A BSDE-based approach for the optimal reinsurance problem under partial information
AA BSDE-based approach for theoptimal reinsurance problemunder partial information
Brachetta M. ∗† [email protected] Ceci, C. ∗ [email protected] Abstract
We investigate the optimal reinsurance problem under the criterion of maximizing the ex-pected utility of terminal wealth when the insurance company has restricted informationon the loss process. We propose a risk model with claim arrival intensity and claim sizesdistribution affected by an unobservable environmental stochastic factor. By filtering tech-niques (with marked point process observations), we reduce the original problem to an equiv-alent stochastic control problem under full information. Since the classical Hamilton-Jacobi-Bellman approach does not apply, due to the infinite dimensionality of the filter, we choosean alternative approach based on Backward Stochastic Differential Equations (BSDEs). Pre-cisely, we characterize the value process and the optimal reinsurance strategy in terms of theunique solution to a BSDE driven by a marked point process.
Keywords:
Optimal reinsurance, partial information, stochastic control, backward stochasticdifferential equations.
JEL Classification codes:
G220, C610.
MSC Classification codes:
1. Introduction
The aim of this paper is to investigate the optimal reinsurance problem when the insurer hasonly limited information at disposal. Insurance business requires very effective tools to man-age risks and reinsurance arrangements are considered incisive to this end. From the opera-tional viewpoint, a risk-sharing agreement helps the insurer reducing unexpected losses, stabi-lizing operating results, increasing business capacity and so on. The existing literature mostlyconcerns classical reinsurance contracts such as the proportional and the excess-of-loss, whichwere widely investigated under a variety of optimization criteria (see [Irgens and Paulsen, 2004],[Liu and Ma, 2009], [Brachetta and Ceci, 2019b] and references therein). All these papers can begathered in two main groups, depending on the underlying risk model: some authors describe theinsurer’s loss process as a diffusion model (this approach is motivated by the Cram´er-Lundbergapproximation); others use jump processes, as in our case.The common ground of the majority of those papers is the complete information setting.However, in the real world the insurer has only a partial information at disposal. In fact, onlythe claims occurrences (times and sizes) are directly observable. Precisely, the claims intensity is amathematical object and it is required by all the risk models, but its realizations are not observedby economic agents (as mentioned in [Grandell, 1991, Chapter 2]). In practice, the insurer relieson an estimation, which is based on the information at disposal. The same applies to the claim ∗ Department of Economics, University of Chieti-Pescara, Viale Pindaro, 42 - 65127 Pescara, Italy. † Corresponding author. a r X i v : . [ q -f i n . M F ] M a y izes distribution, which is estimated by the accident realizations. In [Liang and Bayraktar, 2014]we recognize a noteworthy attempt to introduce a partial information framework. At first, theyintroduce a stochastic factor Y which influences the risk process. As discussed in [Grandell, 1991],this external driver Y represents any environmental alteration reflecting on risk fluctuations (fora discussion in a complete information context see also [Brachetta and Ceci, 2019b]). Then, theysuppose that Y is not observable. Consequently, the intensity is unobservable itself. Since Y isa finite-state Markov chain in that work, the classical Hamilton-Jacobi-Bellman (HJB) approachworks well after the reduction to an equivalent problem with complete information (this resultis achieved by means of the filtering techniques).In our paper we study the optimal reinsurance problem under partial information. The insurerwishes to maximize the expected exponential utility of the terminal wealth, using the informationat disposal. We propose a risk model with claim arrival intensity and claim sizes distributionaffected by an unobservable environmental stochastic factor Y . More specifically, the loss pro-cess is a marked point process with dual predictable projection dependent on Y , extending theCram`er-Lundberg model (where a Poisson process with constant intensity is used). In contrast to[Liang and Bayraktar, 2014], here Y is a general Markov process (including finite-state Markovchains, diffusions and jump-diffusions as special cases). Using filtering techniques with markedpoint process observations, the original problem is reduced to an equivalent stochastic controlproblem under complete information. Since the filter process turns out to be infinite-dimensional,the classical HJB method does not apply and we use a Backward Stochastic Differential Equation(BSDE)-based approach. Precisely, we characterize the value process and the optimal reinsurancestrategy in terms of the solution to a BSDE, whose existence and uniqueness are ensured undersuitable hypotheses. This is a well established approach in the financial literature, indeed sev-eral papers (see e.g. [El Karoui et al., 1997], [Ceci and Gerardi, 2011], [Lim and Quenez, 2011],[Ceci, 2004] and [Ceci, 2012] and references therein) deal with stochastic optimization problemsin finance by means of BSDEs. The recent book [Delong, 2013] applies BSDE techniques also toactuarial problems, extending the classical mathematical tools in this field.Moreover, we model the insurance gross risk premium and the reinsurance premium asstochastic processes. Clearly, they are adapted to the filtration which represents the restrictedinformation, since the insurance and the reinsurance companies choose the premium based onthe information at disposal.Another important peculiarity of our work is that we consider a generic reinsurance contract,which is characterized by the self-insurance function (which represents the insurer’s retainedlosses). Hence the retention level is chosen in the interval [0 , I ], with I ∈ (0 , + ∞ ]. Evidently,the proportional and the excess-of-loss optimal policies can be derived as special cases.Finally, we allow the insurer to invest the surplus in a risk-free asset with interest rate R >
0. The absence of a financial market with a risky asset is not restrictive. In fact, theexisting literature (e.g. [Brachetta and Ceci, 2019b]) have shown that the optimal reinsurancestrategy only depends on the risk-free asset, even in presence of a risky asset, under the standardassumption of independence between the financial and the insurance markets. In this case, theinvestment strategy can be eventually determined using one of the well known results on thistopic.The paper is organized as follows: in Section 2 the model is formulated and the problem isintroduced. In particular, the original problem with partial information is reduced to an equiv-alent problem with complete information via filtering with marked point process observations.Some details about filtering results can be found in the Appendix. In Section 3 we derive acomplete characterization of the value process in terms of a solution to a BSDE, whose exis-tence and uniqueness are discussed. In addition to this, we prove the existence of an optimalreinsurance strategy under suitable conditions. Section 4 is devoted to investigate the structureof the optimal reinsurance strategy. Finally, in Section 5 we investigate some properties of theoptimal reinsurance strategy, such as the Markovianity with respect to the filter process, and wediscuss some relevant examples. In particular, the effect of the safety loading is analyzed and acomparison with the optimal strategy under full information is illustrated.2 . Problem formulation2.1. Model formulation
Let
T > , G , P , G ) is a complete probability spaceendowed with a filtration G . = {G t } t ∈ [0 ,T ] satisfying the usual conditions. This filtration rep-resents all the achievable information, so that the knowledge of G means full information. Weassume that the insurance market is influenced by an external driver Y = { Y t } t ∈ [0 ,T ] , mod-eled as a c`adl`ag Markov process with infinitesimal generator L Y . Clearly, the sigma-algebra F Y generated by Y is included in G , that is F Yt ⊆ G t ∀ t ∈ [0 , T ]. For instance, Y could bea finite-state Markov chain, a diffusion process, a jump-diffusion and so on. This stochasticfactor represents any environmental alteration reflecting on risk fluctuations. In practice, as sug-gested by Grandell, J. (see [Grandell, 1991], Chapter 2), in automobile insurance Y may describeroad conditions, weather conditions (foggy days, rainy days, . . . ), traffic volume, and so on (seealso [Brachetta and Ceci, 2019b]).The insurer’s losses are described by the double sequence { ( T n , Z n ) } n =1 ,... , where • { T n } n ≥ is a sequence of G -stopping times such that T n < T n +1 P -a.s. ∀ n ≥
1, representingthe claims arrival times; • { Z n } n ≥ is a sequence of G T n -measurable and (0 , + ∞ )-valued random variables, which arethe claims amounts.The corresponding random measure m ( dt, dz ) is given by m ( dt, dz ) . = ∑ n ≥ δ ( T n ,Z n ) ( dt, dz ) { T n ≤ T } , (2.1)where δ ( t,z ) denotes the Dirac measure at point ( t, z ). The marked point process m ( dt, dz ) ischaracterized by the next hypotheses.We propose a risk model with both the claims intensity and the claim sizes distributionaffected by the stochastic factor Y . For this purpose, we use the following assumption. Assumption 2.1.
Given a measurable function λ ( t, y ) : [0 , T ] × R → (0 , + ∞ ) , let us define the G -predictable process { λ t . = λ ( t, Y t − ) } t ∈ [0 ,T ] . Suppose that there exists a constant Λ > suchthat < λ ( t, y ) ≤ Λ ∀ ( t, y ) ∈ [0 , T ] × R . (2.2) In addition to this, suppose that there exists a probability transition kernel F Z ( t, y, dz ) from ([0 , T ] × R , B ([0 , T ]) ⊗ B ( R )) into ([0 , + ∞ ) , B ([0 , + ∞ ))) such that E [∫ T ∫ + ∞ z F Z ( t, Y t , dz ) dt ] < + ∞ . (2.3) Then we assume that m ( dt, dz ) admits the following G -dual predictable projection: ν ( dt, dz ) = λ t F Z ( t, Y t − , dz ) dt, (2.4) i.e. for every nonnegative, G -predictable and [0 , + ∞ ) -indexed process { H ( t, z ) } t ∈ [0 ,T ] we havethat E [∫ T ∫ + ∞ H ( t, z ) m ( dt, dz ) ] = E [∫ T ∫ + ∞ H ( t, z ) λ t F Z ( t, Y t , dz ) dt ] . We will denote by F . = {F t } t ∈ [0 ,T ] the filtration generated by m ( dt, dz ), that is F t = σ { m ((0 , s ] × A ) , s ≤ t, A ∈ B ([0 , + ∞ )) } . (2.5)Using the marked point processes theory , it is possible to obtain a precise interpretation of { λ t } t ∈ [0 ,T ] and F Z ( t, y, dz ) separately. For details on this topic see [Br´emaud, 1981]. N t = m ((0 , t ] × [0 , + ∞ )) = ∑ n ≥ { T n ≤ t } the claims arrival process, whichcounts the number of occurred claims. According to the definition of dual predictable projection,choosing H ( t, z ) = H t with { H t } t ∈ [0 ,T ] any nonnegative G -predictable process, we get that E [∫ T ∫ + ∞ H t m ( dt, dz ) ] = E [∫ T H t dN t ] = E [∫ T H t λ t dt ] , i.e., { N t } t ∈ [0 ,T ] is a point process with G -intensity { λ t } t ∈ [0 ,T ] .Moreover, F Z ( t, y, dz ) can be interpreted as the conditional distribution of the claim sizesgiven the knowledge of the stochastic factor. Proposition 2.1. ∀ n = 1 , . . . and ∀ A ∈ B ([0 , + ∞ )) P [ Z n ∈ A | G T − n ] = ∫ A F Z ( T n , Y T − n , dz ) = P [ Z n ∈ A | F YT − n ] P -a.s. , where G T − n is the strict past of the σ -algebra until time T n : G T − n := σ { A ∩ { t < T n } , A ∈ G t , t ∈ [0 , T ] } , and F T − n is defined similarly.Proof. See [Brachetta and Ceci, 2019a, Proposition 1].We define the cumulative claims up to time t ∈ [0 , T ] as follows: C t = N t ∑ n =1 Z n = ∫ t ∫ + ∞ z m ( ds, dz ) . (2.6)Let us observe that our model formulation is able to fit some well known risk models (forwhich the reader can refer to [Rolski et al., 1999] or [Schmidli, 2018]). Example 2.1 (Cram´er-Lundberg Risk Model) . If we consider a constant intensity λ ( t, y ) = λ and a distribution function F Z ( t, y, dz ) = F Z ( dz ) , then we obtain the classical Cram´er-Lundbergrisk model. Example 2.2 (Markov Modulated Risk Model) . Suppose that the stochastic factor Y is a con-tinuous time irreducible Markov process with a finite state space S = { , . . . , M } , with M ≥ .Taking λ ( y ) and F Z ( y, dz ) we obtain the so called Markov Modulated Risk Model. Equivalently,we can associated M constants { λ i } i =1 ,...,M and distribution functions { F iZ ( dz ) } i =1 ,...,M to eachstate of Y . Correspondingly, we can define M independent classical risk models (as in Example2.1), with loss processes { C i } i =1 ,...,M such that Eq. (2.6) becomes C t = ∫ t ∑ i ∈S Y t = i dC is . Eventually, without loss of generality we could assume that λ i ∫ + ∞ zF iZ ( dz ) ≤ λ j ∫ + ∞ zF jZ ( dz ) ∀ i, j ∈ { , . . . , M } , i < j. In the rest of the paper we suppose that the insurer is not able to get access to the completeinformation G . In contrast, at any time t ∈ [0 , T ] she is allowed to observe only these objects: • the occurred claims times, i.e. the jump times of m ( dt, dz ) up to time t ;4 the occurred claims size, i.e. the marks of m ( dt, dz ) up to time t .More formally, the information flow at insurer’s disposal is described by F ⊆ G , defined in Eq.(2.5). In fact, in risk theory the claims intensity is a mathematical object and its realizationsare not directly observed by economic agents (see [Grandell, 1991, Chapter 2]). In practice theinsurer relies on an estimation of the intensity and this is based on the information at disposal,which is made of the accidents realizations. This is the basic idea behind the filtering techniques.We further extend this concept to the claim sizes distribution, which is included in the filter.That is, the insurer estimates the intensity and the size distribution at the same time.In this framework we suppose that the gross risk premium rate { c t } t ∈ [0 ,T ] is an F -predictablenonnegative process (the insurance company chooses the premium based on the information flow)such that E [∫ T c t dt ] < + ∞ . (2.7)The insurer can subscribe a generic reinsurance contract with retention level u ∈ [0 , I ], where I > I = + ∞ ), transferring part of her risks to the reinsurer. More precisely,we model the retained losses using a generic self-insurance function g ( z, u ) : [0 , + ∞ ) × [0 , I ] → [0 , + ∞ ) which characterizes the reinsurance agreement. Remark 2.1.
