
0
wX
0
0
0
0
e1
0
0
0
0 0
0 0
0 0
0 0
0 0
0 0
e2 0
0 2e3
0 0 X þ e2
0
0
0
k
0
0
0
0
0
0
0
2k
0
0
0
0
0
X
0
0
0
0
0
0
0
0
Z
Y
0
0
0
0
0
0
0
g ¼ 2! þ e1 þ f 2 R þ W 3 ;
g 0 ¼ Xðr f Þ þ K þ Y Z þ W 2
0
X
0
0
Z
3
7 Y 7 7 0 0 0 7 7 7 2k 0 0 7 7 0 0 0 7 7 7 0 0 0 7 70 0 0 0 7 7 7 0 0 0 7 7 7 0 0 0 7 7 2ðR e3 Þ 0 0 7 7 7 0 S 0 5 0 0 Q
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2
W1
N0 ¼ 6 4 W2 0
W2 W3 f Q
0
3
7 f Q 5 P 0;
X > 0; Y; Z; W 1 ; W 2 ; W 3 ; K; Q > 0; S > 0; R > 0;
e1 > 0; e2 > 0; e3 > 0:
Q
Now, we may transfer our problem assuming m = 13 and n = 15
x ¼ ðx1 ; x2 ; x3 ; . . . ; xm¼13 Þ ¼ ðK; X; Y; Z; Q ; R; S; W 1 ; W 2 ; W 3 ; e1 ; e2 ; e3 Þ; while the 14 matrices with dimensions (15 15) will derived from the following LMI
N O
O 0
N
¼ F 0 þ x1 F 1 þ x2 F 2 þ x3 F 3 þ þ x13 F 13 0:
Table 1 The 14 matrices appearing in the solution of the respective LMI for the simplest case of the model for the portfolio with the single insurance product.
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A.A. Zimbidis / Applied Mathematics and Computation 232 (2014) 685–697
Table 1 (continued)
The solution of the last LMI will also be a solution for our problem. So, we have the following 14 matrices (see Table 1). Furthermore, we assume seven different scenarios as regards the following parameters: (a) the central value for the investment rate r, (b) the uncertainty level for the investment rate w (c) the volatility factor k, while for the feedback factor we run through all the potential range from zero to unity using a step of 0.1 (so actually we run all seven scenarios for f = 0%, 10%, 20%, . . . , 100%). The choice for the first central value of the investment rate r = 3% corresponds to the typical current figures of European Corporate Bonds of high quality (normally quoted as 6A: A, AA and AAA). The other choice of r = 6% corresponds r + 3%, where the additional 3% may coincide with a minimum typical risk premium for running insurance business. The uncertainty level w = 0.04 corresponds to normal statistics of the relevant rates mentioned before over the last decades while the w = 0.08 has been considered in order to test the model over extreme cases. As regards the volatility factor, again the value k = 3 factor corresponds to typical values of volatility while the k = 5 has been chosen as to test the model over the extreme cases. The results are also presented in the following Tables 2 and 3 and Fig. 2.
Table 2 Results for the stabilization control factor K X 1 under different scenarios. Scenarios
f = 0%
f = 10%
f = 20%
f = 30%
f = 40%
w = 0.04 - r = 0.03 - k = 3 w = 0.04 - r = 0.06 - k = 3 w = 0.08 - r = 0.03 - k = 3 w = 0.08 - r = 0.06 - k = 3 w = 0.04 - r = 0.03 - k = 5 w = 0.04 - r = 0.06 - k = 5 w = 0.08 - r = 0.06 - k = 5
52,3% 54,0% 52,5% 54,2% 52,3% 54,0% 54,2%
47,2% 48,7% 47,4% 48,9% 47,1% 48,7% 48,9%
42,7% 44,0% 42,8% 44,2% 42,6% 44,0% 44,2%
38,6% 39,8% 38,7% 40,0% 38,5% 39,8% 40,0%
34,7% 35,9% 34,9% 36,1% No convergence
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Table 3 Results for the stabilization control factor K X under different scenarios. Scenarios
f = 50%
w = 0.04 - r = 0.03 - k = 3 w = 0.04 - r = 0.06 - k = 3 w = 0.08 - r = 0.03 - k = 3 w = 0.08 - r = 0.06 - k = 3 w = 0.04 - r = 0.03 - k = 5 w = 0.04 - r = 0.06 - k = 5 w = 0.08 - r = 0.06 - k = 5
The algorithm for the solution of the respective LMI does not converge (using a set of 1000 iterations)
f = 60%
f = 70%
f = 80%
f = 90%
f = 100%
Fig. 2. Results for the stabilization control factor K X 1 under different scenarios.
