Optimal proportional reinsurance and investment for stochastic factor models
OOptimal proportional reinsuranceand investment for stochasticfactor models
Brachetta M. ∗ [email protected] Ceci, C. ∗† [email protected] Abstract
In this work we investigate the optimal proportional reinsurance-investment strategy of aninsurance company which wishes to maximize the expected exponential utility of its termi-nal wealth in a finite time horizon. Our goal is to extend the classical Cram´er-Lundbergmodel introducing a stochastic factor which affects the intensity of the claims arrival pro-cess, described by a Cox process, as well as the insurance and reinsurance premia. Usingthe classical stochastic control approach based on the Hamilton-Jacobi-Bellman equation wecharacterize the optimal strategy and provide a verification result for the value function viaclassical solutions of two backward partial differential equations. Existence and uniquenessof these solutions are discussed. Results under various premium calculation principles areillustrated and a new premium calculation rule is proposed in order to get more realisticstrategies and to better fit our stochastic factor model. Finally, numerical simulations areperformed to obtain sensitivity analyses.
Keywords:
Optimal proportional reinsurance, optimal investment, Cox model, stochastic con-trol.
JEL Classification codes:
G220, C610, G110.
MSC Classification codes:
Declarations of interest: none.
1. Introduction
In this paper we investigate the optimal reinsurance-investment problem of an insurance companywhich wishes to maximize the expected exponential utility of its terminal wealth in a finite timehorizon. In the actuarial literature there is an increasing interest in both optimal reinsurance andoptimal investment strategies, because they allow insurance firms to increase financial results andto manage risks. In particular, reinsurance contracts help the reinsured to increase the businesscapacity, to stabilize operating results, to enter in new markets, and so on. Among the traditionalreinsurance arrangements the excess-of-loss and the proportional treaties are of great importance.The former was studied in [Sheng et al., 2014], [Li et al., 2018] and references therein. The latterwas intensively studied by many authors under the criterion of maximizing the expected utility ofthe terminal wealth. Beyond the references contained therein, let us recall some noteworthy pa-pers: in [Liu and Ma, 2009] the authors considered a very general model, also including consump-tion, focusing on well posedness of the optimization problem and on existence of admissible strate-gies; in [Liang et al., 2011] a stock price with instantaneous rate of investment return described byan Ornstein-Uhlenbeck process has been considered ; in [Liang and Bayraktar, 2014] the problemhas been studied in a partially observable framework by introducing an unobservable Markov-modulated risk process; in [Zhu et al., 2015] the surplus is invested in a defaultable financial ∗ Department of Economics, University of Chieti-Pescara, Viale Pindaro, 42 - 65127 Pescara, Italy. † Corresponding author. a r X i v : . [ q -f i n . R M ] J un arket; in [Liang and Yuen, 2016] and [Yuen et al., 2015] multiple dependent classes of insur-ance business are considered. All these works may be considered as attempts to extend both theinsurance risk and the financial market models. In all these articles we can recognize two differentapproaches to dealing with the surplus process of the insurance company: some authors consid-ered it as a diffusion process approximating the pure-jump term of the Cram´er-Lundberg model(see for example [Bai and Guo, 2008, Cao and Wan, 2009, Zhang et al., 2009, Gu et al., 2010,Li et al., 2018] and references therein). This approach is validated by means of the famousCram´er-Lundberg approximation (see [Grandell, 1991]). Other authors (see [Liu and Ma, 2009,Zhu et al., 2015, Liang et al., 2011, Sheng et al., 2014, Yuen et al., 2015] and references therein)took into account the jump term using a compound Poisson