Here we recall some useful properties of the self-insurance function according tothe classical risk theory : • g is increasing in both the variables z, u ; moreover, it is continuous in u ∈ [0 , I ] ; • g ( z, u ) ≤ z ∀ u ∈ [0 , I ] , because the retained loss is always less or equal than the claimamount; • g ( z,
0) = 0 ∀ z ∈ [0 , + ∞ ) , because u = 0 is the full reinsurance; • g ( z, I ) = z ∀ z ∈ [0 , + ∞ ) , because u = I is the null reinsurance. Our general formulation includes standard reinsurance agreements as special cases.
Example 2.3.
Under a proportional reinsurance the insurer transfers a percentage u of anyfuture loss, hence I = 1 and g ( z, u ) = uz, u ∈ [0 , . Under an excess-of-loss policy the reinsurer covers all the losses which overshoot a threshold u ,that is I = + ∞ and g ( z, u ) = z ∧ u, u ∈ [0 , + ∞ ) . In order to continuously buy a reinsurance agreement, the primary insurer pays a reinsur-ance premium { q ut } t ∈ [0 ,T ] , which is an F -predictable nonnegative process satisfying the followingassumption. Assumption 2.2. (Reinsurance premium) We assume that the reinsurance premium admits thefollowing representation: q ut ( ω ) = q ( t, ω, u ) ∀ ( t, ω, u ) ∈ [0 , T ] × Ω × [0 , I ] , for a given function q ( t, ω, u ) : [0 , T ] × Ω × [0 , I ] → [0 , + ∞ ) continuous and decreasing in u , withpartial derivative ∂q ( t,ω,u ) ∂u continuous in u . In the rest of the paper ∂q ( t,ω, ∂u and ∂q ( t,ω,I ) ∂u areinterpreted as right and left derivatives, respectively.In the sequel it is natural to assume that q ( t, ω, I ) = 0 ∀ ( t, ω ) ∈ [0 , T ] × Ω , See [Schmidli, 2008, Chapter 4] or [Schmidli, 2018]. ecause a null protection is not expensive. Moreover, we prevent the insurer from gaining arisk-free profit by assuming that q ( t, ω, > c t ∀ ( t, ω ) ∈ [0 , T ] × Ω . The reinsurance premium associated with a dynamic reinsurance strategy { u t } t ∈ [0 ,T ] will be de-noted by { q ut } t ∈ [0 ,T ] as well, with the obvious meaning depending on context.Finally, we assume the following integrability condition: E [∫ T q t dt ] < + ∞ . As mentioned above, the premia are F -predictable. This is a natural assumption in our con-text, because all the economic agents decisions are based on the common available information,which is described by F .Under these hypotheses, the surplus (or reserve) process associated with a given reinsurancestrategy { u t } t ∈ [0 ,T ] is described by the following SDE: dR ut = [ c t − q ut ] dt − ∫ + ∞ g ( z, u t ) m ( dt, dz ) , R u = R ∈ R + . (2.8)Furthermore, we allow the insurer to invest her surplus in a risk-free asset (bond or bankaccount) with constant rate R >
0. As a consequence, the insurer’s wealth { X ut } t ∈ [0 ,T ] associatedwith a given strategy { u t } t ∈ [0 ,T ] follows this dynamic: dX ut = dR ut + RX ut dt, X u = R ∈ R + . (2.9) Remark 2.2.
It can be verified that the solution to the SDE (2.9) is given by X ut = R e Rt + ∫ t e R ( t − r ) [ c r − q ur ] dr − ∫ t ∫ + ∞ e R ( t − r ) g ( z, u r ) m ( dr, dz ) . (2.10)Now we are ready to formulate the optimization problem of an insurance company whichsubscribes a reinsurance contract with a dynamic retention level { u t } t ∈ [0 ,T ] . The objective is tomaximize the expected utility of the terminal wealth:sup u ∈U E [ U ( X uT ) ] , where U : R → [0 , + ∞ ) is the utility function representing the insurer’s preferences and U theclass of admissible strategies (see Definition 2.1 below). Since only a partial information is avail-able to the insurer and it is described by the filtration F , the retention level u turns out to bean F -predictable process and a control problem with partial information arises.We focus on CARA ( Constant Absolute Risk Aversion ) utility functions, whose general ex-pression is given by U ( x ) = 1 − e − ηx , x ∈ R , where η > u ∈U E [ − e − ηX uT ] . (2.11) Definition 2.1 (Admissible strategies) . We denote by U the set of all the admissible strategies,which are all the F -predictable processes { u t } t ∈ [0 ,T ] with values in [0 , I ] such that E [ e − ηX uT ] < + ∞ . When we want to restrict the controls to the time interval [ t, T ] , we will use the notation U t .
6e can show that U is a nonempty class under suitable hypotheses. Assumption 2.3.
The following conditions hold good: E [ e ηe RT C T ] < + ∞ , (2.12) E [ e ηe RT ∫ T e − Rt q t dt ] < + ∞ . (2.13) Proposition 2.2.
Under Assumption 2.3 every F -predictable process { u t } t ∈ [0 ,T ] with values in [0 , I ] is admissible, that is u ∈ U .Proof. By our hypotheses, taking into account that q ut ≤ q t ∀ t ∈ [0 , T ] and ∀ u ∈ U (see Assump-tion 2.2) and using the well-known inequality ab ≤ ( a + b ) ∀ a, b ∈ R , we have that E [ e − ηX uT ] = E [ e − ηe RT R e − η ∫ T e R ( T − t ) ( c t − q ut ) dt e η ∫ T ∫ + ∞ e R ( T − t ) g ( z,u t ) m ( dt,dz ) ] ≤ E [ e η ∫ T e R ( T − t ) q t dt e ηe RT ∫ T ∫ + ∞ z m ( dt,dz ) ] ≤ ( E [ e ηe RT ∫ T e − Rt q t dt ] + E [ e ηe RT C T ]) < + ∞ , hence Definition 2.1 is satisfied.A sufficient condition for Eq. (2.12) can be obtained by the following lemma with the choice p = 2. Lemma 2.1.
Let p > and assume that there exists an integrable function Φ p : [0 , T ] → (0 , + ∞ ) such that ∫ + ∞ ( e pηe RT z − ) F Z ( t, y, dz ) ≤ Φ p ( t ) ∀ ( t, y ) ∈ [0 , T ] × R . (2.14) Then the following property holds good: E [ e pηe RT C t ] < + ∞ ∀ t ∈ [0 , T ] . (2.15) Proof.
Since { C t } t ∈ [0 ,T ] is a pure-jump process (see Eq. (2.6)), we have that e pηe RT C t = e pηe RT C + ∑ s ≤ t ( e pηe RT C s − e pηe RT C s − ) = 1 + ∑ s ≤ t e pηe RT C s − ( e pηe RT ∆ C s − ) = 1 + ∫ t e pηe RT C s − ∫ + ∞ ( e pηe RT z − ) m ( ds, dz ) . Taking the expectation, by (2.4), (2.2) and (2.14) we get that E [ e pηe RT C t ] = 1 + E [∫ t e pηe RT C s − ∫ + ∞ ( e pηe RT z − ) λ s F Z ( s, Y s , dz ) ds ] ≤ ∫ t E [ e pηe RT C s ] Φ p ( s ) ds. Applying Gronwall’s lemma we finally obtain that E [ e pηe RT C t ] ≤ e Λ ∫ t Φ p ( s ) ds . emark 2.3. Let us denote by m Z ( k ) . = E [ e kZ ] , k ∈ R , the moment generating function of Z .Assuming F Z ( t, y, dz ) = F Z ( dz ) as in Example 2.1, the condition (2.14) is equivalent to m Z ( pηe RT ) < + ∞ . In particular, in view of Lemma 2.1, m Z (2 ηe RT ) < + ∞ implies Eq. (2.12) .As special cases we may consider the following distribution functions: • if Z ∼ Γ( α, ζ ) we have that m Z ( k ) = Γ( α )( ζ − k ) ∀ k < ζ , where Γ denotes the gamma function;hence Eq. (2.12) is fulfilled for any ζ > ηe RT ; • if Z is exponentially distributed, then Z ∼ Γ(1 , ζ ) and hence the same condition ζ > ηe RT applies; • if Z has a truncated normal distribution on the interval [0 , + ∞ ) , then m Z ( k ) = e µk + σ k − N ( − µσ − σk )1 − N ( − µσ ) ∀ k > , where N denotes the standard normal distribution function. Remark 2.4.