As we observe, there is no viable solution when the feedback delay factor f exceeds the critical value 50%. That means, when there is an effort of over-refunding (for the undesired surplus back to the policyholders), the process becomes
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non-stable. Additionally, when the volatility (see the parameter k) or/and the uncertainty level of the investment rate (see the parameter w) are increased then the viable solutions are restricted even more. Under these scenarios (5, 6 and 7) there is no viable solution even for f = 40%. The whole analysis suggests that the initial feedback action should be limited in order to leave enough space for stabilization corrections. That is directly comparable with the results of Zimbidis and Haberman [13] who found similar values for instability level. Finally, an interesting observation is that the summation of the feedback factor f and the stabilization factor K does not exceed 80% (actually the maximum is obtained at 76.1% = 40% + 36,1%). This may be interpreted as: We must always keep a non-refundable portion (at least 20%) of the accumulated surplus in order to secure stabilization of the total process. 5. Conclusions – further research Closing this paper, we present a short resume. The new modeling concept introduced by this research project is the introduction of uncertainty into the insurance pricing process. The framework of uncertainty is further enhanced assuming also some kind of delay. The H-infinity control theory is employed as the basic tool in order to handle the application. Furthermore, H1-control leads to linear matrix inequalities (LMIs) since the basic stability condition relies on a system of LMIs. After constructing a quite general pricing model, we focus to the simplest case that is described in a single dimension. A numerical application is fully investigated by solving the respective LMI using an iterative algorithm (up to 1000 iterations). The numerical results of our application coincide with those of older papers (see [13]) supporting that the feedback factor f (f represents the amount of delayed information integrated into the system) must be restricted below 50% or even lower when considering models with high uncertainty or volatility levels, otherwise the system is not robust stochastically stable. It is also interesting to note the solution for the trivial case where there is no feedback stabilization action and the relevant feedback factor equals zero (f = 0%). Then, the robust stabilization factor K X 1 is slightly higher than 50% (from 52% up to 54%). It also worth noticing (see the last diagram of Fig. 1) that all the solutions are almost parallel and be arranged within a narrow zone-path starting from the interval [52%, 54%] for f = 0% and ending to interval [34%, 36%] for f = 40%. Finally, we should restate the observation with respect to the summation of the feedback and stabilization factors f and K respectively. We found that f + K is less than 80% in all cases. That means we should always keep a certain no-refundable level for the accumulated surplus (in our example this level is almost equal to 20%). Further research is also carried forward. The full numerical solution of the general model (in the multidimensional case) is just a problem of computer programming (certainly a demanding one). An interesting direction of research is the reformulation of the wide range of insurance problems using the tools of H-infinity control and Linear Matrix Inequality theory. References [1] S. Boyd, L. Ghaoui, E. Feron, V. Balakrishan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics, Philadelphia, 1994. [2] W.-H. Chen, Z.-H. Guan, X. Lu, Delay-dependent robust stabilization and H1-control of uncertain stochastic systems with time-varying delay, IMA J. Math. Control Inform. 21 (2004) 345–358. [3] L. Denis, C. Martini, A theoretical framework for the pricing of contingent claims in the presence of model uncertainty, Ann. Appl. Probab. 16 (2006) 827–852. [4] P. Emms, S. Haberman, I. Savoulli, Optimal strategies for pricing general insurance, Insurance: Math. Econ. 40 (2007) 15–34. [5] A.M. Lyapunov, Problème général de la stabilité du movement, Annals of Mathematics Studies, vol. 17, Princeton University Press, Princeton, 1947. [6] A. Martin-Löf, Premium control in an insurance system, an approach using linear control theory, Scandinavian Actuarial J. (1983) 1–27. [7] R. Norberg, Ruin problems with assets and liabilities of diffusion type, Stochastic Process. Appl. 28 (2) (1999) 263–280. [8] R. Orsi, M.A. Rami, J.B. Moore, A finite step projective algorithm for solving linear matrix inequalities, in: Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, USA, 2003, pp. 4979–4985. [9] M.A. Rami, U. Helmke, J.B. Moore, A finite steps algorithm for solving convex feasibility problems, J. Global Optim. 8 (1) (2007) 143–160. [10] M. Vandebroek, J. Dhaene, Optimal premium control in a non-life insurance business, Scandanavian Actuarial J. (1990) 3–13. [11] V.A. Yakubovich, The solution of certain matrix inequalities in automatic control theory, Soviet Math. Dokl. 3 (1962) 620–623 (in Russian, 1961). [12] V.R. Young, Pricing life insurance under stochastic mortality via the instantaneous sharpe ratio, 42, Insurance Math. Econ. 42 (2) (2008) 691–703. [13] A. Zimbidis, S. Haberman, The combined effect of delay and feedback on the insurance pricing process, Insurance Math. Econ. 28 (2) (2001) 263–280. [14] A. Zimbidis, Pricing in a competitive market driven by fractional noise, Variance 5 (1) (2011) 55–67.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Insurance pricing using H1-control Alexandros A. Zimbidis Department of Statistics, Athens University of Economics and Business, 76 Patision St., Athens 104 34, Greece
a r t i c l e
i n f o
Keywords: Insurance pricing H1-control Robust stochastic stabilization Linear matrix inequalities Time varying delayed systems
a b s t r a c t The paper considers a typical insurance system ‘‘suffering’’ from the three standard ‘‘curses’’: (a) the stochastic nature of claims, (b) the inherent delays in claims settlement and reserving process and (c) the uncertainty concept that endows many of its parameters and especially the investment process. We construct a general multidimensional model for pricing simultaneously one, two or more different insurance products. The responsible decision maker uses the incomplete information of claims and aims to balance the system by the means of a feedback mechanism. The robust stabilization controller of the system is obtained by the means of H1-control using typical linear matrix inequalities. Finally, a numerical application is fully investigated providing further insight into the practical problem of pricing assuming the simplest case of a portfolio with a single product. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction The pricing process is one of the basic concerns of an insurance company. The traditional static point of view has already been replaced by the dynamic view using the tools of stochastic control theory. Actually, typical insurance systems have been described by the means of stochastic differential or difference equations while the pricing process has been established as an optimization problem under a certain objective function. In general the ‘‘insurance pricing puzzle’’ is quite complicated and difficult. There are several issues and modelling technicalities such as: the stochastic nature of the parameters, the inherent delay of the available information, the uncertainty of the economic environment and many others. An insurance company starting with an initial capital it continuously receives premiums while pays claims at specific time points and always targets to a positive (but not huge) balance in any future time. Past research papers have investigated the pricing problem of an insurance system allowing for stochastic variables or delays or other technicalities. Here, we mention only a few of these research efforts as an indicative guide to the huge problem of pricing. Martin-Löf [6] was amongst the pioneers for using the tools of control theory in the insurance pricing context. Vandebroek and Dhaene [10] apply the control theory techniques in the non-life insurance business. Norberg [7], using the standard Brownian motion as the basic tool for modelling, formulates the problem of pricing and reserving within a continuous time framework. His results coincide with the typical results of the traditional approach from risk theory. Zimbidis and Haberman [13] examine the combined effect of delay and feedback on the pricing process. Emms et al. [4] explore the optimal premium strategies within a competitive insurance market using the tools of a standard Brownian motion. Young [12] investigates the pricing of life contracts when mortality is modelled stochastically via the instantaneous Sharpe ratio. Zimbidis [14] further develops the concept of competitive insurance markets using the fractional Brownian motion as the modelling tool for the claim process. Finally, we provide an interesting reference for the modeling of uncertainty in the financial setting. Denis and Martini [3] introduce the uncertainty concept in the financial pricing of a contingent claim. They E-mail address: [email protected] http://dx.doi.org/10.1016/j.amc.2014.01.105 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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incorporate uncertainty via a family of martingale measures and calculate the superreplication price of a European contingent claim. In this paper, we investigate the pricing problem considering all the three complications mentioned before: Stochastic nature, delay and uncertainty. The new modelling concept is the uncertainty. We solve the complications by using the tools of H1-control. Additionally, we make an allowance for pricing a mixed portfolio of different insurance products. We start from the very basics, where we consider an insurance company with an initial reserve, an incoming premium flow plus incoming investment proceeds while also an additional outcome of claim payments. The pricing rule (premium determination) is based on past claim experience and over an initial balancing feedback mechanism evaluating the total accumulated surplus (refunding a certain portion of surplus back to the policyholders by reducing the initial level of premium). Now, although the typical description mentioned above indicates premium as input and claims as output variables, in the typical system design we should consider (the vice versa situation) claims as input variable while premiums as output variable. Then, our target is to simultaneously stabilize the premium level charged to the policyholders while also stabilize the state variable: the accumulated surplus fund of the insurance company. The stabilization may be obtained by the attachment of an additional feedback mechanism. That may be described with equations and Fig. 1 below. [Surplus Fund at time t] = [Initial Fund of the Company at time 0] + [Investment Proceeds up to time t] + [Premiums received up to time t] [Claims paid up to time t]. [Premium received at time t] = [Average of paid claims per time-unit up to time t] [% of the Surplus Fund at time t]. The paper is organised as follows: Section 2 contains a short introductory guide to (H infinity) H1-control and linear matrix inequalities (LMI’s). Section 3 describes the basic structure for the pricing process and the set of required assumptions. Furthermore, we analyze the problem in the multidimensional case allowing for pricing two or more different insurance products. Section 4 presents the theoretical solution of the model for the general multidimensional case while provides a detailed numerical application and practical considerations as regards the insurance pricing for the simplest case of a portfolio with a single product. Finally, Section 5 concludes the paper. 2. H‘-control and linear matrix inequalities Control theory and especially optimal control theory has played an important role in many scientific areas and practical problems over the past decades. The last two or three decades, control theory has been fully applied to actuarial problems. A new direction of research for control theory the very last years is the H1-control. Actually, H1-control is optimal control design when considering the worst exogenous input for a closed loop system. So, H1-control offers an ideal framework to investigate problems under uncertain (but somehow bounded) parameters and conditions. Below, we provide a short note on the relevant theory as regards the H1-control (see [2] for more details) and linear matrix inequalities (LMIs) (see [1] for more details) that is the powerful tool for solving the respective stability problems. Before going further, we formalize the notation for the matrices i.e. Let A represents a matrix, then A ð P; 6; Þ 0 denotes that A is symmetric positive definite (symmetric positive semi-definite, symmetric negative definite, symmetric negative semi-definite). 2.1. H1-control (or H infinity control) We assume an uncertain linear stochastic delayed and controlled differential system,
dxðtÞ ¼ ðAðtÞxðtÞ þ Ad ðtÞxðt sðtÞÞ þ BðtÞuðtÞ þ Bt tðtÞÞdt þ ðEðtÞxðtÞ þ Ed ðtÞxðt sðtÞÞ þ Et tðtÞÞdWðtÞ t P 0;
ð1Þ
zðtÞ ¼ CxðtÞ þ C d xðt sðtÞÞ þ DuðtÞ t P 0;
ð2Þ
zðtÞ ¼ /ðtÞ t 2 ½l; 0;
ð3Þ
where xðtÞ 2 Rn is the state variable,
Fig. 1. Control system design of an insurance system.
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687
uðtÞ 2 Rm is the controlled input variable, mðtÞ 2 Rp is the disturbance input variable (zero or positive valued & square integrable), zðtÞ 2 Rq is the controlled output variable, s(t) is time-varying bounded delay time satisfying the following conditions
d sðtÞ 6 h < 1 dt
0 6 sðtÞ 6 l
/(t) is any given initial data in the Rn -valued family F0 measurable stochastic process f(s) with supl6s60 EjfðsÞj2 < 1, where E[. . .] stands for the expectation operator with respect to the given probability measure P. W = {W(t); t P 0} is a scalar Brownian motion defined on a complete probability space (X, F, P) with a natural filtration fFt gtP0 . Bt, Et C, Cd, D are known constant matrices A(t), Ad(t), B(t), E(t), Ed(t) are matrix-valued functions with time-varying uncertainties as