risk model with constant intensity,that is the classical Cram´er-Lundberg model. On the one hand this is the standard model fornonlife insurance and it is simple enough to perform calculations, on the other it is too simpleto be realistic (as noticed by [Hipp, 2004]).As observed by Grandell, J. in [Grandell, 1991], more reasonable risk models should allow the in-surance firm to consider the so called size fluctuations as well as the risk fluctuations , which referrespectively to variations of the number of policyholders and to modifications of the underlyingrisks.This paper aims at extending the classical risk model by modelling the claims arrival processas a doubly stochastic Poisson process with intensity affected by an exogenous stochastic process { Y t } t ∈ [0 ,T ] . This environmental factor lead us to a reasonably realistic description of any riskmovement (see [Grandell, 1991], [Schmidli, 2018]). For example, in automobile insurance Y maydescribe road conditions, weather conditions (foggy days, rainy days, . . . ), traffic volume, and soon. While in [Liang and Bayraktar, 2014] the authors considered a Markov-modulated compoundPoisson process with the (unobservable) stochastic factor described by a finite state Markov chain,we consider a stochastic factor model where the exogenous process follows a general diffusion.An additional feature is that the insurance and the reinsurance premia are not evaluated usingpremium calculation principles, contrary to the majority of the literature; moreover, they turnout to be stochastic processes depending on Y . Furthermore, we highlight that under the mostfrequently used premium calculation principles (expected value and variance premium principles)some problems arise: firstly, the optimal reinsurance strategy turns out to be deterministic (thisis a limiting factor because the main goal of our paper is to consider a stochastic factor model);secondly, the optimal reinsurance strategy does not explicitly depend on the claims intensity.In order to fix these problems, we will introduce a new premium calculation principle, which iscalled intensity-adjusted variance premium principle .Finally, the financial market is more general than those usually considered in the litera-ture, since it is composed by a risk-free bond and a risky asset with Markovian rate of returnand volatility. For instance, in [Bai and Guo, 2008], [Cao and Wan, 2009], [Zhang et al., 2009]and [Liang and Bayraktar, 2014] the authors used a geometric Brownian model, in [Gu et al., 2010]and [Sheng et al., 2014] a CEV model. Nevertheless, some authors considered other general mod-els: in [Irgens and Paulsen, 2004] and [Li et al., 2018] the risky asset follows a jump-diffusionprocess with constant parameters, in [Liang et al., 2011] the instantaneous rate of investmentreturn follows an Ornstein-Uhlenbeck process, in [Zhu et al., 2015] the authors used the Hestonmodel, in [Xu et al., 2017] the authors introduced a Markov-modulated model for the financialmarket. However, in these papers the authors considered the classical risk model with constantintensity for the claims arrival process.Using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman equa-tion we characterize the optimal strategy and provide a verification result for the value functionvia classical solutions of two backward partial differential equations (see Theorem 6.1). Moreoverwe provide a class of sufficient conditions for existence and uniqueness of classical solutions to thePDEs involved (see Theorems 8.1 and 8.2). Results under various premium calculation principlesare discussed, including the intensity-adjusted variance premium principle . Finally, numericalsimulations are performed to obtain sensitivity analyses of the optimal strategies.The paper is organized as follows: in Section 2 we formulate the main assumptions anddescribe the optimization problem; Section 3 contains the derivation of the Hamilton-Jacobi-2ellman equation. In Section 4 we characterize the optimal reinsurance strategy, discussing inSubsections 4.1 and 4.2 how the general results apply to special premium calculation principles(expected value, variance premium and intensity-adjusted variance principles). In Section 5we provide the optimal investment strategy. Section 6 contains the Verification Theorem. InSection 7 we illustrate some numerical results and sensitivity analyses. In Section 8 existence anduniqueness theorems are discussed for the PDEs involved in the problem. Finally, in Appendix Athe reader can find some proofs of secondary results.
