Thiele's Differential Equation Based on Markov Jump Processes with Non-countable State Space
TThiele’s Differential Equation Based on Markov JumpProcesses with Non-countable State Space
Emmanuel Coffie ∗ Sindre Duedahl † Frank Proske ‡ February 22, 2021
Abstract
In modern life insurance, Markov processes in continuous time on a finite or atleast countable state space have been over the years an important tool for the mod-elling of the states of an insured. Motivated by applications in disability insurance,we propose in this paper a model for insurance states based on Markov jump pro-cesses with more general state spaces. We use this model to derive a new type ofThiele’s differential equation which e.g. allows for a consistent calculation of reservesin disability insurance based on two-parameter continuous time rehabilitation rates.
Key words : Life insurance, Thiele’s differential equation, Markov processes withgeneral state spaces, rehabilitation rates, insurance reserves.
Over the years, finite-state Markov chains have played a significant role in multi-statemodelling of life insurance risks. Typical insurance applications of finite-state Markovchains pertain to e.g. endowment insurance, life annuities or pension contracts. In fact,there is a rich literature devoted to finite- or countable-state Markov chain modellingof life insurance risks with respect to calculation of reserves. For instance, Henriksen etal. in [HNSS14] employ the finite-state Markov chain framework to model (in additionto insurance risk) behavioural risk in the sense of e.g. surrender or free policy risk andexamine the effects of such types of risks on prospective reserves.On the other hand, Norberg [Nor95] considers the case of a force of interest modelledby a time-continuous homogeneous Markov process with finite-state space and appliesthis model to the computation of prospective reserves for some standard insurance poli-cies. As for a variety of other important applications of finite-state (or countable-state) ∗ Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH, UK. Email:emmanuel.coffi[email protected]. † Danske Bank, N-0250, Aker Brygge, Oslo, Norway. Email: [email protected]. ‡ Department of Mathematics, University of Oslo, N-0316, Blindern, Oslo, Norway. Email:[email protected]. a r X i v : . [ q -f i n . R M ] F e b arkov processes or chains to issues as e.g. unit-linked insurance policies in life insur-ance, we refer the reader to [Kol13] or [MS07] and the references therein.Despite of the wide applicability of Markov processes with countable state spaces ininsurance risk modelling for reserve calculations, such processes may not be sufficient toconsistently describe crucial life insurance risks associated with certain modern life in-surance policies. For example, life insurance risks in connection with disability insurancebased on two-parameter continuous-time rehabilitation rates or ”random spouse” con-tracts cannot be adequately modelled by Markov processes with countable state spaces,but by those on more general state spaces.The application of Markov processes with more general state spaces could be inparticular relevant for immunocompromised policyholders, who have a high exposure todisease recidivism over time and who need to earn benefits whilst undergoing recovery,rehabilitation or medical treatment.In this paper, we use Markov jump processes on more general spaces to model insur-ance risks and to establish a new type of Thiele’s differential equation for the computationof insurance reserves.Our paper is organised as follows: In Subsections 1.1 and 1.2, we introduce themathematical setting of this article and some insurance notation needed later on. InSection 2, we derive Thiele’s differential equation in the framework of Markov jumpprocesses on more general spaces. Finally, in Section 3, we discuss some insurancepolicies which necessitate and justify the use of Markov processes with non-countablestate spaces in risk modelling. Further, an example of numerical implementation ispresented. In this Subsection, we pass in review some mathematical notions and results which wewill need throughout the paper. See e.g. [EK09] or [Blu07] as for results on Markovprocesses.
Definition 1.1.1.
Given a Polish space S , let D ( S ) be the space of c`adl`ag functionsfrom the interval [0 , ∞ ) into S (i.e. the space of functions f : [0 , ∞ ) → S , which areright continuous with existing left sided limits). We also use the symbol S for the Borel σ -algebra on S . Also, define PC([0 , ∞ ) , S ) to be the subpace of D ( S ) determined bythe additional requirement that functions are piecewise-constant, i.e. constant on thehalf-open intervals between jump discontinuities.In the next definition, we introduce the regular insurance model in our settings. See[Kol13] in the case of finite-state Markov chains. Definition 1.1.2.