AðtÞ ¼ A þ DAðtÞ; Ad ðtÞ ¼ Ad þ DAd ðtÞ; BðtÞ ¼ B þ DBðtÞ; EðtÞ ¼ E þ DEðtÞ; Ed ðtÞ ¼ Ed þ DEd ðtÞ where A, Ad, B, E and Ed are known real constant matrices while DAðtÞ; DAd ðtÞ; DBðtÞ; DEðtÞ and DEd ðtÞ are unknown matrices representing time-varying parameters uncertainties. These uncertainties are norm-bounded and may be described as follows
½DAðtÞ DAd ðtÞ DBðtÞ DEðtÞ DEd ðtÞ ¼ HFðtÞ½Na Nad Nb Ne Ned where H; N a ; N ad ; N b ; N e and N ed are known real constant matrices and F(t) is an unknown matrix function with Lebesgue measurable elements, satisfying the condition
F s ðtÞ FðtÞ I; where F s ðtÞ stands for the transpose matrix of FðtÞ. Furthermore, we provide the formal definitions for robust stability and robust performance for the system (1)-(3). Definition 1. The system (1)–(3) with u(t) = 0, t(t) = 0 is said to be robust stochastically stable if there exists a positive constant q such that
lim E
Z
T!1
T
xs ðtÞ xðtÞ dt < q sup EjfðsÞj2
ð4Þ
l6s60
0
for all admissible uncertainties DAðtÞ; DAd ðtÞ; DEðtÞ and DEd ðtÞ Definition 2. Given a scalar c > 0, the unforced stochastic system (1)–(3) with u(t) = 0 is said to be robust stochastically stable with disturbance attenuation c if it is robust stochastically stable in the sense of definition 1 and under zero initial conditions,
E½
Z
1
zs ðtÞ zðtÞ dt < c2
0
Z
1
W s ðtÞ WðtÞ dt
ð5Þ
0
Below, we provide two basic theorems that present the solutions to the robust stabilization problem for system (1)-(3). The first theorem corresponds to the special case of a zero disturbance input variable, t(t) = 0. The second theorem solves the general format of the problem using the validity of Theorem 1. Both detailed proofs (for Theorems 1 and 2) are described in Chen et al. [2]. So, we have the following Theorems 1 and 2: Theorem 1. Consider the system (1)-(3) with t(t) = 0. Then for given scalars l > 0, h < 1, this system is robust stochastically d stabilizable for any time-delay s(t) satisfying 0 6 sðtÞ 6 l; dt sðtÞ 6 h if for some prescribed scalar d, there exist matrices X 0; Y; Z; W 1 ; W 2 ; W 3 ; K; Q 0; S 0; R 0 and scalars e1 > 0; e2 > 0; e3 > 0 such that the following linear matrix inequalities (LMIs) (6) and (7) hold:
2
X1 6 Xs 6 2
6 6 0 6 6 s 6X 6 6 lZ 6 6 s 4 L11 s
L21
X2 X3
L11
0
lZ s lY s
0
X
ð1 dÞAd S
0
ð1 dÞSs As
ð1 hÞS
0
0
L12
0 lY
0 0
S 0
0 lQ
0 0
0
Ls12
0
0
U1
0
Ls22
0
0
0
d
L21
3
0 7 7 7 L22 7 7 7 0 7 0 7 0 7 7 7 0 5
U2
ð6Þ
688
A.A. Zimbidis / Applied Mathematics and Computation 232 (2014) 685–697
2
W1
W2
6 s 4 W2
7 dAd Q 5 0
W3 d Q s As
0
3
0
ð7Þ
Q
d
(Note: some of the ZEROS in (6) and (7) correspond to blocks of zeros),
where
X1 ¼ Z þ Z s þ lW 1 ; X2 ¼ XðAs þ dAsd Þ þ K s Bs þ Y Z s þ lW 2 ; X3 ¼ Y Y s þ e1 HHs þ d2 Ad R Asd þ lW 3 ; n
U1 ¼ diag e1 I; e2 I;
o
l 1h
n
U2 ¼ diag X e2 HHs ;
e3 I ;
l 1h
o ðR e3 HHs Þ ;
h
L11 ¼ XN a s þ KNb s XN e s h
l
L12 ¼ SNad s SNed s h
L21 ¼ XEs
l 1h
h
L22 ¼ SEsd
l 1h
1h
l 1h
i XN e s ;
i SNed s ;
i XEs ; SEsd
i
and the stabilizing control law is described as
uðtÞ ¼ K X 1 xðtÞ:
ð8Þ
Theorem 2. Consider the system (1)-(3). Then for given scalars l > 0, h < 1, this system is robust stochastically stabilizable with disturbance attenuation c > 0 for any time-delay s(t) satisfying 0 6 sðtÞ 6 l; dtd sðtÞ 6 h if for some prescribed scalar d, there exist matrices X 0; Y; Z; W 1 ; W 2 ; W 3 ; K; Q 0; S 0; R 0 and scalars e1 > 0; e2 > 0; e3 > 0 such that LMI (7) and the following LMI holds:
2
X1 Xs2
6 6 6 6 0 6 6 6 0 6 6 s 6 X C þ DK s 6 6 Ls11 6 6 6 Ls21 6 6 Xs 4
lZ
X2 X3
0
0
XC s þ KDs
L11
L21
X
ð1 dÞAd S
Bt
0
0
0
0
L12
L22
0
3
lZ s 7 lY s 7 7
ð1 dÞSs As
ð1 hÞS
0
Bst
0
c2 I
0
0
L23
0
0
0
I
0
0
0
0
d
0
Cd
Ss
s
SC d
0
0
Ls12
0
0
U1
0
0
0
0
Ls22
Ls23
0
0
U2
0
0
0
0
0
0
0
0
S
0
lY s
0
0
0
0
0
0
lQ
7 7 7 7 7 7 7 0 7 7 7 7 7 7 7 5
ð9Þ
(some of the ZEROS in (9) correspond to blocks of zeros), where X1, X2, X3, U1, U2, L11, L12, L21, L22, are defined in previous l s Theorem 1 and L23 ¼ Est 1h Et . Then, the stabilizing control law is described as
uðtÞ ¼ K X 1 xðtÞ: 2.2. Linear matrix inequalities (LMI’s) The general format of a typical Linear Matrix Inequality is the following
ð10Þ
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A.A. Zimbidis / Applied Mathematics and Computation 232 (2014) 685–697 m X xi F i 0;
FðxÞ ¼ F 0 þ
ð11Þ
i¼1
where x ¼ ðx1 ; x2 ; x3 ; . . . ; xm Þ 2 Rm is the variable to be determined, while F i ¼ F si 2 Rnxn i ¼ 0; 1; 2; . . . ; m, additionally, we can state that the LMI above may be easily transformed to a set of n inequalities in x. The relationship of LMIs and dynamic systems has been early recognised by Lyapunov [5]. He demonstrated that the stability of the basic differential equation
d xðtÞ ¼ A xðtÞ dt is obtained when there is a matrix P such that the following conditions hold
P 0 and As P þ PA 0: Now, we may easily transform the conditions above into the standard format of a LMI using a matrix of matrices as below
P
0
0 ðAs P þ PAÞ
0:
Furthermore, Yakubovich [11] was the first who formally established the importance of the LMI’s in the solution of the control problems. The solution for LMI’s has been initially based on the ellipsoid algorithm and on the interior-point methods. These two basic approaches have been further exploited over the last years and also inspire other similar ones. In this paper, we use a quite recent algorithm proposed by Orsi et al. [8] and Rami et al. [9] that is based on the classical alternating projection method. Actually, they use this algorithm to solve ‘‘convex feasibility problems where the constraints are given by the intersection of two convex cones in a Hilbert space’’. Then, as an application, they derive the solution of an LMI problem calculating a sequence of matrix eigenvalue–eigenvector decompositions. Below, we provide a short description of the relevant algorithm that is fully described in the papers mentioned above (and especially in Orsi et al. [8, p. 4982]), aiming to find x 2 Rm satisfying a strict LMI (11) (or the non-strict version where the symbol ‘‘’’ is replaced by ‘‘’’). Before provide the formal description of the algorithm, we establish two special functionals ‘‘_ ’’ and ‘‘^ ’’ from Sn the set of real symmetric matrices to Rp and vice versa, where p ¼ nðnþ1Þ , as below: 2
0
s11
B B s12 B S¼B B s13 B @ s1n
s12
s13
s22
s23
s23
s33
s2n
s3n
s1n
1
C s2n C _ C s s3n C C ! S ¼ ðs11 ; s12 ; s13 ; ; s1n ; s22 ; s23 ; ; s2n ; s33 ; ; s3n ; :; snn Þ C A snn 0
s11 B B s12 ^ B S ¼ ðs11 ; s12 ; s13 ; . . . ; s1n ; s22 ; s23 ; . . . ; s2n ; s33 ; . . . ; s3n ; . . . ; snn Þs ! S ¼ B B s13 B @
s12
s13
s22
s23
s23
s33
s1n
s2n
s3n
s1n
C s2n C C s3n C C C A snn
2.3. Algorithm for solving the strict LMI (11) Data: A set of m + 1 symmetric matrices F i ¼ F si 2 Rnxn i ¼ 0; 1; 2; . . . ; m Initialization: Choose initial conditions and calculate the basic matrix G
-1
, as below
Choose values for the parameters q, t such that q > 0 and t 2 (0, 2) (e.g. q = 1 & t = 1,99). Choose any x0 2 R and x 2 Rm . Choose any real symmetric matrix S 2 Rnxn (that is a slack variable). Define matrix Q as Q :¼ ½F^0 ; F^1 ; . . . ; F^m 2 Rp x ðmþ1Þ ; p ¼ nðnþ1Þ . 2 Calculate matrix G :¼ Q Q s þ 12 diagð^I þ 1Þ, where I is the n n identity matrix and 1 2 Rp while 1 ¼ ð1; 1; . . . ; 1Þs . (vi) Calculate the inverse matrix G1.