Let us consider the special case of complete information. We denote by { S ut } t ∈ [0 ,T ] the insurer’s wealth in a full information framework, that is S ut = R e Rt + ∫ t e R ( t − r ) [ ¯ c r − ¯ q ur ] dr − ∫ t ∫ + ∞ e R ( t − r ) g ( z, u r ) m ( dr, dz ) , where the G -predictable processes { ¯ c t } t ∈ [0 ,T ] and { ¯ q t } t ∈ [0 ,T ] denote the insurance and the reinsur-ance premium, respectively. In order to simplify the comparison, the full and the partial informa-tion frameworks are defined in a similar way. U G denotes the class of admissible strategies andit is defined as in Definition 2.1, replacing F with G and X ut with S ut . Under Assumption 2.3,as in Proposition 2.2, we can prove the admissibility of every G -predictable process. Hence, sinceany F -predictable process is also G -predictable, we get U ⊆ U G . We take the same insurancepremia c t = ¯ c t and reinsurance premia q ut = ¯ q ut ∀ u ∈ U . In this simple context, we can readilyget that E [ e − ηX uT ] = E [ e − ηS uT ] ∀ u ∈ U , and, as a consequence, inf u ∈U G E [ e − ηS uT ] ≤ inf u ∈U E [ e − ηS uT ] = inf u ∈U E [ e − ηX uT ] . In words, the complete information allows the insurer to improve her result. However, we pointout that such an expected result is no longer easy to prove in general (for example when thepremia do not coincide).In Section 5 we will compare the optimal strategies under partial information with those undercomplete information in some special cases.
In the previous subsection we have introduced the partially observable problem. In order to studyit, we need to reduce it to an equivalent problem with complete information. This can be achievedby deriving the compensator m π ( dt, dz ) of the random measure given in Eq. (2.1), that is theinsurer’s loss process, with respect to its internal filtration F , which represents the information atdisposal to the insurance and the reinsurance companies. This result can be obtained by solvinga filtering problem with marked point process observations. It is well known that the filter, thatis the conditional distribution of Y t given the σ -algebra F t , for any t ∈ [0 , T ], provides the bestmean-squared estimate of the unobservable stochastic factor Y from the available information.8recisely, the filter is the F -adapted c`adl`ag process { π t ( f ) } t ∈ [0 ,T ] taking values in the space ofprobability measures on R defined by π t ( f ) = E [ f ( t, Y t ) | F t ] , for any measurable function f : [0 , T ] × R → R such that E [ | f ( t, Y t ) | ] < + ∞ ∀ t ∈ [0 , T ].By applying [Ceci and Colaneri, 2012, Proposition 2.2], we can derive m π ( dt, dz ). Lemma 2.2.
The random measure m ( dt, dz ) given in (2.1) has F -dual predictable projection m π ( dt, dz ) given by π t − ( λF Z ( dz )) dt, that is, the following expression holds for any A ∈ B ([0 , + ∞ )) m π ( dt, A ) . = π t − ( λ ( t, · ) F Z ( t, · , A )) dt, (2.16) where π t ( λ ( t, · ) F Z ( t, · , A )) = E [ λ ( t, Y t ) F Z ( t, Y t , A ) | F t ] and π t − denotes the left version of theprocess π t . Remark 2.5.
By definition of dual predictable projection, for every nonnegative, F -predictableand [0 , + ∞ ) -indexed process { H ( t, z ) } t ∈ [0 ,T ] we have that E [∫ T ∫ + ∞ H ( t, z ) m ( dt, dz ) ] = E [∫ T ∫ + ∞ H ( t, z ) λ t F Z ( t, Y t , dz ) dt ] = E [∫ T ∫ + ∞ H ( t, z ) π t − ( λF Z ( dz )) dt ] . By Remark 2.5 we can rewrite the classical premium calculation principles adapting them toour dynamic and partially observable context via the filter process . Example 2.4 (Premium calculation principles) . Under the expected value principle, the expectedrevenue covers the expected losses plus a profit which is proportional to the expected losses: c t = (1 + θ i ) ∫ + ∞ z π t − ( λF Z ( dz )) ,q ut = (1 + θ ) ∫ + ∞ ( z − g ( z, u t )) π t − ( λF Z ( dz )) , (2.17) where θ > θ i > represent the safety loadings.Under the variance premium principle, the expected gain is proportional to the variance of thelosses instead: c t = ∫ + ∞ z π t − ( λF Z ( dz )) + θ i ∫ + ∞ z π t − ( λF Z ( dz )) ,q ut = ∫ + ∞ ( z − g ( z, u t )) π t − ( λF Z ( dz )) + θ ∫ + ∞ ( z − g ( z, u t )) π t − ( λF Z ( dz )) , (2.18) for some safety loadings θ > θ i > . Observe that in these examples the premium at time t depends on the estimate of the compensator of the loss process given the available informationimmediately before time t , that is π t − ( λF Z ( dz )) dt .A formal derivation of these premium calculation rules in a dynamic context can be found in[Brachetta and Ceci, 2019b] and [Brachetta and Ceci, 2019a]. Filtering problems with marked point process observations have been widely investigated inthe literature, see [Br´emaud, 1981] and more recently [Ceci and Gerardi, 2006] and [Ceci, 2006].See also [Ceci and Colaneri, 2012] and [Ceci and Colaneri, 2014] for jump-diffusion observations.Here, starting from the existing literature, we derive an explicit formula for the filter under See [Young, 2006] for the original formulation in a static framework. Y . Precisely, we assume Y to be a c`adl`ag Markovprocess, but we do not assign any specific dynamics to Y . More details can be found in Appendix.Let us denote by L Y the Markov generator of Y with domain D Y , that is for every function f ∈ D Y ⊆ C b ([0 , T ] × R ) f ( t, Y t ) = f ( t , y ) + ∫ tt L Y f ( s, Y s ) ds + M Yt , t ∈ [0 , T ] , for some F Y -martingale { M Yt } t ∈ [0 ,T ] and ( t , y ) ∈ [0 , T ] × R . Assumption 2.4.
We assume the following standard hypotheses: • for any initial value ( t , y ) ∈ [0 , T ] × R the martingale problem for the operator L Y is wellposed on the space of c`adl`ag trajectories (this is true, for instance, when Y is the uniquestrong solution of a SDE for any initial values ( t , y ) ∈ [0 , T ] × R ); • L Y f ∈ C b ([0 , T ] × R ) for any f ∈ D Y ; • D Y is an algebra dense in C b ([0 , T ] × R ) . For simplicity, we assume no common jump times between Y and m ( dt, dz ) (we should specifythe dynamic for Y to remove such a simplification). Proposition 2.3.
Under Assumption 2.4, letting y ∈ R be a fixed initial value for Y at time t = 0 , the filter π can be obtained by the following recursive procedure • π ( f ) = f (0 , y ) , ∀ t ∈ (0 , T ) π t ( f ) = E [ f ( t, Y t ) e − ∫ t λ ( r,Y r ) dr | Y = y ] E [ e − ∫ t λ ( r,Y r ) dr | Y = y ] ; • at a jump time T n , n ≥ : π T n ( f ) = W ( T n , π T − n , Z n ) . = dπ T − n ( λF Z f ) dπ T − n ( λF Z ) ( Z n ) , (2.19) where dπ t − ( λF Z f ) dπ t − ( λF Z ) ( z ) denotes the Radon-Nikodym derivative of the measure π t − ( λF Z ( dz ) f ) with respect to π t − ( λF Z ( dz )) ; • between two consecutive jump times, t ∈ ( T n , T n +1 ) , n ≥ : π t ( f ) = E n [ f ( t, Y t ) e − ∫ ts λ ( r,Y r ) dr ] | s = T n E n [ e − ∫ ts λ ( r,Y r ) dr ] | s = T n , where E n denotes the conditional expectation given the distribution Y T n equal to π T n .Proof. The results are derived in Appendix.Similarly to [Ceci and Gerardi, 2006, Section 3.3], by Proposition 2.3 we can write a recursivealgorithm to approximate the filter. We conclude the section with some special cases. Thefollowing results are discussed in Appendix.
Remark 2.6 (Known jump size distribution and unknown intensity) . In the special case where F Z ( t, y, dz ) = F Z ( dz ) , that is, the insurance company has complete knowledge on the claim sizedistribution and partial information on the claim arrival intensity. Eq. (2.19) reduces to π T n ( f ) = W ( T n , π T − n ) = π T − n ( λf ) π T − n ( λ ) , (2.20) See [Ethier and Kurtz, 1986] for details about martingale problems. ee Example A.1 in Appendix.If Y takes values in a discrete set S = { , , . . . } , defining the functions f i ( y ) := y = i , i ∈ S ,the filter is completely described via the knowledge of π t ( i ) := π t ( f i ) = P ( Y t = i | F t ) , i ∈ S ,because for every function f we have that π i ( f ) = ∑ i ∈S f ( i ) π t ( i ) . Eq. (2.19) reads as π T n ( i ) = d ( λ ( T n , i ) F Z ( T n , i, dz ) π T − n ( i )) d ( ∑ j ∈S λ ( T n , j ) F Z ( T n , j, dz ) π T − n ( j )) ( Z n ) , (2.21) which, in the special case F Z ( t, y, dz ) = F Z ( dz ) , simplifies to π T n ( i ) = W i ( T n , π T − n ) . = λ ( T n , i ) π T − n ( i ) ∑ j ∈S λ ( T n , j ) π T − n ( j ) , (2.22) see Example A.3 in Appendix. Remark 2.7 (Markov Modulated Risk Model with infinitely many states) . If Y takes valuesin a discrete set S = { , , . . . } , the random measure m ( dt, dz ) in (2.1) has F -dual predictableprojection given by m π ( dt, dz ) = ∑ i ∈S π t − ( i ) λ ( t, i ) F Z ( t, i, dz ) dt. In particular, under the Markov Modulated Risk Model (see Example 2.2) we get m π ( dt, dz ) = M ∑ i =1 π t − ( i ) λ ( i ) F iZ ( dz ) dt. and by Eq. (2.22) π T n ( i ) = W i ( π T − n ) = λ i π T − n ( i ) ∑ Mj =1 λ j π T − n ( j ) , (2.23) see Example A.2 in Appendix. [Liang and Bayraktar, 2014] consider this case under the assump-tion that the claim distribution for any state i = 1 , . . . M admits density, that is F iZ ( dz ) = f i ( z ) dz .
3. The BSDE approach
As usual in stochastic control problems, we introduce the dynamic problem associated to (2.11).For the sake of notational simplicity, we study the corresponding minimization problem for thefunction e − ηx . Precisely, for any admissible control u ∈ U let us define the Snell envelope: J ut . = ess inf ¯ u ∈U ( t,u ) E [ e − ηX ¯ uT | F t ] , (3.1)where U ( t, u ) denotes the class U restricted to the controls ¯ u such that ¯ u s = u s ∀ s ≤ t , for agiven arbitrary control u ∈ U .Let us introduce the discounted wealth { ¯ X ut . = e − Rt X ut } t ∈ [0 ,T ] , that is¯ X ut = R + ∫ t e − Rs [ c s − q us ] ds − ∫ t ∫ + ∞ e − Rs g ( z, u s ) m ( ds, dz ) , t ∈ [0 , T ] . (3.2)Then, by Eq. (2.10) we get J ut = e − η ¯ X ut e RT V t , (3.3)where we define the value process V t . = ess inf ¯ u ∈U t E [ e − ηe RT ( ¯ X ¯ uT − ¯ X ¯ ut ) | F t ] , (3.4)11ith U t denoting the class of admissible controls restricted to the time interval [ t, T ] (see Defini-tion 2.1).By Eqs. (3.2) and (3.3) it is easy to show that J ut = e − η ( ¯ X ut − ¯ X It ) e RT e − η ¯ X It e RT V t = e η ( ¯ X It − ¯ X ut ) e RT J It , (3.5)and V t = e η ¯ X It e RT J It , (3.6)where J It denotes the Snell envelope associated to u = I (null reinsurance).The goal of this section is to dynamically characterize the value process by using a BSDE-based approach. The BSDE method works well in non-Markovian settings, where the classicalstochastic control approach based on the Hamilton-Jacobi-Bellman equation does not apply. Sev-eral papers (see e.g. [El Karoui et al., 1997], [Ceci and Gerardi, 2011], [Lim and Quenez, 2011]and references therein) deal with stochastic optimization problems in finance by means of BSDEs.For insurance applications the reader can refer to the recent textbook [Delong, 2013]. Moreover,this approach is also well suited to solve stochastic control problems under partial informationin presence of an infinite-dimensional filter process (see e.g. [Ceci, 2004] and [Ceci, 2012], wherepartially observed power utility maximization problems in financial markets are solved by apply-ing this approach). Proposition 3.1.