A regular insurance model consists of the following objects:1. A measurable space ( S, S ) called the state space , where S is Polish and S is theBorel σ -algebra.2. A filtered probability space (Ω , F , { F t } t ≥ , P ).2. A kernel of positive measure , i.e. a map[0 , T ] × S × S (cid:51) ( t, x, B ) (cid:55)→ q t ( x, B ) ∈ R such that for every t ∈ [0 , T ] , B ∈ S , x (cid:55)→ q t ( x, B ) is S − B ( R )-measurable, andfor every t ∈ [0 , T ] , x ∈ S , B (cid:55)→ q t ( x, B ) is a positive measure.4. A Markov jump process on S , i.e. X : Ω → D T ( S ) whose paths are almost surelyin PC([0 , ∞ ) , S ), such that X is F t -adapted and has the Markov property withrespect to F t and P , and such that q is the jump intensity function, i.e. for x ∈ S , P t,t + h ( x, B ) = q t ( x, B ) h + o ( h ) (1)for h (cid:38)
0, where P t,s ( x, B ) is a transition function of X , i.e. P t,s ( x, B ) (cid:44) P [ X s ∈ B | X t = x ] . We use the notation X t for the random variable given by X t ( ω ) = X ( ω )( t ) for t ∈ [0 , ∞ ) , ω ∈ Ω.5. A measurable function B : [0 , ∞ ) × S → R which is of bounded variation (BV) inthe first variable.6. A BV function b : [0 , ∞ ) × S × S → R . Remark . By the conditional probability expression in item 4 above, we mean thefollowing: Since X t is a random map into a Polish space there exist (see [KS91], pp. 84-85and references cited there) regular conditional probabilities , i.e. maps ν t : S × F → [0 , S (cid:51) g (cid:55)→ ν t ( g, Γ) is measurable for every Γ ∈ F .2. F (cid:51) Γ (cid:55)→ ν t ( g, Γ) is a probability measure for each g ∈ S , and3. P ( A ∩ X − t ( B )) = (cid:82) h ∈ B ν t ( h, A ) P ( X − t ( dh )).From the third equality it follows for all measurable maps Y : Ω → S and f ∈ L ( P Y − )that E ( f ( Y ) | X t ) = (cid:90) h ∈ S f ( h ) ν t ( X t , Y − ( dh )) ,P X − t -a.s. To see this, let φ : S → R be measurable and assume moreover that φ, f both have finite image, i.e. φ = d (cid:88) j =1 α j A j , f = v (cid:88) j =1 β j B j E ( φ ( X t ) f ( Y )) = d (cid:88) i =1 v (cid:88) j =1 α i β j E ( A i ( X t ) B j ( Y ))= d (cid:88) i =1 v (cid:88) j =1 α i β j P ( Y − ( B j ) ∩ X − t ( A i ))= d (cid:88) i =1 α i (cid:90) h ∈ A i v (cid:88) j =1 ν t ( h, Y − ( B j )) P ( X − t ( dh ))= d (cid:88) i =1 α i (cid:90) h ∈ A i (cid:90) g ∈ S f ( g ) ν t ( h, Y − ( dg )) P ( X − t ( dh ))= E (cid:18) φ ( X t ) (cid:90) g ∈ S f ( g ) ν ( X t , Y − ( dg )) (cid:19) We then infer the general case of L functions by approximation with simple functionsand continuity of the expectation functional (or alternatively the monotone class theo-rem). Or see e.g. [GS77] regarding the above relation.In particular, we have that E ( f ( X s ) | X t = g ) = (cid:90) h ∈ S f ( h ) ν t ( g, X − s ( dh )) . (2) Remark . In the sequel, the symbol B ( t, g ), which we sometimes denote by B g ( t )should be interpreted as the accumulated payment stream up to time t if X s = g for all s ≤ t . Since it is assumed that payments depend only on the time and state at eachmoment, the accumulated payments over an interval I is given by (cid:82) I dB X t ( t ).The symbol b ( t, g, h ), which we also denote by b gh ( t ), should be interpreted as animmediately incurred payment at the time t of a transition from g to h , i.e. the sum ofall payments from such events over an interval I is given by (cid:90) I b gh ( t ) dN gh ( t ) , where N gh ( t ) is the number of jumps performed by X from g to h up to time t , inother words: N gh ( t ) = (cid:40) { s | s ≤ t, X t − = g, X t = h } , g (cid:54) = h , g = h Remark . The assumptions made here imply that a Markov process having a givenfunction as its jump intensity can always be constructed, see [Ebe15]. The constructioninvolves decomposing the problem in two parts: The jumping times J , J , . . . Y , Y , . . . such that X t = Y i for t ∈ [ J i , J i +1 ].The conditional distribution of the next jumping time is given by the survival func-tion : P ( J i +1 − J i > t | J , . . . , J i , Y , . . . , Y i ) = e − (cid:82) t λ Ji + s ( Y i ) ds , (3)where λ is the total jump rate given by λ t ( x ) = q t ( x, S \ { x } ) . It will also be the case that P [ Y i ∈ B | J , . . . , J i , Y , . . . Y i − ] = π J i ( Y i − , B ) , where π t is a family of probability measures parametrized by S , given by: π t ( x, B ) = q t ( x, B ) λ t ( x ) . We recall the
Kolmogorov-Chapman equation and the backward and forward Kolmogorov equations from Markov process theory. See e.g. [EK09] or [Blu07].