(i) (ii) (iii) (iv) (v)
1
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Then we may start the algorithm described with the following steps Step 1: Calculate and replace via the formula x0 :¼ ð1 tÞ x0 þ t maxfq; x0 g. Step 2: Find an eigenvalue–eigenvector decomposition of S. Let S ¼ V D V s with D ¼ diagðd1 ; d2 ; . . . ; dn Þ the respective matrix of eigenvalues while V the respective matrix of eigenvectors. Step 3: Define a new matrix D ¼ diagðmaxfq; d1 g ; maxfq; d2 g ; . . . ; maxfq; dn g Þ Step 4: Calculate and replace matrix S via the formula S ¼ V ðð1 tÞD þ tDÞ V s . Step 5: Calculate a :¼ G1 ðQ ½x0 ; xs s ^ SÞ. Step 6: Calculate and replace x0 and x via the formula ½x0 ; xs s :¼ ½x0 ; xs s Q s a. ^ Step 7: Define T :¼ 0; 5 a . Step 8: Redefine matrix T by duplicating all the elements of the first diagonal. Step 9: Replace matrix S via the formula S :¼ S T Step 10: If x0 > 0 and the minimum eigenvalue of S is greater than zero then the solution is x=x0 otherwise we return to step 1 and continue the algorithm. 3. The pricing process 3.1. Portfolio with a single product – one dimension We consider a typical insurance system with an initial reserve, incoming variable premiums determined by a feedback mechanism according to past experience and claims driven by a drifted Brownian motion. The reserve is invested in a safe asset with a variable yet uncertain force of interest. Then, the reserve of the company obeys the following stochastic differential equation.
dRðtÞ ¼ ½Reserve at time ðt þ dtÞ ½Reserve at time ðtÞ ¼ ½Investment income earned in ðt; t þ dtÞ þ ½Premiums in ðt; t þ dtÞ ½Claims in ðt; t þ dtÞ ⁄ (Claims also include the element of administration expenses) or using standard notation the reserve obeys the following relationship
dRðtÞ ¼ rðtÞRðtÞdt þ p ðtÞdt ½mðtÞdt þ rðtÞdWðtÞ;
ð12Þ
while the premium is established via the following feedback mechanism
p ðtÞ ¼ pðtÞ f Rðt lðtÞÞ; where R(t): is the reserve at time t, rðtÞ : is the force of interest at time t, p ðtÞ: actual premium rate at time t, p(t): standard premium rate at time t, f: profit sharing factor, lðtÞ: time delay for claims settlement at time t, mðtÞ: average claim rate at time t, rðtÞ: volatility claim rate at time t, further assume that WðtÞ: standard Brownian motion.
ð13Þ
rðtÞ ¼ k mðtÞ
Combining equations (12) and (13) and after some algebra we finally obtain the system
dRðtÞ ¼ ½rðtÞRðtÞ f Rðt lðtÞÞ þ pðtÞ mðtÞdt þ ½kmðtÞdWðtÞ;
ð14Þ
p ðtÞ ¼ pðtÞ f Rðt lðtÞÞ:
ð15Þ
Following the reasoning of Zimbidis and Haberman [13], parameter (f) – the feedback factor – normally lies in the interval of [0, 1]. The practical justification is more or less obvious. An insurance company should be prepared to refund a smaller or larger portion of the accumulated surplus but certainly reluctant to exceed the 100%. Furthermore, a negative feedback factor corresponds to unreasonable and unnecessary premium loading. So, f normally lies in [0, 1]. Additionally, supporting Eq. (13) we also assume that there is no distribution of surplus to shareholders or anyone else (e.g. the tax authorities) except to the policyholders.