Under Assumption 2.3 we have that E [( sup t ∈ [0 ,T ] J It ) ] < + ∞ . (3.7) Proof.
By Eq. (3.2) for u = I (null reinsurance) we have that¯ X It = R + ∫ t e − Rs c s ds − ∫ t ∫ + ∞ e − Rs z m ( ds, dz ) . By definition of V t (see Eq. (3.4)), since u = I ∈ U ≤ V t ≤ E [ e − ηe RT ( ¯ X IT − ¯ X It ) | F t ] ≤ E [ e ηe RT ( C T − C t ) | F t ] P -a.s. ∀ t ∈ [0 , T ] . Analogously, by definition of J It (see Eq. (3.3)) we immediately get0 ≤ J It = e − η ¯ X It e RT V t ≤ e ηC t e RT E [ e ηe RT ( C T − C t ) | F t ]= E [ e ηe RT C T | F t ] P -a.s. ∀ t ∈ [0 , T ] . It follows that J It ≤ E [ e ηe RT C T | F t ] . = m t , where { m t } t ∈ [0 ,T ] is an F -martingale. By Doob’s martingale inequality, we have that E [( sup t ∈ [0 ,T ] J It ) ] ≤ E [( sup t ∈ [0 ,T ] m t ) ] ≤ E [ m T ]= 4 E [ e ηe RT C T ] < + ∞ . { J It } t ∈ [0 ,T ] solves a BSDE driven by the compensatedjump measure m ( dt, dz ) − π t − ( λF Z ( dz )) dt . In order to derive this BSDE, we need the followingadditional hypotheses. Strengthening the assumptions is useful for deriving the BSDE at thisstage, but in the Verification Theorem (see Theorem 3.1 below) we will come back to the weakerAssumption 2.3. Assumption 3.1.
The following conditions are satisfied: E [ e ηpe RT C T ] < + ∞ ∀ p ≥ , (3.8) E [ e ηpe RT ∫ T e − Rs q s ds ] < + ∞ ∀ p ≥ . (3.9) Remark 3.1.
Under the classical premium calculation principles (2.17) and (2.18) , Eq. (3.9) is fulfilled if we take the claim sizes distribution F Z ( t, y, dz ) = F Z ( dz ) such that ∫ + ∞ z F Z ( dz ) < + ∞ , In fact, in this case q t is a bounded process and hence Eq. (3.9) is clearly satisfied. Proposition 3.2 (Bellman’s optimality principle) . Under Assumption 3.1 the following state-ments hold good:1. { J ut } t ∈ [0 ,T ] is an F -sub-martingale for any u ∈ U ;2. { J u ∗ t } t ∈ [0 ,T ] is an F -martingale if and only if u ∗ ∈ U is an optimal control.Proof. By [Lim and Quenez, 2011, Prop. 4.1], the result is valid if ∀ u ∈ U and ∀ p ≥ E [ sup s ∈ [ t,T ] e − ηpX ut,x ( s ) ] < + ∞ ∀ ( t, x ) ∈ [0 , T ] × R , where { X ut,x ( s ) } s ∈ [ t,T ] denotes the solution to Eq. (2.9) with initial condition ( t, x ) ∈ [0 , T ] × R .We observe that e − ηpX ut,x ( s ) ≤ e ηpe Rs ∫ st e − Rr q ur dr e ηpe Rs C s ≤ ( e ηpe Rs ∫ st e − Rr q ur dr + e ηpe Rs C s ) P -a.s. ∀ t ∈ [0 , T ] , hence ∀ ( t, x ) ∈ [0 , T ] × R we get E [ sup s ∈ [ t,T ] e − ηpX ut,x ( s ) ] ≤ ( E [ e ηpe RT ∫ T e − Rs q s ds ] + E [ e ηpe RT C T ] ) < + ∞ . Remark 3.2.
Under Assumption 3.1 we can apply Bellman’s optimality principle (see Proposi-tion 3.2). Since u = I ∈ U , { J It } t ∈ [0 ,T ] is an F -sub-martingale. Consequently, by Doob-Meyerdecomposition and the martingale representation theorems , it admits the following expression: J It = ∫ t ∫ + ∞ Γ( s, z ) ( m ( ds, dz ) − π s − ( λF Z ( dz )) ds ) ) + A t , (3.10) where by (3.7) Γ( t, z ) is a [0 , + ∞ ) -indexed F -predictable process such that E [∫ T ∫ + ∞ | Γ( s, z ) | π s − ( λF Z ( dz )) ds ] < + ∞ , and { A t } t ∈ [0 ,T ] is an increasing F -predictable process such that E [ ∫ T A s ds ] < + ∞ . E.g. see [Br´emaud, 1981, Theorem T8]. emma 3.1 (Snell envelope decomposition) . Under Assumption 3.1, for any u ∈ U the Snellenvelope { J ut } t ∈ [0 ,T ] admits the following representation: dJ ut = dM ut + e η ( ¯ X It − ¯ X ut ) e RT [ A t − f ( t, Γ( t, z ) , J It , u t ) ] dt, (3.11) where M ut . = ∫ t e η ( ¯ X Is − − ¯ X us − ) e RT ∫ + ∞ Γ( s, z ) e − ηe R ( T − s ) ( z − g ( z,u s )) ( m ( ds, dz ) − π s − ( λF Z ( dz )) ds ) ) + ∫ t J Is − e η ( ¯ X Is − − ¯ X us − ) e RT ∫ + ∞ ( e − ηe R ( T − s ) ( z − g ( z,u s )) − )( m ( ds, dz ) − π s − ( λF Z ( dz )) ds ) ) (3.12) is an F -martingale and f ( t, Γ( t, z ) , J It , u t ) . = − J It − ηe R ( T − t ) q ut − ∫ + ∞ ( J It − + Γ( t, z ) )( e − ηe R ( T − t ) ( z − g ( z,u t )) − ) π t − ( λF Z ( dz )) . (3.13) Proof.
Since J ut = e η ( ¯ X It − ¯ X ut ) e RT J It by Eq. (3.5), we focus on the computation of the latter term.By the product rule for stochastic integrals we get that d ( e η ( ¯ X It − ¯ X ut ) e RT J It ) = e η ( ¯ X It − − ¯ X ut − ) e RT dJ It + J It − d ( e η ( ¯ X It − ¯ X ut ) e RT )+ d (∑ s ≤ t ∆ J Is ∆ e η ( ¯ X Is − ¯ X us ) e RT ) . (3.14)Let us evaluate (3.14) item by item. Using the expression (3.10) we can easily obtain the firstterm. By Eq. (3.2) we get¯ X It − ¯ X ut = ∫ t e − Rs q us ds − ∫ t ∫ + ∞ e − Rs ( z − g ( z, u s )) m ( ds, dz ) . (3.15)Hence by Itˆo’s formula we have that d ( e η ( ¯ X It − ¯ X ut ) e RT ) = ηe RT e η ( ¯ X It − ¯ X ut ) e RT e − Rt q ut dt + d (∑ s ≤ t e η ( ¯ X Is − − ¯ X us − ) e RT ( e ηe RT ( ( ¯ X Is − ¯ X us ) − ( ¯ X Is − − ¯ X us − ) ) − )) = ηe RT e η ( ¯ X It − ¯ X ut ) e RT e − Rt q ut dt + e η ( ¯ X It − − ¯ X ut − ) e RT ∫ + ∞ ( e − ηe R ( T − t ) ( z − g ( z,u t )) − ) m ( dt, dz ) . By the last equation we also find out that d (∑ s ≤ t ∆ J Is ∆ e η ( ¯ X Is − ¯ X us ) e RT ) = e η ( ¯ X It − − ¯ X ut − ) e RT ∫ + ∞ Γ( t, z ) ( e − ηe R ( T − t ) ( z − g ( z,u t )) − ) m ( dt, dz ) . Let us come back to (3.14). We have just obtained that d ( e η ( ¯ X It − ¯ X ut ) e RT J It ) = e η ( ¯ X It − − ¯ X ut − ) e RT [∫ + ∞ Γ( t, z ) ( m ( dt, dz ) − π t − ( λF Z ( dz )) dt ) ) + dA t ] + J It − ηe RT e η ( ¯ X It − ¯ X ut ) e RT e − Rt q ut dt + J It − e η ( ¯ X It − − ¯ X ut − ) e RT ∫ + ∞ ( e − ηe R ( T − t ) ( z − g ( z,u t )) − ) m ( dt, dz )+ e η ( ¯ X It − − ¯ X ut − ) e RT ∫ + ∞ Γ( t, z ) ( e − ηe R ( T − t ) ( z − g ( z,u t )) − ) m ( dt, dz ) . d ( e η ( ¯ X It − ¯ X ut ) e RT J It )= e η ( ¯ X It − − ¯ X ut − ) e RT ∫ + ∞ Γ( t, z ) e − ηe R ( T − t ) ( z − g ( z,u t )) ( m ( dt, dz ) − π t − ( λF Z ( dz )) dt ) ) + J It − e η ( ¯ X It − − ¯ X ut − ) e RT ∫ + ∞ ( e − ηe R ( T − t ) ( z − g ( z,u t )) − )( m ( dt, dz ) − π t − ( λF Z ( dz )) dt ) ) + e η ( ¯ X It − − ¯ X ut − ) e RT dA t + J It − ηe RT e η ( ¯ X It − ¯ X ut ) e RT e − Rt q ut dt + e η ( ¯ X It − − ¯ X ut − ) e RT ∫ + ∞ ( J It − + Γ( t, z ) )( e − ηe R ( T − t ) ( z − g ( z,u t )) − ) π t − ( λF Z ( dz )) dt. By definition of { M ut } t ∈ [0 ,T ] and { f ( t, Γ( t, z ) , J It , u t ) } t ∈ [0 ,T ] (see Eqs. (3.12) and (3.13), respec-tively), we obtain the expression (3.11).In order to complete the proof, we need to show that { M ut } t ∈ [0 ,T ] is an F -martingale for any u ∈ U , that is E [∫ T e η ( ¯ X Is − − ¯ X us − ) e RT ∫ + ∞ | Γ( s, z ) | e − ηe R ( T − s ) ( z − g ( z,u s )) π s − ( λF Z ( dz )) ds ] < + ∞ , E [∫ T J Is − e η ( ¯ X Is − − ¯ X us − ) e RT ∫ + ∞ ⏐⏐⏐ e − ηe R ( T − s ) ( z − g ( z,u s )) − ⏐⏐⏐ π s − ( λF Z ( dz )) ds ] < + ∞ . In the rest of the proof
C > E [∫ T e η ( ¯ X Is − − ¯ X us − ) e RT ∫ + ∞ | Γ( s, z ) | e − ηe R ( T − s ) ( z − g ( z,u s )) λ s F Z ( s, Y s , dz ) ds ] ≤ E [ e ηe RT ∫ T e − Rs q s ds ∫ T ∫ + ∞ | Γ( s, z ) | λ s F Z ( s, Y s , dz ) ds ] ≤ C E [ e ηe RT ∫ T e − Rs q s ds ] + C E [∫ T ∫ + ∞ | Γ( s, z ) | π s − ( λF Z ( dz )) ds ] < + ∞ . Now let us evaluate the second expectation. By Remark 2.5, Eq. (3.15) and Eq. (3.7) E [∫ T J Is − e η ( ¯ X Is − − ¯ X us − ) e RT ∫ + ∞ ⏐⏐⏐ e − ηe R ( T − s ) ( z − g ( z,u s )) − ⏐⏐⏐ λ s F Z ( s, Y s , dz ) ds ] ≤ Λ E [∫ T J Is − e ηe RT ∫ T e − Rr q r dr ds ] ≤ C ( E [∫ T | J Is − | ds ] + E [ e ηe RT ∫ T e − Rs q s ds ]) < + ∞ . Definition 3.1.