Lemma 1.2.1.
For t < u < s , g ∈ S and Γ ∈ S ,i) P t,t ( g, Γ) = Γ ( g ) .ii) P t,s ( g, Γ) = (cid:82) S P t,u ( g, dh ) P u,s ( h, Γ) .iii) ∂∂t P t,s ( g, Γ) = λ t ( g ) P t,s ( g, Γ) − (cid:82) S \{ g } q t ( g, dh ) P t,s ( h, Γ) .iv) ∂∂s P t,s ( g, dh ) = − P t,s ( g, dh ) λ s ( h ) + (cid:82) S \{ h } P t,s ( g, dk ) q s ( k, dh ) . In this Subsection, we recall some notions from life insurance which can be e.g. foundin [Kol13] or [Nor95] in the setting of finite-state Markov chains or processes.5 efinition 2.1.1.
A (deterministic) discount function v is a continuous function v :[0 , ∞ ) × [0 , ∞ ) → (0 , ∞ ). Often but not always, v derived from a technical interest rate r which is usually positive, i.e. v ( s, t ) = e − r ( t − s ) < Definition 2.1.2.
Given a regular insurance model and a discount function v , the present value of future cashflows (liabilities) is defined as V ( t ) = (cid:90) [ t, ∞ ) v ( t, s ) (cid:0) dB X s ( s ) + b X s − ,X s ( s ) dN X s − ,X s ( s ) (cid:1) (4)or (cid:90) [ t, ∞ ) v ( t, s ) dB ( s ) (5)where B is the total cashflow up to time t given by dB X t ( t ) + b X t − ,X t ( t ) dN X t − ,X t ( t ) . Since X t , t ≥ t , henceforth called N X ( t ), also is non-decreasing and piecewise constant, both B X · and N X are BV (i.e. of bounded variation) and the differentials are interpreted inthe sense of Lebesgue-Stieltjes integration. V can in fact be rewritten as: V ( t ) = (cid:90) [ t, ∞ ) v ( t, s ) dB X s ( s ) + (cid:88) i ≥ v ( J i ) b Y i − ,Y i ( J i ) . (6) Definition 2.1.3.
The prospective reserve is the map S × [0 , ∞ ) (cid:51) ( g, t ) (cid:55)→ V g ( t ) givenby V g ( t ) = E [ V ( t ) | X t = g ] . Using the notation and results of the previous Sections, we are now able to derivethe following new type of Thiele’s differential equation for the calculation of insurancereserves.
Theorem 2.2.1.
Assume the regular insurance model in Definition 1.1.2 and supposethat the discount function is of the form v ( t, s ) = e − (cid:82) st r ( u ) du . Then, dV g ( t ) = ( λ t ( g ) + r ( t )) V g ( t ) dt − dB g ( t ) − (cid:90) S ( b gh ( t ) + V h ( t )) q t ( g, dh ) dt. Proof.