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3.2. Portfolio with two products – two dimensions Now, we consider a typical insurance system with two products. The extra complication is the interaction (incoming or outgoing reserve portion) between the two reserves. Keeping unchanged the other characteristics of the model, we may identify that the individual reserve of each company obeys the following stochastic differential equation dR(t) = [Reserve at time (t + dt)] [Reserve at time (t)] = [profit sharing coming from the other product in (t, t + dt)] [profit sharing given to the other product in (t, t + dt)] + [Investment income earned in (t,t + dt)] + [Premiums in (t, t + dt)] [Claims in (t, t + dt)] or extending the previous standard notation we obtain
dR1 ðtÞ ¼ ½r1 ðtÞR1 ðtÞ þ q21 ðtÞR2 ðt l2 ðtÞÞ q12 ðtÞR1 ðt l1 ðtÞÞ f1 R1 ðt l1 ðtÞÞ þ p1 ðtÞ m1 ðtÞdt þ ½k1 ðtÞm1 ðtÞdWðtÞ; p 1 ðtÞ ¼ p1 ðtÞ f1 R1 ðt l1 ðtÞÞ dR2 ðtÞ ¼ ½r2 ðtÞR2 ðtÞ þ q12 ðtÞR1 ðt l1 ðtÞÞ q21 ðtÞR2 ðt l2 ðtÞÞ f2 R2 ðt l2 ðtÞÞ þ p2 ðtÞ m2 ðtÞdt þ ½k2 ðtÞm2 ðtÞdWðtÞ; p 2 ðtÞ ¼ p2 ðtÞ f2 R2 ðt l2 ðtÞÞ; where the subscript 1 and 2 correspond to the first and second product respectively while q12 ðtÞ: proportion of reserve transferred from the first to the second product at time t, q21 ðtÞ: proportion of reserve transferred from the second to the first product at time t, Using the vector representation we obtain the following vector equations
" dR ðtÞ ¼ " þ
p ðtÞ ¼
#
dR1 ðtÞ
¼
dR2 ðtÞ 1
0
0
1
p 1 ðtÞ p 2 ðtÞ
("
#"
0
0 #)
r 2 ðtÞ ("
m1 ðtÞ
dt þ
m2 ðtÞ
¼
#"
r 1 ðtÞ
1 0 0 1
p1 ðtÞ p2 ðtÞ
#
" þ
R2 ðtÞ k1
0
0
k2
þ
R1 ðtÞ
f1 0
#"
0 f2
#"
q12 ðtÞ f1
q21 ðtÞ
q12 ðtÞ
q21 ðtÞ f2
m1 ðtÞ m2 ðtÞ
#)
R1 ðt l1 ðtÞÞ
#
R2 ðt l2 ðtÞÞ
" þ
1 0 0 1
#"
p1 ðtÞ
#
p2 ðtÞ
dWðtÞ;
R1 ðt l1 ðtÞÞ : R2 ðt l2 ðtÞÞ
3.3. Portfolio with n-products – multidimensional case Now, we may easily extend the model to a mixed portfolio with n products – dimensions. The respective vector representation of the general multidimensional model is as follows
80 3 2 r 1 ðtÞ 0 ... 0 1 dR1 ðtÞ > > > 7 >B 7 6 C6 > 6 dR ðtÞ 7 > 6 dR ðtÞ 7 r 2 ðtÞ . . . 0 C 7 6 2 7 B 7 C B 6 ... 7 > ... . . . . . . . . . A6 . . . 7 > @ 5 > 5 4 4 > > : 0 0 . . . r n ðtÞ dRn ðtÞ dRn ðtÞ 12 3 0 q11 ðtÞ f1 q21 ðtÞ ... qn1 ðtÞ R1 ðt l1 ðtÞÞ C6 7 B 7 B q ðtÞ 6 q22 ðtÞ f2 . . . qn2 ðtÞ C C6 R2 ðt l2 ðtÞÞ 7 B 12 C6 7 þB C6 7 B C6 7 B ... ... ... ... ... A4 5 @ q1n ðtÞ q2n ðtÞ . . . qnn ðtÞ fn Rn ðt ln ðtÞÞ 12 39 0 0 1 0 . . . 0 k1 m1 ðtÞ > 0 > > > C6 7> B B B 0 1 . . . 0 C6 m2 ðtÞ 7= B 0 k2 C6 7 B B þB þ fB C6 7 dt B . . . . . . . . . . . . C6 :::::::::::: 7> B ... ... > A4 5> @ @ > > ; 0 0 . . . 1 mn ðtÞ 0 0 2
dR1 ðtÞ
3
0 1 0 . . . 0 12 p1 ðtÞ 3 C6 7 B 7 B 0 6 1 ... 0 C C6 p2 ðtÞ 7 B þB C6 7 B . . . . . . . . . . . . C6 . . . 7 A4 5 @ ... ... ...
0 0 ... 1 12 3 0 m1 ðtÞ C6 7 7 6 0 C C6 m2 ðtÞ 7 C6 7dWðtÞ 7 6 ... C A4 . . . 5
. . . kn
mn ðtÞ
pn ðtÞ
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qii ðtÞ ¼ 1 ½qi1 ðtÞ þ qi2 ðtÞ þ þ qi;i1 ðtÞ þ qi;iþ1 ðtÞ þ þ qi;n ðtÞ; 2
p 1 ðtÞ
3
0
1
0
...
0
12
p1 ðtÞ
3
0
7 B C6 7 B 6 6 p ðtÞ 7 B 0 6 p ðtÞ 7 B 0 1 ... 0 C 7 C 7 B 6 B 6 2 2 7¼B C6 7þB p ðtÞ ¼ 6 7 B C6 7 B 6 6 . . . 7 B . . . . . . . . . . . . C6 . . . 7 B . . . 5 @ A4 5 @ 4 p n ðtÞ
0
0
...
1
0
...
f2
...
...
...
0
. . . fn
f1
pn ðtÞ
0
0
12
R1 ðt l1 ðtÞÞ
3
C6 7 7 6 0 C C6 R2 ðt l2 ðtÞÞ 7 C6 7 C6 7 7 6 ... C . . . A4 5 Rn ðt ln ðtÞÞ
4. The general theoretical solution and numerical application 4.1. The theoretical solution As regards the theoretical solution, it is straight-forward when applying Theorem 2.We must only match the notation and symbols accordingly. So, if we assume the following
3 dR1 ðtÞ 7 6 6 dR ðtÞ 7 6 2 7 7; xðtÞ ¼ 6 7 6 6 ... 7 5 4
3 p1 ðtÞ 7 6 6 p ðtÞ 7 6 2 7 7; uðtÞ ¼ 6 7 6 6 ... 7 5 4
2
3 m1 ðtÞ 7 6 6 m ðtÞ 7 6 2 7 7; tðtÞ ¼ 6 7 6 6 ... 7 5 4
2
2
pn ðtÞ
dRn ðtÞ
mn ðtÞ
1
0
C B C B B 0 r2 . . . 0 C C B A¼B C; B... ... ... ...C C B A @ 0 0 . . . rn
B B B B Ad ðtÞ ¼ B B B @
0
r1
0
0
1
0
...