We introduce the following classes of stochastic processes: • L is the space of c`adl`ag F -adapted processes { ˆ J t } t ∈ [0 ,T ] such that E [∫ T | ˆ J t | dt ] < + ∞ . (3.16) • ˜ L is the space of [0 , + ∞ ) -indexed F -predictable processes { ˆΓ( t, z ) , z ∈ [0 , + ∞ ) } t ∈ [0 ,T ] suchthat E [∫ T ∫ + ∞ | ˆΓ( t, z ) | π t − ( λF Z ( dz )) dt ] < + ∞ . (3.17)15 roposition 3.3. Let { u ∗ t } t ∈ [0 ,T ] be an optimal control for the optimization problem (3.4) . Un-der Assumption 3.1 ( J It , Γ( t, z )) ∈ L × ˜ L is a solution to the following BSDE: J It = ξ − ∫ Tt ∫ + ∞ Γ( s, z ) ( m ( ds, dz ) − π s − ( λF Z ( dz )) ds ) ) − ∫ Tt ess sup u ∈U f ( s, Γ( s, z ) , J Is , u s ) ds, (3.18) where { f ( t, Γ( t, z ) , J It , u t ) } t ∈ [0 ,T ] is defined in (3.13) and ξ = e − ηX IT .Moreover, f ( t, Γ( t, z ) , J It , u t ) attains its maximum in u ∗ t , that is f ( t, Γ( t, z ) , J It , u ∗ t ) = ess sup u ∈U f ( t, Γ( t, z ) , J It , u t ) . (3.19) Proof.
For any admissible control u ∈ U , by Bellman’s optimality principle (Proposition 3.2) { J ut } t ∈ [0 ,T ] is an F -sub-martingale and thus by Eq. (3.11) we readily get ∀ u ∈ U A t ≥ f ( t, Γ( t, z ) , J It , u t ) P -a.s. ∀ t ∈ [0 , T ] . (3.20)Let { u ∗ t } t ∈ [0 ,T ] be an optimal control for the problem (3.4). By Bellman’s optimality principle { J u ∗ t } t ∈ [0 ,T ] is an F -martingale and by Lemma 3.1 this is true if only if A t = f ( t, Γ( t, z ) , J It , u ∗ t ) . Combining this result with (3.20) leads toess sup u ∈U f ( t, Γ( t, z ) , J It , u t ) ≥ f ( t, Γ( t, z ) , J It , u ∗ t ) = A t ≥ ess sup u ∈U f ( t, Γ( t, z ) , J It , u t ) , which implies Eq. (3.19). Now, using the Doob-Meyer representation (3.10), we conclude that( J It , Γ( t, z )) is a solution to (3.18), with the terminal condition easily derived by Eq. (3.3). Remark 3.3.
The process { f ( t, Γ( t, z ) , J It , u ∗ t ) } t ∈ [0 ,T ] (see Eq. (3.19) ) is non negative. Indeed,by Eq. (3.13) we immediately get f ( t, Γ( t, z ) , J It , u ∗ t ) ≥ f ( t, Γ( t, z ) , J It , I ) = 0 . Remark 3.4.
The process { J It } t ≥ completely determines the predictable random field { Γ( t, z ) , z ∈ [0 , + ∞ ) } t ≥ outside a null set with respect to the measure π t − ( λF Z ( dz ))( ω ) P ( dω ) dt . In fact, if ( J It , Γ( t, z )) and ( J It , Γ ( t, z )) satisfy the BSDE (3.18) , on the jump times of the random measure m ( dt, dz ) we necessarily have that Γ( T n , Z n ) = ∆ J IT n = Γ ( T n , Z n ) ∀ n ≥ . Hence, for any t ∈ [0 , T ] and C ∈ B ([0 , + ∞ ))0 = E [∫ t ∫ C | Γ( s, z ) − Γ ( s, z ) | m ( ds, dz ) ] = E [∫ t ∫ C | Γ( s, z ) − Γ ( s, z ) | π s − ( λF Z ( dz )) ds ] = 0 , (3.21) and this implies that Γ( t, z ) = Γ ( t, z ) π t − ( λF Z ( dz ))( ω ) P ( dω ) dt -a.e.. Recalling that V t = e η ¯ X It e RT J It (see Eq. (3.6)), using the Bellman’s optimality principle wehave connected the value process (3.4) to the solution of the BSDE (3.18). For this purpose, wemade extensive use the hypotheses included in Assumption 3.1. Now a verification argument isneeded. To this end, we will assume the weaker conditions given in Assumption 2.3. Proposition 3.4 (A general Verification Theorem) . Under Assumption 2.3, let us suppose thatthere exists an F -adapted process { D t } t ∈ [0 ,T ] such that . { D t e − η ¯ X ut e RT } t ∈ [0 ,T ] is an F -sub-martingale for any u ∈ U and an F -martingale for some u ∗ ∈ U ;2. D T = 1 .Then D t = V t and u ∗ is an optimal control.Proof. Using the terminal condition and the sub-martingale property, we have that for any t ∈ [0 , T ] E [ e − η ¯ X uT e RT | F t ] ≥ D t e − η ¯ X ut e RT ∀ u ∈ U , hence D t ≤ E [ e − ηe RT ( ¯ X uT − ¯ X ut ) | F t ] , which implies D t ≤ V t . Moreover, for u ∗ ∈ U we have that D t = E [ e − ηe RT ( ¯ X u ∗ T − ¯ X u ∗ t ) | F t ] ≥ V t . The two inequalities imply the thesis.
Theorem 3.1 (Verification Theorem) . Suppose that Assumption 2.3 is fulfilled. Let ( ˆ J t , ˆΓ( t, z )) ∈L × ˜ L be a solution to the BSDE (3.18) and let u ∗ = { u ∗ t } t ∈ [0 ,T ] be an F -predictable processsuch that ess sup u ∈U f ( t, ˆΓ( t, z ) , ˆ J t , u t ) = f ( t, ˆΓ( t, z ) , ˆ J t , u ∗ t ) ∀ t ∈ [0 , T ] . (3.22) Then { D t . = e η ¯ X It e RT ˆ J t } t ∈ [0 ,T ] is the value process of the optimal reinsurance problem, that is D t = V t (see Eq. (3.4) ), and u ∗ ∈ U is an optimal control.Proof. In view of the general Verification Theorem introduced in Proposition 3.4, let us considerthe stochastic process { D t e − η ¯ X ut e RT } t ∈ [0 ,T ] . Since e − η ¯ X ut e RT D t = e η ( ¯ X It − ¯ X ut ) e RT ˆ J t , by definition of D t , using the BSDE (3.18) and imitating the proof of Lemma 3.1, we have that d ( e − η ¯ X ut e RT D t ) = d ˆ M ut + e η ( ¯ X It − ¯ X ut ) e RT [ ess sup w ∈U f ( t, ˆΓ( t, z ) , ˆ J t , w t ) − f ( t, ˆΓ( t, z ) , ˆ J t , u t ) ] dt, where ˆ M ut is defined in Eq. (3.12) and f is given in Eq. (3.13) by replacing ( J It , Γ( t, z )) with( ˆ J t , ˆΓ( t, z )). In order to prove that { ˆ M ut } t ∈ [0 ,T ] is an F -martingale ∀ u ∈ U , we replicate thecalculations of the proof of Lemma 3.1. By Assumption 2.3 we obtain that E [∫ T e η ( ¯ X Is − − ¯ X us − ) e RT ∫ + ∞ | ˆΓ( s, z ) | e − ηe R ( T − s ) ( z − g ( z,u s )) λ s F Z ( s, Y s , dz ) ds ] ≤ C E [ e ηe RT ∫ T e − Rs q s ds ] + C E [∫ T ∫ + ∞ | ˆΓ( s, z ) | π s − ( λF Z ( dz )) ds ] < + ∞ , where C > E [∫ T ˆ J s − e η ( ¯ X Is − − ¯ X us − ) e RT ∫ + ∞ ⏐⏐⏐ e − ηe R ( T − s ) ( z − g ( z,u s )) − ⏐⏐⏐ λ s F Z ( s, Y s , dz ) ds ] ≤ ˜ C E [∫ T | ˆ J s | ds ] + ˜ C E [ e ηe RT ∫ T e − Rs q s ds ] < + ∞ , where ˜ C > u ∈ U ess sup w ∈U t f ( t, ˆΓ( t, z ) , ˆ J t , w t ) ≥ f ( t, ˆΓ( t, z ) , ˆ J t , u t ) , hence { e − η ¯ X ut e RT D t } t ∈ [0 ,T ] turns out to be an F -sub-martingale.Now let us consider the F -predictable process { u ∗ t } t ∈ [0 ,T ] satisfying Eq. (3.22). In this case theprevious inequality reads as an equality by definition of u ∗ , hence { e − η ¯ X u ∗ t e RT D t } t ∈ [0 ,T ] is an F -martingale. Finally, D T = e η ¯ X IT e RT ˆ J T = 1 . As announced, our statement follows by Proposition 3.4.
Remark 3.5.
Let us notice that f given in Eq. (3.13) is continuous in u ∈ [0 , I ] and underAssumption 2.3 every F -predictable process is admissible by Proposition 2.2. As a consequence,an optimal control exists as long as the BSDE (3.18) admits a solution ( ˆ J t , ˆΓ( t, z )) ∈ L × ˜ L .Precisely, there exists a measurable function u ∗ ( t, ω, γ ( · ) , j ) , with t ∈ [0 , T ] , ω ∈ Ω , γ : [0 , + ∞ ) → R , j ∈ [0 , + ∞ ) , such that f ( t, ω, γ ( · ) , j, u ∗ ( t, ω, γ, j )) = max u ∈ [0 ,I ] f ( t, ω, γ ( · ) , j, u ) (3.23) and u ∗ t = u ∗ ( t, ˆΓ( t, z ) , ˆ J t − ) is an optimal control. This topic will be developed further in Section 4. (3.18) In this section we deal with the solution to the BSDE (3.18), that provides our value process(3.4) in view of Theorem 3.1. Precisely, we discuss its existence and uniqueness.
Lemma 3.2.
Suppose that Eq. (2.12) is fulfilled. The final condition ξ = e − ηX IT of the BSDE (3.18) is square-integrable.Proof. Recalling that q It = 0 ∀ t ∈ [0 , T ] and g ( z, I ) = z ∀ z ∈ [0 , + ∞ ), by Eq. (2.10) we havethat e − ηX IT = e − ηR e RT e − η ∫ T e R ( T − r ) c r dr e η ∫ T ∫ + ∞ e R ( T − r ) z m ( dr,dz ) ≤ e ηe RT C T P -a.s. . The statement immediately follows by Eq. (2.12).Now we handle the problem of existence and uniqueness of a solution to (3.18).
Definition 3.2.