2. Problem formulation
Assume that (Ω , F , P , {F t } ) is a complete probability space endowed with a filtration {F t } t ∈ [0 ,T ] ,shortly denoted with {F t } , satisfying the usual conditions. We introduce the stochastic factor Y = { Y t } t ∈ [0 ,T ] as the solution of the following SDE: dY t = b ( t, Y t ) dt + γ ( t, Y t ) dW ( Y ) t Y ∈ R (2.1)where { W ( Y ) t } t ∈ [0 ,T ] is a standard Brownian motion on (Ω , F , P , {F t } ). This stochastic factorrepresents any environmental alteration reflecting on risk fluctuations. For instance, as suggestedby Grandell, J. (see [Grandell, 1991], Chapter 2), in automobile insurance Y may describe roadconditions, weather conditions (foggy days, rainy days, . . . ), traffic volume, and so on.We suppose that there exists a unique strong solution to (2.1) such that E [∫ T | b ( t, Y t ) | dt + ∫ T γ ( t, Y t ) dt ] < ∞ (2.2)sup t ∈ [0 ,T ] E [ | Y t | ] < ∞ (2.3)(for instance, it is true if the coefficients of the SDE (2.1) satisfy the classical Lipschitz and sub-linear growth conditions, see [Gihman and Skorohod, 1972]) and denote by L Y its infinitesimalgenerator: L Y f ( t, y ) = b ( t, y ) ∂f∂y ( t, y ) + 12 γ ( t, y ) ∂ f∂y ( t, y ) f ∈ C , ((0 , T ) × R ) . Let us introduce a strictly positive measurable function λ ( t, y ) : [0 , T ] × R → (0 , + ∞ ) and definethe process { λ t . = λ ( t, Y t ) } t ∈ [0 ,T ] for all t ∈ [0 , T ]. Under the hypothesis that E [∫ T λ u du ] < ∞ (2.4)we denote by { N t } t ∈ [0 ,T ] the claims arrival process, which is a conditional Poisson process having { λ t } t ∈ [0 ,T ] as intensity. More precisely, we have that for all 0 ≤ s ≤ t ≤ T and k = 0 , , . . . P [ N t − N s = k | F YT ∨ F s ] = (∫ ts λ u du ) k k ! e − ∫ ts λ u du , where {F Yt } t ∈ [0 ,T ] denotes the filtration generated by Y . Then it is easy to show that N t − ∫ t λ s ds is an {F t } -martingale .Now we define the cumulative claims up to time t as follows: C t = N t ∑ i =1 Z i t ∈ [0 , T ] , See e.g. [Br´emaud, 1981, II] F -random variables { Z i } i =1 ,... represents theamount of the claims. In the sequel we will assume that all the { Z i } i =1 ,... are distributed like ar.v. Z , independent on { N t } t ∈ [0 ,T ] and { Y t } t ∈ [0 ,T ] , with distribution function F Z ( dz ) such that F Z ( z ) = 1 ∀ z ≥ D , with D ∈ R + (eventually D = + ∞ ). Moreover, Z satisfies some suitableintegrability conditions (see (2.19) below).Consider the random measure associated with the marked point process { C t } t ∈ [0 ,T ] defined asfollows m ( dt, dz ) = ∑ t ∈ [0 ,T ]:∆ C t ̸ =0 δ ( t, ∆ C t ) ( dt, dz )= ∑ n ≥ δ ( T n ,Z n ) ( dt, dz ) { T n ≤ T } , (2.5)where { T n } n =1 ,... denotes the sequence of jump times of { N t } t ∈ [0 ,T ] , then the process { C t } t ∈ [0 ,T ] satisfies C t = ∫ t ∫ D zm ( ds, dz ) . (2.6)The following Lemma will be useful in the sequel. Lemma 2.1.
The random measure m ( dt, dz ) given in (2.5) has dual predictable projection ν given by the following: ν ( dt, dz ) = dF Z ( z ) λ t dt (2.7) i.e. for every nonnegative, {F t } -predictable and [0 , D ] -indexed process { H ( t, z ) } t ∈ [0 ,T ] E [∫ T ∫ D H ( t, z ) m ( dt, dz ) ] = E [∫ T ∫ D H ( t, z ) dF Z ( z ) λ t dt ] . Proof.
See Appendix A.
Remark 2.1.