0
...
0
1
C B C B B 0 1 ... 0 C C B BðtÞ ¼ B C; B... ... ... ...C C B A @ 0 0 ... 1 3 p 1 ðtÞ 7 6 6 7 6 p2 ðtÞ 7 7 6 zðtÞ6 7; 6 ... 7 7 6 5 4
pn ðtÞ 2
0
1 0 B B B 0 1 B D¼B B... ... B @ 0 0
0
q11 ðtÞ f1 q12 ðtÞ
0
q21 ðtÞ
...
q1n ðtÞ
q2n ðtÞ
1
1
C C ... 0 C C C; ... ...C C A ... 1
0
...
0
0
C C C A
. . . qnn ðtÞ fn
1
0
k1
C C 0 C C C; ... ... C C A . . . 1
B B B 0 B Et ¼ B B ... B @ 0
0
...
f1 B B B 0 B Cd ¼ B B ... B @ 0
C
C qn2 ðtÞ C C
...
1 . . . ...
1
qn1 ðtÞ
...
q22 ðtÞ f2 . . .
...
B B B 0 B Bt ¼ B B ... B @ 0 ...
AðtÞ ¼ A þ DAðtÞ
0 f2 ... 0
0
0 k2 ... 0
...
0
1
C C 0 C C C ... ... C C A . . . kn ...
1
C C 0 C C C ... ... C C A . . . fn ...
EðtÞ ¼ 0 ; Ed ðtÞ ¼ 0 ; C ¼ 0 ; Nad ¼ 0 ; Nb ¼ 0 ; Ne ¼ 0 ; Ned ¼ 0
DAd ðtÞ ¼ 0;
DBðtÞ ¼ 0;
DEðtÞ ¼ 0;
DEd ðtÞ ¼ 0
DAðtÞ ¼ HFðtÞN a where H; N a are known real constant matrices and F(t) is an unknown matrix function with Lebesgue measurable elements, satisfying the condition F sðtÞ FðtÞ I. The optimal solution is obtained via the application of LMIs (9) and (7). As regards the limitations imposed by the hypotheses described in Theorem 2, we argue the following: (1) The delay occurring in an insurance system is certainly a bounded variable, so there is no problem to accept the inequality, 0 6 sðtÞ 6 l. (2) But the second inequality dtd sðtÞ < h ¼ 1 imposes a certain restriction. If we consider the discrete version of the specific restriction, we may rewrite it as Ds(t) < Dt. That means, the delay should not be enhanced faster than the time elapsed. Otherwise, the system becomes unstable. The detailed solution is calculated in the following subsection for the single dimension.
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4.2. Numerical application for a single dimension (product) Now, we apply the theoretical solution for the single product case and solve our problem, assuming the following xðtÞ ¼ RðtÞ; uðtÞ ¼ pðtÞ; tðtÞ ¼ mðtÞ
AðtÞ ¼ rðtÞ ¼ r þ DrðtÞ; EðtÞ ¼ 0;
Ed ðtÞ ¼ 0;
Nad ¼ 0;
Nb ¼ 0;
Ad ðtÞ ¼ f ; Et ¼ k;
Ne ¼ 0;
BðtÞ ¼ 1;
C ¼ 0;
Bt ðtÞ ¼ 1;
C d ¼ f ;
D ¼ 1;
Ned ¼ 0:
Substituting in matrices (9) and (7) and since our problem has only one dimension (there is no meaning for the symbol of transpose ()s), we obtain, 3 2 2Z þ lW 1 g0 0 0 K XN a 0 0 0 0 X lZ 7 6 6 g0 g ð1 dÞf S 1 0 0 0 0 0 0 0 lY 7 7 6 6 0 ð1 dÞf S ð1 hÞS 0 f S 0 0 0 0 0 0 0 7 7 6 7 6 l 2 7 6 0 1 0 c 0 0 0 0 k k 0 0 1h 7 6 7 6 6 K 0 f S 0 1 0 0 0 0 0 0 0 7 7 6 6 XN a 0 0 0 0 e1 0 0 0 0 0 0 7 7 6 70 6 6 0 0 0 0 0 0 e2 0 0 0 0 0 7 7 6 l 6 0 0 0 0 0 0 0 1h e3 0 0 0 0 7 7 6 7 6 6 0 0 0 k 0 0 0 0 X þ e2 H2 0 0 0 7 7 6 7 6 l l 6 0 0 0 k 1h 0 0 0 0 0 1h ðR e3 H2 Þ 0 0 7 7 6 7 6 0 5 X 0 0 0 0 0 0 0 0 0 S 4 lZ lY 0 0 0 0 0 0 0 0 0 lQ
g ¼ 2! þ e1 H2 þ d2 f 2 R þ lW 3 ; 2
W1
6 4 W2 0
W2 W3
g 0 ¼ Xðr df Þ þ K þ Y Z þ lW 2 ;
3
0
7 df Q 5 0
df Q
Q
Furthermore assuming
A ¼ r;
Et ¼ k;
C d ¼ f ;
Na ¼ w;
l ¼ 1; h ¼ 0:5; c ¼ 1; H ¼ 1; FðtÞ ¼ 1; d ¼ 1: We obtain
2 6 6 6 6 6 6 6 6 6 6 6 6 N¼6 6 6 6 6 6 6 6 6 6 6 6 4
2Z þ W 1
g0
0
0
K
wX
0
0
0
g0
g
0
1
0
0
0
0
0
0
0
0; 5S
0
f S
0
0
0
0
0
1
0
1
0
0
0
0
k 0
K
0
f S
0
1
0
0
0
wX
0
0
0
0
e1
0
0
0
0 0
0 0
0 0
0 0
0 0
0 0
e2 0
0 2e3
0 0 X þ e2
0
0
0
k
0
0
0
0
0
0
0
2k
0
0
0
0
0
X
0
0
0
0
0
0
0
0
Z
Y
0
0
0
0
0
0
0
g ¼ 2! þ e1 þ f 2 R þ W 3 ;
g 0 ¼ Xðr f Þ þ K þ Y Z þ W 2
0
X
0
0
Z
3
7 Y 7 7 0 0 0 7 7 7 2k 0 0 7 7 0 0 0 7 7 7 0 0 0 7 70 0 0 0 7 7 7 0 0 0 7 7 7 0 0 0 7 7 2ðR e3 Þ 0 0 7 7 7 0 S 0 5 0 0 Q
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2
W1
N0 ¼ 6 4 W2 0
W2 W3 f Q
0
3
7 f Q 5 P 0;
X > 0; Y; Z; W 1 ; W 2 ; W 3 ; K; Q > 0; S > 0; R > 0;
e1 > 0; e2 > 0; e3 > 0:
Q
Now, we may transfer our problem assuming m = 13 and n = 15
x ¼ ðx1 ; x2 ; x3 ; . . . ; xm¼13 Þ ¼ ðK; X; Y; Z; Q ; R; S; W 1 ; W 2 ; W 3 ; e1 ; e2 ; e3 Þ; while the 14 matrices with dimensions (15 15) will derived from the following LMI
N O
O 0
N
¼ F 0 þ x1 F 1 þ x2 F 2 þ x3 F 3 þ þ x13 F 13 0:
Table 1 The 14 matrices appearing in the solution of the respective LMI for the simplest case of the model for the portfolio with the single insurance product.