For any t ∈ [0 , T ] and ω ∈ Ω we denote by Θ( t, ω ) the space of all the functions γ : [0 , + ∞ ) → R such that ∫ + ∞ | γ ( z ) | π t − ( λF Z ( dz )) < + ∞ . In the sequel we use this short notation:
A . = { ( t, ω, γ ( · ) , j, u ) ∈ [0 , T ] × Ω × Θ( t, ω ) × [0 , + ∞ ) × [0 , I ] } . Correspondingly, we take¯
A . = { ( t, ω, γ ( · ) , j ) ∈ [0 , T ] × Ω × Θ( t, ω ) × [0 , + ∞ ) } . Theorem 3.2.
Suppose that the following hypotheses are fulfilled: the condition (2.12) is fulfilled; • the function q ( t, ω, u ) given in Assumption 2.2 is bounded;There exists a unique solution ( ˆ J t , ˆΓ( t, z )) ∈ L × ˜ L which solves the BSDE (3.18) .Proof. In order to apply the results of [Confortola and Fuhrman, 2013], let us notice that theclasses introduced in Definition 3.1 and Definition 3.2 are equivalent to those of the cited paper,except for the absence of a parameter β >
0; in fact, in our framework there is no need of this,because the compensator of the counting process { N t } t ∈ [0 ,T ] is { π t − ( λ ) } t ∈ [0 ,T ] and it is boundedby Λ > f be an F -predictable process defined on A by f ( t, ω, γ ( · ) , j, u ) . = − jηe R ( T − t ) q ut − ∫ + ∞ ( j + γ ( z ) )( e − ηe R ( T − t ) ( z − g ( z,u )) − ) π t − ( λF Z ( dz )) . (3.24)Since q ut is bounded by hypothesis, using the condition (2.2) and taking γ, γ ′ ∈ Θ( t, ω ) and j, j ′ ∈ [0 , + ∞ ), we have that f satisfies a Lipschitz condition uniformly in t, ω, u : | f ( t, ω, γ ′ ( · ) , j ′ , u ) − f ( t, ω, γ ( · ) , j, u ) | = | j ηe R ( T − t ) q ut + ∫ + ∞ ( j + γ ( z )) ( e − ηe R ( T − t ) ( z − g ( z,u )) − ) π t − ( λF Z ( dz )) − j ′ ηe R ( T − t ) q ut − ∫ + ∞ ( j ′ + γ ′ ( z )) ( e − ηe R ( T − t ) ( z − g ( z,u )) − ) π t − ( λF Z ( dz )) |≤ L | j − j ′ | + ⏐⏐⏐⏐∫ + ∞ ( γ ( z ) − γ ′ ( z ) )( e − ηe R ( T − t ) ( z − g ( z,u )) − ) π t − ( λF Z ( dz )) ⏐⏐⏐⏐ ≤ L | j − j ′ | + ∫ + ∞ | γ ( z ) − γ ′ ( z ) | π t − ( λF Z ( dz )) ≤ L | j − j ′ | + Λ (∫ + ∞ | γ ( z ) − γ ′ ( z ) | π t − ( λF Z ( dz )) ) ∀ t ∈ [0 , T ] , ω ∈ Ω , u ∈ [0 , I ] , for a suitable constant L >
0. It can be proved that sup u ∈ [0 ,I ] f ( t, ω, γ ( · ) , j, u ) preserves thisproperty, in fact ⏐⏐⏐⏐⏐ sup u ∈ [0 ,I ] f ( t, ω, γ ( · ) , j, u ) − sup u ∈ [0 ,I ] f ( t, ω, γ ′ ( · ) , j ′ , u ) ⏐⏐⏐⏐⏐ ≤ sup u ∈ [0 ,I ] | f ( t, ω, γ ( · ) , j, u ) − f ( t, ω, γ ′ ( · ) , j ′ , u ) |≤ L | j − j ′ | + Λ (∫ + ∞ | γ ( z ) − γ ′ ( z ) | π t − ( λF Z ( dz )) ) ∀ t ∈ [0 , T ] , ω ∈ Ω . Further, let us observe that f ( t, ω, , , u ) = 0 ∀ ( t, ω, u ) ∈ [0 , T ] × Ω × [0 , I ] and the BSDEterminal condition is square-integrable by Lemma 3.2. We can deduce that Hypothesis 3.1of [Confortola and Fuhrman, 2013] is fulfilled. Hypothesis 4.5 is satisfied as well, because ofRemark 3.5. Finally, our statement is a consequence of [Confortola and Fuhrman, 2013, Theorem3.4].Let us summarize the results of this section in the following theorem. Theorem 3.3.
Suppose that Assumption 2.3 is fulfilled and the reinsurance premium is bounded.Then ( J It , Γ( t, z )) ∈ L × ˜ L is the unique solution of the BSDE (3.18) . Moreover, let u ∗ ( t, ω, γ ( · ) , j ) be the maximizer of Eq. (3.23) , then u ∗ t = u ∗ ( t, Γ( t, z ) , J It − ) is an optimal control and the valueprocess in Eq. (3.4) admits the representation { V t . = e η ¯ X It e RT J It } t ∈ [0 ,T ] . roof. The BSDE (3.18) admits a unique solution by Theorem 3.2 and the existence of an optimalcontrol is guaranteed by Remark 3.5. This in turn implies that ( J It , Γ( t, z )) ∈ L × ˜ L is theunique solution of Eq. (3.18) by Proposition 3.3. Finally, the expression of the value process isobtained by Theorem 3.1.
4. The optimal reinsurance strategy
Eq. (3.23) suggests a natural way to find an optimal strategy. This is the main topic of thissection.
Proposition 4.1.
Assume g ( z, u ) differentiable in u ∈ [0 , I ] . Let f be defined by Eq. (3.24) andsuppose that it is strictly concave in u . Let the function u ∗ ( t, ω, γ, j ) be defined as follows: u ∗ ( t, ω, γ ( · ) , j ) = ⎧⎪⎨⎪⎩ t, ω, γ ( · ) , j ) ∈ A ˆ u ( t, ω, γ ( · ) , j ) ( t, ω, γ ( · ) , j ) ∈ ( A ∪ A I ) C I ( t, ω, γ ( · ) , j ) ∈ A I , (4.1) where A . = { ( t, ω, γ ( · ) , j ) ∈ ¯ A | − j ∂q t ∂u ≤ ∫ + ∞ ( j + γ ( z ) ) e − ηe R ( T − t ) z ∂g ( z, ∂u π t − ( λF Z ( dz )) } ,A I . = { ( t, ω, γ ( · ) , j ) ∈ ¯ A | − j ∂q It ∂u ≥ ∫ + ∞ ( j + γ ( z ) ) ∂g ( z, I ) ∂u π t − ( λF Z ( dz )) } , and < ˆ u ( t, ω, γ ( · ) , j ) < I is the solution to − j ∂q ut ∂u = ∫ + ∞ ( j + γ ( z ) ) e − ηe R ( T − t ) ( z − g ( z,u )) ∂g ( z, u ) ∂u π t − ( λF Z ( dz )) , (4.2) for any ( t, ω, γ ( · ) , j ) ∈ ( A ∪ A I ) C . Then u ∗ ( t, ω, γ ( · ) , j ) is the unique maximizer of f , that isEq. (3.23) is valid.Proof. Since f is continuous on the compact set [0 , I ], it admits a maximum. Moreover, itis concave and the uniqueness of the maximizer is guaranteed. Now let us evaluate the firstderivative of f : ∂f ( t, ω, γ ( · ) , j, u ) ∂u = − jηe R ( T − t ) ∂q ut ∂u − ∫ + ∞ ( j + γ ( z ) ) ηe R ( T − t ) e − ηe R ( T − t ) ( z − g ( z,u )) ∂g ( z, u ) ∂u π t − ( λF Z ( dz )) . (4.3)Since A = { ( t, ω, γ ( · ) , j ) ∈ ¯ A | ∂f ( t, ω, γ ( · ) , j, ∂u ≤ } ,A I = { ( t, ω, γ ( · ) , j ) ∈ ¯ A | ∂f ( t, ω, γ ( · ) , j, I ) ∂u ≥ } , by definition (see Eq. (4.3)), using the concavity of f we have that ∂f∂u is decreasing in u ∈ [0 , I ],hence A ∩ A = ∅ . Now there are only three possible cases. If ( t, ω, γ ( · ) , j ) ∈ A , f is decreasingin u ∈ [0 , I ] and the maximizer is u = 0. Similarly, if ( t, ω, γ ( · ) , j ) ∈ A I , f is increasing in u ∈ [0 , I ] and the maximizer is u = I . Finally, if ( t, ω, γ ( · ) , j ) ∈ ( A ∪ A I ) C , the maximizercoincides with the unique stationary point ˆ u ( t, ω, γ ( · ) , j ) ∈ (0 , I ), that is the solution to Eq.(4.2). 20 orollary 4.1. Assume g ( z, u ) differentiable in u ∈ [0 , I ] . Let f be defined by Eq. (3.24) and suppose that it is strictly concave in u . Suppose that Assumption 2.3 is fulfilled and let ( ˆ J t , ˆΓ( t, z )) ∈ L × ˜ L be a solution to the BSDE (3.18) . Let us define the control { u ∗ t . = u ∗ ( t, ω, ˆΓ( t, z ) , ˆ J t − ) } t ∈ [0 ,T ] , with the function u ∗ ( t, ω, γ, j ) given in Eq. (4.1) , that is u ∗ ( t, ω, ˆΓ( t, z ) , ˆ J t − ) = ⎧⎪⎨⎪⎩ t, ω ) ∈ ˜ A ˆ u ( t, ω, ˆΓ( t, z ) , ˆ J t − ) ( t, ω ) ∈ ( ˜ A ∪ ˜ A I ) C I ( t, ω ) ∈ ˜ A I , (4.4) where ˜ A . = { ( t, ω ) ∈ [0 , T ] × Ω | − ˆ J t − ∂q t ∂u ≤ ∫ + ∞ ( ˆ J t − + ˆΓ( t, z ) ) e − ηe R ( T − t ) z ∂g ( z, ∂u π t − ( λF Z ( dz )) } , ˜ A I . = { ( t, ω ) ∈ [0 , T ] × Ω | − ˆ J t − ∂q It ∂u ≥ ∫ + ∞ ( ˆ J t − + ˆΓ( t, z ) ) ∂g ( z, I ) ∂u π t − ( λF Z ( dz )) } , and < ˆ u ( t, ω, Γ( t, z ) , J t − ) < I is the solution to − ˆ J t − ∂q ut ∂u = ∫ + ∞ ( ˆ J t − + ˆΓ( t, z ) ) e − ηe R ( T − t ) ( z − g ( z,u )) ∂g ( z, u ) ∂u π t − ( λF Z ( dz )) . (4.5) Then { u ∗ t } t ∈ [0 ,T ] is an optimal control.Proof. By Proposition 2.2 u ∗ ∈ U . Since Eq. (3.23) holds by Proposition 4.1, then u ∗ is anoptimal control.Here we provide sufficient conditions for the concavity of f , which is the main hypothesis ofProposition 4.1. Proposition 4.2.
Suppose that the reinsurance premium q ut and the self-insurance function g ( z, u ) are linear or convex in u ∈ [0 , I ] . Then the function f given in Eq. (3.24) is strictlyconcave in u .Proof. It follows directly by Eq. (3.24).The following remark stress that the two hypotheses of the previous proposition are notmerely technical conditions.
Remark 4.1.
Both the classical premium calculation principles (2.17) and (2.18) and the pro-portional as well as the excess-of-loss reinsurance agreements satisfy the hypotheses of Proposition4.2. In the next section we provide the explicit form of the optimal strategy in some special cases.
Remark 4.2.