Let J t = J i where i = min { k ∈ N | J k ≥ t } . Since P [ J t ≥ u | X t = g ] = P [ X t + v = g ∀ v ∈ (0 , u ) | X t = g ] = e − (cid:82) [ t,u ) λ v ( g ) dv (7)6he real-valued random variable J t has a conditional density given by P ( J t ∈ ds | X t = g ) = λ s ( g ) e − (cid:82) [ t,s ) λ u ( g ) du ds. So V g ( t ) = E [ V ( t ) | X t = g ] = (cid:90) [ t, ∞ ) P ( J t ∈ ds | X t = g ) E [ V ( t ) | J t = s, X t = g ]= (cid:90) [ t, ∞ ) λ s ( g ) e − (cid:82) [ t,s ) λ u ( g ) du (cid:40)(cid:90) [0 ,s ] v ( t, u ) dB g ( u ) + E [( b gY + V Y ( s )) | J = s ] (cid:41) ds = V (1) g ( t ) + V (2) g ( t ) , where V (1) g ( t ) = (cid:90) [ t, ∞ ) λ s ( g ) e − (cid:82) [ t,s ) λ u ( g ) du (cid:90) [ t,s ] v ( t, u ) dB g ( u ) ds. By Fubini’s theorem, this is equal to (cid:90) [ t, ∞ ) v ( t, u ) (cid:90) [ u, ∞ ) λ s ( g ) e − (cid:82) [ t,s ) λ y ( g ) dy dsdB g ( u ) = (cid:90) [ t, ∞ ) v ( t, u ) e − (cid:82) [ t,u ) λ y ( g ) dy dB g ( u )= (cid:90) [ t, ∞ ) e − (cid:82) [ t,u ) λ y ( g )+ r ( u ) dy dB g ( u ) . This means that V (2) g ( t ) = (cid:90) [ t, ∞ ) v ( t, s ) λ s ( g ) e − (cid:82) [ t,s ) λ u ( g ) du (cid:90) S \{ g } ( b gh ( s ) + V h ( s )) π s ( g, dh ) ds = (cid:90) [ t, ∞ ) v ( t, s ) e − (cid:82) [ t,s ) λ u ( g ) du (cid:90) S \{ g } ( b gh ( s ) + V h ( s )) q s ( g, dh ) ds. Computing the derivatives of V (1) g ( t ), resp. V (2) g ( t ) with respect to t , we get dV (1) g ( t ) = − dB g ( t ) + ( λ t ( g ) + r ( t )) V (1) g ( t )and ddt V (2) g ( t ) = − (cid:90) S \{ g } ( b gh ( t ) + V h ( t ) q t ( g, dh ) + ( λ t ( g ) + r ( t )) V (2) g ( t ) , so 7 V g ( t ) = ( λ t ( g ) + r ( t )) V g ( t ) dt − dB g ( t ) − (cid:90) S \{ g } ( b gh ( t ) + V h ( t ) q t ( g, dh ) . Finally, we aim at discussing some examples from life insurance which show the needof risk modelling by using Markov processes on more general state spaces.
Example 2.2.2. ( The discrete case )If S is a countable set with the discrete topology and we assume that q t ( x, · ) is aBorel measure on S for every t ≥ x ∈ S , the model is reduced to the one describedin [Kol13]. Values of the q -measure on singletons are identical to the jump intensities( µ ij ( t ) = q t ( i, { j } ), and the Thiele equation is reduced to the familiar form: dV g ( t ) = ( λ t ( g ) + r ( t )) V g ( t ) dt − dB g ( t ) − (cid:88) h ∈ S,h (cid:54) = g ( b gh ( t ) + V h ( t )) µ gh ( t ) dt. Example 2.2.3. ( Disability insurance with rehabilitation )Assume a state space for the insured consisting of three states, S = {∗ , (cid:5) , †} , interpreted respectively as healthy, disabled, and deceased. A Markov model withthis state space is an unsatisfactory model for disability insurance, since it implies thatthe jump intensity from the disabled state to the healthy state (rehabilitation) is solelya function of time. It is clear that a model with any hope of being realistic would haveto take into account the dependence of the rehabilitation intensity on the time elapsedsince the last transition into the disabled state. The solution to this problem (compareto examples in [Kol13] in the case of finite-state Markov chains.) lies in replacing S bythe state space S (cid:48) (cid:44) {∗ , †} ∪ ( {(cid:5)} × [0 , ∞ )) , interpreted as follows: ∗ is the healthy state, † is death, as before. ( (cid:5) , t ) means that theinsured is disabled and that the jump to disability occurred at the time point t .The prescription that the rehabilitation intensity µ (cid:5)∗ ( t, τ ) should be a function notonly of time t , but also of the time τ since the last jump to (cid:5) , is realized by defining q by q t ( ∗ , {(cid:5)} × H ) = µ ∗(cid:5) ( t ) H ( t ) ,q t (( (cid:5) , s ) , {∗} ) = µ (cid:5)∗ ( t, t − s ) [0 , ∞ ) ( t − s )and with the death intensity (from healthy or disabled state) defined in the usualway. 