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Table 1 (continued)
The solution of the last LMI will also be a solution for our problem. So, we have the following 14 matrices (see Table 1). Furthermore, we assume seven different scenarios as regards the following parameters: (a) the central value for the investment rate r, (b) the uncertainty level for the investment rate w (c) the volatility factor k, while for the feedback factor we run through all the potential range from zero to unity using a step of 0.1 (so actually we run all seven scenarios for f = 0%, 10%, 20%, . . . , 100%). The choice for the first central value of the investment rate r = 3% corresponds to the typical current figures of European Corporate Bonds of high quality (normally quoted as 6A: A, AA and AAA). The other choice of r = 6% corresponds r + 3%, where the additional 3% may coincide with a minimum typical risk premium for running insurance business. The uncertainty level w = 0.04 corresponds to normal statistics of the relevant rates mentioned before over the last decades while the w = 0.08 has been considered in order to test the model over extreme cases. As regards the volatility factor, again the value k = 3 factor corresponds to typical values of volatility while the k = 5 has been chosen as to test the model over the extreme cases. The results are also presented in the following Tables 2 and 3 and Fig. 2.
Table 2 Results for the stabilization control factor K X 1 under different scenarios. Scenarios
f = 0%
f = 10%
f = 20%
f = 30%
f = 40%
w = 0.04 - r = 0.03 - k = 3 w = 0.04 - r = 0.06 - k = 3 w = 0.08 - r = 0.03 - k = 3 w = 0.08 - r = 0.06 - k = 3 w = 0.04 - r = 0.03 - k = 5 w = 0.04 - r = 0.06 - k = 5 w = 0.08 - r = 0.06 - k = 5
52,3% 54,0% 52,5% 54,2% 52,3% 54,0% 54,2%
47,2% 48,7% 47,4% 48,9% 47,1% 48,7% 48,9%
42,7% 44,0% 42,8% 44,2% 42,6% 44,0% 44,2%
38,6% 39,8% 38,7% 40,0% 38,5% 39,8% 40,0%
34,7% 35,9% 34,9% 36,1% No convergence
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Table 3 Results for the stabilization control factor K X under different scenarios. Scenarios
f = 50%
w = 0.04 - r = 0.03 - k = 3 w = 0.04 - r = 0.06 - k = 3 w = 0.08 - r = 0.03 - k = 3 w = 0.08 - r = 0.06 - k = 3 w = 0.04 - r = 0.03 - k = 5 w = 0.04 - r = 0.06 - k = 5 w = 0.08 - r = 0.06 - k = 5
The algorithm for the solution of the respective LMI does not converge (using a set of 1000 iterations)
f = 60%
f = 70%
f = 80%
f = 90%
f = 100%
Fig. 2. Results for the stabilization control factor K X 1 under different scenarios.
As we observe, there is no viable solution when the feedback delay factor f exceeds the critical value 50%. That means, when there is an effort of over-refunding (for the undesired surplus back to the policyholders), the process becomes
A.A. Zimbidis / Applied Mathematics and Computation 232 (2014) 685–697
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non-stable. Additionally, when the volatility (see the parameter k) or/and the uncertainty level of the investment rate (see the parameter w) are increased then the viable solutions are restricted even more. Under these scenarios (5, 6 and 7) there is no viable solution even for f = 40%. The whole analysis suggests that the initial feedback action should be limited in order to leave enough space for stabilization corrections. That is directly comparable with the results of Zimbidis and Haberman [13] who found similar values for instability level. Finally, an interesting observation is that the summation of the feedback factor f and the stabilization factor K does not exceed 80% (actually the maximum is obtained at 76.1% = 40% + 36,1%). This may be interpreted as: We must always keep a non-refundable portion (at least 20%) of the accumulated surplus in order to secure stabilization of the total process. 5. Conclusions – further research Closing this paper, we present a short resume. The new modeling concept introduced by this research project is the introduction of uncertainty into the insurance pricing process. The framework of uncertainty is further enhanced assuming also some kind of delay. The H-infinity control theory is employed as the basic tool in order to handle the application. Furthermore, H1-control leads to linear matrix inequalities (LMIs) since the basic stability condition relies on a system of LMIs. After constructing a quite general pricing model, we focus to the simplest case that is described in a single dimension. A numerical application is fully investigated by solving the respective LMI using an iterative algorithm (up to 1000 iterations). The numerical results of our application coincide with those of older papers (see [13]) supporting that the feedback factor f (f represents the amount of delayed information integrated into the system) must be restricted below 50% or even lower when considering models with high uncertainty or volatility levels, otherwise the system is not robust stochastically stable. It is also interesting to note the solution for the trivial case where there is no feedback stabilization action and the relevant feedback factor equals zero (f = 0%). Then, the robust stabilization factor K X 1 is slightly higher than 50% (from 52% up to 54%). It also worth noticing (see the last diagram of Fig. 1) that all the solutions are almost parallel and be arranged within a narrow zone-path starting from the interval [52%, 54%] for f = 0% and ending to interval [34%, 36%] for f = 40%. 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