When, ∀ ( t, y ) ∈ [0 , T ] × R , the distribution F Z ( t, y, dz ) admits a density function f Z ( t, y, z ) , the differentiability of g in Proposition 4.1 can be weakened by the hypothesis of g differentiable in u ∈ [0 , I ] for almost every z ∈ [0 , + ∞ ) .
5. Some properties of the optimal reinsurance strategy
In this Section we investigate some properties of the optimal reinsurance strategy. In Subsection5.1 we prove that if the premia satisfy the Markovian property in the filter process, then thesame property applies to the optimal strategy. This means that the optimal strategy dependson the estimate of the environmental stochastic factor distribution given the available informa-tion. Next, in Subsection 5.2 we perform a sensitivity analysis and in Subsection 5.3 we givea comparison result with the full information case for some relevant examples. In particular,we extend the comparison made in [Liang and Bayraktar, 2014] for the Markov modulated riskmodel under the proportional reinsurance to the case of Y having infinitely many states and tothe excess of loss reinsurance contract. 21 .1. Markovianity in the filter process Assuming premia at time t depending on the filter process π t − , as in the classical premiumcalculation principles, see Example 2.4. Let P ( R ) be the space of probability measures on R endowed with the weak topology. Let us observe that { π t } t ∈ [0 ,T ] is an F -Markov process takingvalues in P ( R ) (see Eq. (A.1)). Then the value process { V t } t ∈ [0 ,T ] , given in (3.4), is such that V t = v ( t, π t − ), with v ( t, π ) measurable function on the space [0 , T ] × P ( R ).Let us observe that by Eq. (3.3) we have that J IT n − J IT − n = Γ( T n , Z n )= e − η ¯ X ITn e RT V T n − e − η ¯ X IT − n e RT V T − n = e − η ¯ X IT − n e RT ( V T n e ηZ n e R ( T − Tn ) − V T − n ) . Denote by W ( t, π, z ) : [0 , T ] × P ( R ) × [0 , + ∞ ) → P ( R ) a the measurable function suchthat (2.19) is fulfilled, that is π T n ( f ) = W ( T n , π T − n , Z n )( f ), ∀ f ∈ D Y . Then we have that V T n = v ( T n , W ( T n , π T − n , Z n )), so by Remark 3.4 we can writeΓ( t, z ) = e − η ¯ X It − e RT ( v ( t, W ( t, π t − , z )) e ηze R ( T − t ) − v ( t, π t − ) ) π t − ( λF Z ( dz ))( ω ) P ( dω ) dt -a.e. . and, as a consequence, ( J It − + Γ( t, z ) ) e − ηze R ( T − t ) = v ( t, W ( t, π t − , z )) e − η ¯ X It − e RT π t − ( λF Z ( dz ))( ω ) P ( dω ) dt -a.e. . (5.1)We are now able to prove that the reinsurance optimal strategy is a filter-feedback control,this means that at time t only depends on the estimate of the distribution of the environmentalstochastic factor immediately before time t . Proposition 5.1.
Assume that the premia c t and q ut , ∀ u ∈ [0 , I ] , at time t depend on the filterprocess π t − . Then the optimal reinsurance strategy is Markovian in the filter process, that is u ∗ t = u ∗ ( t, π t − ) , with u ∗ ( t, π ) being a measurable function of ( t, π ) ∈ [0 , T ] × P ( R ) .Proof. Recall that the optimal reinsurance strategy is the maximizer of f ( t, Γ( t, z ) , J It − , u ) (givenin Eq. (3.13)) over the class of admissible controls. By J It − = e − η ¯ X It − e RT v ( t, π t − ) , (5.2)and Eq. (5.1), one gets that f ( t, Γ( t, z ) , J It − , u ) = e − η ¯ X It − e RT h ( t, π t − , u ) ∀ u ∈ [0 , I ] , (5.3)where h ( t, π t − , u ) . = − v ( t, π t − ) ηe R ( T − t ) q ut + ∫ + ∞ v ( t, W ( t, π t − , z ))( e ηe R ( T − t ) z − e ηe R ( T − t ) g ( z,u ) ) π t − ( λF Z ( dz )) . (5.4)Hence our result follows by measurability selection theorems. Remark 5.1.
Notice that, assuming premia c t and q ut , ∀ u ∈ [0 , I ] , at time t depending of thefilter process π t − , the pair { ( ¯ X It , π t ) } t ∈ [0 ,T ] is an F -Markov process and J It = ˜ v ( t, ¯ X It , π t − ) with ˜ v ( t, x, π ) . = e − ηxe RT v ( t, π ) . In the Markov modulated risk model, that is when Y is a continuoustime Markov chain taking values in S = { , . . . , M } , the pair { ( ¯ X It , π t ) } t ∈ [0 ,T ] is an ( M +1) -dimensional F -Markov process and ˜ v ( t, x, π ) can be characterized in terms of the associatedHJB-equation, if it is regular enough. Concerning that point, see [Liang and Bayraktar, 2014],where the problem is discussed in a Markov modulated risk model and the authors make use ofa generalized HJB equation, introducing a weaker notion of differentiability. In the general casethis approach is not suitable since the filter is an infinite-dimensional process and this motivatesthe characterization in terms of BSDEs, as proposed in this paper. .2. Effect of the safety loading In this subsection we determine the effect of the reinsurance safety loading θ > g ( z, u ) = zu , u ∈ [0 ,
1] (seeExample 2.3) and under the expected value principle (see Example 2.4). We will show that thegreater is the value of θ , which implies a greater reinsurance premium, the greater will be theoptimal retention level. This is consistent with the classical law of demand in economics andwith existing results. Let B ( t, π ) . = ∫ + ∞ v ( t,W ( t,π,z )) v ( t,π ) zπ ( λF Z ( dz )) ∫ + ∞ zπ ( λF Z ( dz )) − , (5.5) D ( t, π ) . = ∫ + ∞ v ( t,W ( t,π,z )) v ( t,π ) e ηe R ( T − t ) z zπ ( λF Z ( dz )) ∫ + ∞ zπ ( λF Z ( dz )) − . (5.6) Proposition 5.2.
In the proportional reinsurance, under the expected value principle the optimalreinsurance strategy increases with respect the reinsurance safety loading θ . Furthermore, it isgiven by u ∗ ( t, π t − ) = ⎧⎪⎨⎪⎩ < θ ≤ B ( t, π t − )ˆ u ( t, π t − ) B ( t, π t − ) < θ ≤ D ( t, π t − )1 θ ≥ D ( t, π t − ) , (5.7) where B ( t, π ) and D ( t, π ) are defined in (5.5) and (5.6) , respectively, and < ˆ u ( t, π t − ) < I isthe unique solution to (1 + θ ) ∫ + ∞ zπ t − ( λF Z ( dz )) = ∫ + ∞ v ( t, W ( t, π t − , z )) v ( t, π t − ) e ηe R ( T − t ) zu zπ t − ( λF Z ( dz )) . = G ( u ) . (5.8) Proof.
Under the expected value principle, see Eq. (2.17), we have that h ( t, π, u ) = − v ( t, π ) ηe R ( T − t ) (1 + θ ) ∫ + ∞ z (1 − u ) π ( λF Z ( dz ))+ ∫ + ∞ v ( t, W ( t, π, z ))( e ηe R ( T − t ) z − e ηe R ( T − t ) zu ) π ( λF Z ( dz )) (5.9)is strictly concave in u ∈ [0 , I ] and, taking into account Eq. (5.3), f is so. Using Eq. (5.1)and Eq. (5.2) we notice that Eq. (4.5) can be rewritten as (5.8). The right hand term in thisequation is an increasing function on u ∈ [0 , u ( t, π ) increases with respect to θ (seeFigure 1).Finally, using Corollary 4.1 we get the explicit form of the optimal strategy and the resultreadily follows. In this subsection we compare the optimal strategy under partial information to the one with fullinformation. In some special cases (see Propositions 5.3 and 5.4 below) we can prove that theoptimal retention level under partial information is smaller than the one in the full informationcase. This means that the insurer who takes into account a partial information framework tendsto buy an additional protection with respect to the (theoretical) case of complete information.We consider the case of unknown time-homogeneous jump intensity (i.e. λ ( t, y ) = λ ( y ))and known claims size distribution (i.e. F Z ( t, y, dz ) = F Z ( dz )). Moreover, we suppose that thestochastic factor Y takes value in a discrete set S = { , , . . . } . Let us recall (see Remarks 2.623 igure 1: The effect of θ on the reinsurance strategy. and 2.7) that in this case the filter is described by the sequence π t ( i ) = P ( Y t = i |F t ) , i ∈ S and W ( t, π t − , z ) = W ( π t − ) = { W i ( π t − ) , i ∈ S} by Eq. (2.23), where W i ( π t − ) = λ ( i ) π t − ( i ) ∑ j ∈S λ ( j ) π t − ( j ) , i ∈ S . (5.10)Without loss of generality we assume λ (1) ≤ λ (2) ≤ . . . and following the same lines as in[Liang and Bayraktar, 2014, Lemma 4.1] (where the case with a finite set S is discussed), seealso [Buerle and Rieder, 2007, Theorem 5.6], we can prove that v ( t, W ( π )) ≥ v ( t, π ) , ∀ t ∈ [0 , T ] , π ∈ {{ π i } i ∈S ; ∑ i ∈S π i = 1 , π i ∈ [0 , } (5.11)with W ( π ) = { W i ( π ) , i ∈ S} . Proposition 5.3.
Let the assumptions of this subsection be satisfied. Under the proportionalreinsurance and the premium calculation principles in Example 2.4, the optimal reinsurancestrategy under partial information is always less or equal to the one under full information.Proof.
We analyze two premium calculation principles and, correspondingly, we divide the proofin two parts.
Expected value principle
Under the expected value principle (see Eq. (2.17)) and a proportional reinsurance (i.e. g ( z, u ) = uz, u ∈ [0 ,
1] by Example 2.3), using Proposition 4.1 and Corollary 4.1 we easily obtainthat the optimal reinsurance strategy is given by u ∗ t = ⎧⎪⎪⎨⎪⎪⎩ θ ≤ v ( t,W ( π t − )) v ( t,π t − ) θ ) E [ Z ] ≥ ∫ + ∞ v ( t,W ( π t − )) v ( t,π t − ) ze ηe R ( T − t ) z F Z ( dz )ˆ u ( t, π t − ) otherwise, (5.12)where ˆ u is the unique solution to(1 + θ ) E [ Z ] = v ( t, W ( π t − )) v ( t, π t − ) ∫ + ∞ ze ηe R ( T − t ) uz F Z ( dz ) . = h ( t, π t − , u ) . (5.13) The result essentially follows from the stochastic dominance of Poisson processes with increasing intensitiesand by this inequality: ∑ j ∈S π j α j β j ≤ ∑ j ∈S π j α j ∑ j ∈S π j β j , where { α j } j ∈S is an increasing sequence and { β j } j ∈S is increasing, while the nonnegative sequence { π j } j ∈S is such that ∑ j ∈S π j = 1. igure 2: Functions h and h giving ˆ u t and ˆ u f ( t ) under the expected value principle. In the full information case, from [Brachetta and Ceci, 2019b, Lemma 4.2], full reinsuranceis never optimal, the optimal reinsurance strategy is a deterministic function of time and it isgiven by u ∗ ,f ( t ) = { θ ) E [ Z ] ≥ ∫ + ∞ ze ηe R ( T − t ) z F Z ( dz )ˆ u f ( t ) otherwise, (5.14)where ˆ u f is the unique solution to(1 + θ ) E [ Z ] = ∫ + ∞ ze ηe R ( T − t ) uz F Z ( dz ) . = h ( t, u ) (5.15)Let us consider the equations (5.13) and (5.15) defined for all u ∈ R , then equations (5.12)and (5.14) can be written as u ∗ t = 0 ∧ ˆ u t ∨ u ∗ ,f ( t ) = ˆ u f ( t ) ∨ , respectively. Since h ( t, π t − , u ) = v ( t, W ( π t − )) v ( t, π t − ) h ( t, u ) ≥ h ( t, u ) ∀ u ∈ [0 , , and both sides are increasing in u , in order to have h ( t, π t − , ˆ u t ) = 1 + θ = h ( t, ˆ u f ( t )) , we must have that ˆ u t ≤ ˆ u f ( t ) (see Figure 2), which implies our statement. Variance premium principle
Now let us denote H ( u ) . = E [ Z ] + 2 θ (1 − u ) E [ Z ] . By Corollary 4.1 we obtain that under the variance premium principle the optimal reinsurancestrategy is given by u ∗ t = { θ E [ Z ] ≤ ( v ( t,W ( π t − )) v ( t,π t − ) − E [ Z ]ˆ u ( t, π t − ) otherwise, (5.16)where ˆ u is the unique solution to H ( u ) = h ( t, π t − , u ) . (5.17)25 igure 3: Functions h and h giving u ∗ t and u ∗ ,f ( t ) under the variance premium principle. In the full information case (see [Brachetta and Ceci, 2019b, Lemma 4.3]) the optimal strategy isa deterministic function of time and is given by u ∗ ,f ( t ) = ˆ u f ( t ), where ˆ u f is the unique solutionto H ( u ) = h ( t, u ) . Hence the inequality u ∗ t ≤ u ∗ ,f ( t ) immediately follows by the same arguments as in the expectedvalue principle case (see Figure 3). Proposition 5.4.