8 xample 2.2.4. ( Random spouse )If we are trying to model an insurance contract giving an annuity payment for aspouse who is left behind when the insured dies, and which is payable continuouslyuntil the death of the spouse, the usual route is to employ a two-life model where thestate space is a product space of the respective state spaces for the insured and thespouse. However in practice there exist arrangements where the insurer does not knowthe age of the spouse or whether there even is a spouse at the initiation of the contract,but instead learns of this upon the death of the insured. Nevertheless the insurer has tocompute a reserve, which thus has to take into account the random nature of the maritalstatus. In some model definitions, including some mandated as minimum requirementsfor technical provisions of Norwegian life insurers (see e.g. [Fin21]), the probability ofthe existence of a spouse, and the distribution of the age of the spouse, conditional onhis or her existence, are given by functions which depend on the age of the insured atthe time of death. Since the subsequent evolution of the system involves the mortalityof the spouse which is a function of age, and thus depends on the preceding history,Markov chains on a finite or discrete state space are not well-suited for this. The currentframework gives a natural resolution to this problem via the following setup: S = {∗} ∪ ( {†} × R )where the continuous state variable is interpreted as the age difference between theinsured and the spouse revealed at the time of death. Assume given the age-dependentprobability g ( t ) of observing a spouse at the time of death of the insured, and assumethat the conditional distribution of the age difference is given by a probability measure φ on ( R , B ( R )). This setup is realized by defining q as follows: q t ( ∗ , {†} × H )) = µ ∗† ( t ) g ( t ) φ ( H )where µ ∗† is just the usual mortality rate. We will give a proof-of-concept numerical implementation method for the life insurancemodel, focusing on example 2.2.3. Assuming B g is differentiable with respect to time,we write ˙ B g ( t ) = ddt B g ( t ) . Theorem 2.2.1 implies ddt V g ( t ) = ( λ t ( g ) + r ( t )) V g ( t ) − ˙ B g ( t ) − (cid:90) S ( b gh ( t ) + V h ( t )) q t ( g, dh )In our disability example, we have the below identities with δ g ( · ) denoting the Diracmeasure at g ; 9 t ( ∗ , A ) = µ ∗† ( t ) δ † ( A ) + µ ∗(cid:5) ( t ) δ ( (cid:5) ,t ) ( A ) q t (( (cid:5) , s ) , A ) = µ (cid:5)∗ ( t, t − s ) [0 . ∞ )] ( t − s ) δ ∗ ( A ) + µ (cid:5)† ( t ) δ † ( A ) . So (cid:90) S ( b ∗ h ( t ) + V h ( t )) q t ( ∗ , dh ) = ( b ∗† ( t )+ =0 (cid:122) (cid:125)(cid:124) (cid:123) V † ( t )) µ ∗† ( t ) + ( b ∗ , ( (cid:5) ,t ) ( t ) + V ( (cid:5) ,t ) ( t )) µ ∗(cid:5) ( t ) . On the other hand for g = ( (cid:5) , s ) we get that (cid:90) S ( b ( (cid:5) ,s ) h ( t )+ V h ( t )) q t (( (cid:5) , s ) , dh ) = ( b ( (cid:5) ,s ) ∗ ( t )+ V ∗ ( t )) µ (cid:5)∗ ( t, t − s ) [0 , ∞ ) ( t − s )+( b ( (cid:5) ,s ) † ( t )+ =0 (cid:122) (cid:125)(cid:124) (cid:123) V † ( t )) µ (cid:5)† ( t ) . Moreover λ t ( ∗ )def= q t ( ∗ , S − {∗} ) = µ ∗† ( t ) + µ ∗(cid:5) ( t )and λ t (( (cid:5) , s ))def= q t (( (cid:5) , s ) , S − { ( (cid:5) , s ) } ) = µ (cid:5)∗ ( t, t − s ) [0 , ∞ ) ( t − s ) + µ (cid:5)† ( t ) . Hence ddt V ∗ ( t ) = ( µ ∗† ( t )+ µ ∗(cid:5) ( t )+ r ( t )) V ∗ ( t ) − ˙ B ∗ ( t ) −{ ( b ∗† ( t )+ =0 (cid:122) (cid:125)(cid:124) (cid:123) V † ( t )) µ ∗† ( t )+( b ∗ , ( (cid:5) ,t ) ( t )+ V ( (cid:5) ,t ) ( t )) µ ∗(cid:5) ( t ) } , (8) ddt V ( (cid:5) ,s ) ( t ) = ( µ (cid:5)∗ ( t, t − s ) [0 , ∞ ) ( t − s ) + µ (cid:5)† ( t ) + r ( t )) V ( (cid:5) ,s ) ( t ) , − ˙ B ( (cid:5) ,s ) ( t ) − ( b ( (cid:5) ,s ) ∗ ( t ) + V ∗ ( t )) µ (cid:5)∗ ( t, t − s ) [0 , ∞ ) ( t − s ) − ( b ( (cid:5) ,s ) † ( t )+ =0 (cid:122) (cid:125)(cid:124) (cid:123) V † ( t )) µ (cid:5)† ( t ) . (9)The discretized version of (8) based on an Euler approximation scheme for ordinarydifferential equations (see [But03]) is the recurrence relation given by V ∗ ( t n − ) = V ∗ ( t n ) − ( t n − t n − )[( µ ∗† ( t n ) + µ ∗(cid:5) ( t n ) + r ( t n )) V ∗ ( t n ) − ˙ B ∗ ( t n ) − { ( b ∗† ( t n )) µ ∗† ( t n ) + ( b ∗ , ( (cid:5) ,t n ) ( t n ) + V ( (cid:5) ,t n ) ( t n )) µ ∗(cid:5) ( t n ) } ] . Similarly for (9) and k ≤ nV ( (cid:5) ,t k ) ( t n − ) = V ( (cid:5) ,t k ) ( t n ) − ( t n − t n − )[( µ (cid:5)∗ ( t n , t n − t k ) [0 , ∞ ) ( t n − t k )+ µ (cid:5)† ( t n )+ r ( t n )) V ( (cid:5) ,t k ) ( t n ) − ˙ B ( (cid:5) ,t k ) ( t n ) − { ( b ( (cid:5) ,t k ) ∗ ( t n ) + V ∗ ( t n )) µ (cid:5)∗ ( t n , t n − t k ) + b ( (cid:5) ,t k ) ( t n ) µ (cid:5)† ( t n ) } .
10e will implement this assuming that the disability insurance pays 1 $ per year aslong as the insured is in the disabled state, but only until the age of retirement whichwe set to 67 years. We also assume a constant force of interest r . The recursion schemeis then reduced to V ∗ ( t n − ) = V ∗ ( t n ) − ( t n − t n − )[( µ ∗† ( t n ) + µ ∗(cid:5) ( t n ) + r ) V ∗ ( t n ) − V ( (cid:5) ,t n ) ( t n ) µ ∗(cid:5) ( t n )] , (10) V ( (cid:5) ,t k ) ( t n − ) = V ( (cid:5) ,t k ) ( t n ) − ( t n − t n − )[( µ (cid:5)∗ ( t n , t n − t k ) + µ (cid:5)† ( t n ) + r ) V ( (cid:5) ,t k ) ( t n ) − − V ∗ ( t n ) µ (cid:5)∗ ( t n , t n − t k )] (11)with the boundary condition V g (67) = 0 , g ∈ S. Here we use transition rates of Gompertz-Makeham type, except for the rehabilitationrate which is somewhat more involved. Specifically we set µ ∗† ( t ) = 0 . . t − . ,µ ∗(cid:5) ( t ) = 0 . . t − . . Further, we model the two-parameter continuous-time rehabilitation rate as follows: µ (cid:5)∗ ( t, τ ) = µ (cid:5)∗ ( t )(1 − µ (cid:5)† ( t + s )) [0 , ∞ ) ( t − s ) , where µ (cid:5)∗ ( t ) = 0 . − . t and t is a parameter to be adjusted to the specific situation. In our example t isgiven by the age of the insured at the start of the contract. Displayed in Figures 1 and 2 are the reserve plots of the case illustrated in example 2.2.3.We observe the prospective reserve in the active state declines with increasing age of theinsured in Figure 1. In Figure 2, we note the prospective reserve in the disabled stateof the insured is significantly impacted. In other words, the prospective reserve in thedisabled state increases with the age of the insured from onset of disability.Comparing our result with that of
Example 2.4.2 in [Kol13], it is obvious our modeloutperforms the classical model by incorporating the influence of rehabilitation on thedisabled state of the insured for the computation of prospective reserves. This applicationjustifies the use of Markov processes with non-countable state spaces in connection withour proposed model. 11igure 1: Plot of reserve in the active state12igure 2: Plot of reserve in the disabled state13 cknowledgements
One of the authors (S.D.) wants to thank Tor Vidvei for interesting discussions onapplication problems which led to the idea behind this paper.
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