Let the assumptions of this subsection be satisfied and F Z ( dz ) admit density.Under the excess of loss reinsurance and the expected value principle the optimal reinsurancestrategy is always less or equal to the one under full information.Proof. Consider the excess-of-loss reinsurance (see Example 2.3) and the expected value principle(see Eq. (2.17)). In the full information case, see [Brachetta and Ceci, 2019a, Proposition 8],the optimal reinsurance strategy satisfies(1 + θ )(1 − F Z ( u )) = e ηe R ( T − t ) g ( z,u )) (1 − F Z ( u )) , so that it is given by u ∗ ,f ( t ) = 1 η e − R ( T − t ) log(1 + θ ) . In the partial information framework, taking into account Remark 4.2, Eq. (4.5) reads as(1 + θ )(1 − F Z ( u )) = v ( t, W ( π t − )) v ( t, π t − ) e ηe R ( T − t ) u (1 − F Z ( u )) , and we find out that u ∗ t = 1 η e − R ( T − t ) log ( (1 + θ ) v ( t, π t − ) v ( t, W ( π t − ) ) . It is easy to see that u ∗ t ≤ u ∗ ,f ( t ) by the inequality (5.11).
6. Conclusions
This paper extends the existing results on optimal reinsurance in many directions. We introducea general risk model where both the claims arrival intensity and the claim size distribution are af-fected by an environmental stochastic factor Y , which is modeled as a general Markov process (in26Liang and Bayraktar, 2014] it was a finite state Markov chain and in [Brachetta and Ceci, 2019a]a real-valued diffusion process). This model formulation allows the insurer to take into accountrisk fluctuations. However, it is well known that the insurer has only a partial information atdisposal. Namely, she only observes the claims arrival times and the corresponding amount.Hence Y is supposed to be unobservable and as a consequence claims arrival intensity and theclaim size distribution has to be inferred from the observations. Considering general premiumand reinsurance contract, we solve the optimization problem characterizing the value processand the optimal strategy in terms of a solution to a BSDE. Our results show that the insurerwould react to risk fluctuations by modifying the reinsurance policy. Some examples for classicalreinsurance agreements are illustrated. By analyzing the effect of safety loading on the optimalstrategy we determine the price that the insurer deems reasonable for the reinsurer assumingpart of her risks. Finally, we show that insurer with partial information is more conservativewith respect the insurer with complete information. Acknowledgements
The authors are partially supported by the GNAMPA Research Project 2019 (
Problemi di con-trollo ottimo stocastico con osservazione parziale in dimensione infinita ) of INdAM (IstitutoNazionale di Alta Matematica). We are also grateful to anonymous referees for their helpfulcomments.
Declaration of interest
None.
A. Filtering with marked point processes observations
Here, we recall the main results on filtering with marked point processes observations. UnderAssumption 2.4, the filter can be characterized as the unique strong solution of the so calledKushner-Stratonovich equation. We refer to [Ceci, 2006] and [Ceci and Colaneri, 2012] for adetailed proof.
Theorem A.1 (KS-equation) . Under Assumption 2.4, the filter π is the unique strong solutionto the Kushner-Stratonovich equation, for any bounded function f ∈ D Y dπ t ( f ) = π t ( L Y f ) dt + ∫ + ∞ w πt ( f, z )( m ( dt, dz ) − π t − ( λF Z ( dz )) dt ) , π ( f ) = f (0 , y ) , (A.1) where w πt ( f, z ) . = dπ t − ( λF Z f ) dπ t − ( λF Z ) ( z ) − π t − ( f ) + dπ t − ( ¯ L f ) dπ t − ( λF Z ) ( z ) . (A.2) Here ¯ L is an operator which takes into account possible common jump times between Y and m ( dt, dz ) , while dπ t − ( λF Z f ) dπ t − ( λF Z ) ( z ) and dπ t − ( ¯ L f ) dπ t − ( λF Z ) ( z ) denote the Radon-Nikodym derivatives of themeasures π t − ( λF Z ( dz ) f ) and π t − ( ¯ L f ( dz )) with respect to π t − ( λF Z ( dz )) , respectively. The filtering equation has a natural recursive structure. In fact, between two consecutivejump times, t ∈ ( T n − , T n ), the equation reads as: dπ t ( f ) = [ π t ( L f ) + π t ( f ) π t ( λ ) − π t ( λf )] dt, (A.3)where L f . = L Y f − ¯ L f and coincides with L Y if there are not common jump times betweenstate and observations.At a jump time T n : π T n ( f ) = dπ T − n ( λF Z f ) dπ T − n ( λF Z ) ( Z n ) + dπ T − n ( ¯ L f ) dπ T − n ( λF Z ) ( Z n ) . π T n ( f ) is completely determined by the observed data ( T n , Z n ) and by the knowledgeof π t in the interval t ∈ [ T n − , T n ).Let us observe that between two consecutive jump times the filter solves a non-linear deter-ministic equation (see Eq. (A.3)). We are able to provide a computable solution by means ofa linearized method (see [Ceci and Gerardi, 2006, Lemma 3.1]). For simplicity, we assume nocommon jump times between Y and m ( dt, dz ) in the sequel. Proposition A.1.
Let ρ n a process with values in the set of positive finite measures on R solutionto the linear equation dρ nt ( f ) = ρ nt ( L Y f − λf ) dt, ρ nT n − ( f ) = π T n − ( f ) , t ∈ ( T n − , T n ) . Then the process ρ nt ( f ) ρ nt (1) , t ∈ ( T n − , T n ) , solves Eq. (A.3) . Moreover the following representation holds ρ nt ( f ) = E n − [ f ( t, Y t ) e − ∫ ts λ ( r,Y r ) dr ] | s = T n − , where E n − denotes the conditional expectation given the distribution Y T n − equal to π T n − . Finally, Proposition 2.3 is a direct consequence of Proposition A.1 and of the strong unique-ness of solution to the Kushner-Stratonovich equation (A.1).In the last part of the section we discuss same special cases.
Example A.1 (Known jump size distribution and unknown intensity) . Let F Z ( t, y, dz ) = F Z ( dz ) , then the filtering equation (A.3) reduces to dπ t ( f ) = π t ( L Y f ) dt + π t − ( λf ) − π t − ( f ) π t − ( λ ) π t − ( λ ) ( dN t − π t − ( λ ) dt ) , where N t = m ((0 , t ] × [0 , + ∞ )) = ∑ n ≥ { T n ≤ t } is the claims arrival process. Between twoconsecutive jump times, t ∈ ( T n − , T n ) : dπ t ( f ) = [ π t ( L Y f ) − π t ( λf ) + π t ( f ) π t ( λ )] dt, while at a jump time T n : π T n ( f ) = W ( T n , π T − n ) . = π T − n ( λf ) π T − n ( λ ) , which coincides with Eq. (2.20) in Remark 2.6. Example A.2 (Markov Modulated Risk Model with infinitely many states) . Now we consider thecase where Y is a continuous time Markov chain taking values in a discrete set S = { , , . . . } and { a ij } i ∈S ,j ∈S its generator matrix. Here, a ij > , i ̸ = j , gives the intensity of a transition fromstate i to state j , and it is such that ∑ j ≥ ,j ̸ = i a ij = − a ii . Defining the functions f i ( y ) := y = i , i ∈ S , the filter is completely described via the knowledge of π t ( i ) := π t ( f i ) = P ( Y t = i | F t ) , i ∈ S , because for every function f we have that π t ( f ) = ∑ i ∈S f ( t, i ) π t ( i ) . The process ( π t ( i )) i ∈S is characterized via the following system of equations dπ t ( i ) = ∑ j ∈S a ji π t ( j ) dt + ∫ + ∞ w πt ( i, z )( m ( dt, dz ) − ∑ j ∈S λ ( t, j ) F Z ( t, j, dz ) π t − ( j ) dt ) , i ∈ S , (A.4)28 here w πt ( i, z ) = d ( λ ( t, i ) F Z ( t, i, dz ) π t − ( i )) d ( ∑ j ∈S λ ( t, j ) F Z ( t, j, dz ) π t − ( j )) ( z ) − π t − ( i ) , and we deduce Eq. (2.21) in Remark 2.6.When F Z ( t, i, dz ) admits density f Z ( t, i, z ) , i ∈ S , it simplifies to w πt ( i, z ) = λ ( t, i ) f Z ( t, i, z ) π t − ( i ) ∑ j ∈S λ ( t, j ) f Z ( t, j, z ) π t − ( j ) − π t − ( i ) . In particular when S is a finite set, the system (A.4) is finite. This case has been consideredin [Liang and Bayraktar, 2014], with the simplification of λ ( t, i ) and f Z ( t, j, z ) not dependent ontime. Example A.3 (Markov Modulated Risk Model with known jump size distribution and unknownintensity) . In the special case where F Z ( t, y, dz ) = F Z ( dz ) , the system (A.4) reduces to dπ t ( i ) = ∑ j ∈S a ji π t ( j ) dt + [ λ ( t, i ) π t − ( i ) ∑ j ∈S λ ( t, j ) π t − ( j ) − π t − ( i ) ] ( dN t − ∑ j ∈S λ ( t, j ) π t − ( j ) dt ) , i ∈ S . (A.5) Between two consecutive jump times, t ∈ ( T n − , T n ) : dπ t ( f ) = [ ∑ j ∈S a ji π t ( j ) − λ ( t, i ) π t ( i ) + π t ( i ) ∑ j ∈S λ ( t, j ) π t ( j )] dt, at a jump time T n : π T n ( i ) = W i ( T n , π T − n ) . = λ ( T n , i ) π T − n ( i ) ∑ j ∈S λ ( T n , j ) π T − n ( j ) . This latter formula provides Eq. (2.22) in Remark 2.6.In particular when S is a finite set, the system (A.5) is finite. References [Brachetta and Ceci, 2019a] Brachetta, M. and Ceci, C. (2019a). Optimal excess-of-loss reinsur-ance for stochastic factor risk models.
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