Let us observe that for any {F t } -predictable and [0 , D ] -indexed process { H ( t, z ) } t ∈ [0 ,T ] such that E [∫ T ∫ D | H ( t, z ) | dF Z ( z ) λ t dt ] < ∞ the process M t = ∫ t ∫ D H ( s, z ) ( m ( ds, dz ) − dF Z ( z ) λ s ds ) t ∈ [0 , T ] turns out to be an {F t } -martingale. If in addition E [∫ T ∫ D | H ( t, z ) | dF Z ( z ) λ t dt ] < ∞ , then { M t } t ∈ [0 ,T ] is a square integrable {F t } -martingale and E [ M t ] = E [∫ t ∫ D | H ( t, z ) | dF Z ( z ) λ t dt ] ∀ t ∈ [0 , T ] . Moreover, the predictable covariation process of { M t } t ∈ [0 ,T ] is given by ⟨ M ⟩ t = ∫ T ∫ D | H ( t, z ) | dF Z ( z ) λ t dt that is { M t − ⟨ M ⟩ t } t ∈ [0 ,T ] is an {F t } -martingale . For these results and other related topics see e.g. [Bass, 2004]. emark 2.2. Let {G t } t ∈ [0 ,T ] be the filtration defined by G t = F t ∨ F YT . Then m ( dt, dz ) definedin (2.5) has {G t } -dual predictable projection ν given in (2.7) . In fact, first observe that { λ t } t ∈ [0 ,T ] is {F t } -adapted by definition, hence it is {G t } -adapted. Now notice that { λ t } is the {G t } -intensityof { N t } t ∈ [0 ,T ] because for any ≤ s ≤ t ≤ T E [ N t | G s ] = N s + E [ N t − N s | G s ]= N s + ∑ k ≥ k (∫ ts λ u du ) k k ! e − ∫ ts λ u du = N s + ∫ ts λ u du and this implies that E [ N t − ∫ t λ u du | G s ] = N s − ∫ s λ u du. Then our statement follows by the proof of Lemma 2.1 (see Appendix A) by replacing {F t } -predictable and [0 , D ] -indexed processes with {G t } -predictable and [0 , D ] -indexed processes. In this framework we suppose that the gross risk premium rate is affected by the stochasticfactor, i.e. we describe the insurance premium as a stochastic process { c t . = c ( t, Y t ) } t ∈ [0 ,T ] , where c : [0 , T ] × R → (0 , + ∞ ) is a nonnegative measurable function such that E [∫ T c ( t, Y t ) dt ] < ∞ . (2.8)The insurance company can continuously purchase a proportional reinsurance contract, trans-ferring at each time t ∈ [0 , T ] a percentage u t of its own risks to the reinsurer, who receives areinsurance premium q t given by the definition below. Definition . (Proportional reinsurance premium) Let us define a function q ( t, y, u ) : [0 , T ] × R × [0 , → [0 , + ∞ ), continuous w.r.t. the triple ( t, y, u ), having continuous partial derivatives ∂q ( t,y,u ) ∂u , ∂ q ( t,y,u ) ∂u in u ∈ [0 ,
1] and such that1. q ( t, y,
0) = 0 for all ( t, y ) ∈ [0 , T ] × R , because a null protection is not expensive;2. ∂q ( t,y,u ) ∂u ≥ t, y, u ) ∈ [0 , T ] × R × [0 , q ( t, y, > c ( t, y ) for all ( t, y ) ∈ [0 , T ] × R , because the cedant is not allowed to gain aprofit without risk.In the rest of the paper ∂q ( t,y, ∂u and ∂q ( t,y, ∂u should be intended as right and left derivatives,respectively. Moreover, we assume the following integrability condition: E [∫ T q ( t, Y t , u ) dt ] < ∞ ∀ u ∈ [0 , . (2.9)Then the reinsurance premium associated with a reinsurance strategy { u t } t ∈ [0 ,T ] (which is theprotection level chosen by the insurer) is defined as { q t . = q ( t, Y t , u t ) } t ∈ [0 ,T ] .In addition, we will use the hypothesis that the insurance gross premium and the reinsurancepremium will never diverge too much (being approximately influenced by the stochastic factorin the same way), that is there exists a positive constant K such that | q ( t, Y t , u ) − c ( t, Y t ) | ≤ K P -a.s. ∀ t ∈ [0 , T ] , u ∈ [0 , . (2.10)5nder 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, Y t ) − q ( t, Y t , u t ) ] dt − (1 − u t ) dC t = [ c ( t, Y t ) − q ( t, Y t , u t ) ] dt − ∫ D (1 − u t ) z m ( dt, dz ) R u = R ∈ R + (2.11)Let us observe that by Remark 2.1, since E [∫ T ∫ D u r zλ r dF Z ( z ) dr ] ≤ E [ Z ] E [∫ T λ r dr ] < ∞ , the process ∫ t ∫ D (1 − u s ) z ( m ( ds, dz ) − λ s dF Z ( z ) ds ) turns out to be an {F t } -martingale.Furthermore, we allow the insurer to invest its surplus in a financial market consisting of arisk-free bond { B t } t ∈ [0 ,T ] and a risky asset { P t } t ∈ [0 ,T ] , whose dynamics on (Ω , F , P , {F t } ) are,respectively, dB t = RB t dt B = 1 (2.12)with a fixed R >
0, and dP t = P t [ µ ( t, P t ) dt + σ ( t, P t ) dW ( P ) t ] P > { W ( P ) t } t ∈ [0 ,T ] is a standard Brownian motion independent of { W ( Y ) } t ∈ [0 ,T ] and the ran-dom measure m ( dt, dz ) . Let us assume that there exists a unique strong solution to (2.13) suchthat E [∫ T | P t µ ( t, P t ) | dt + ∫ T P t σ ( t, P t ) dt ] < ∞ (2.14)sup t ∈ [0 ,T ] E [ P t ] < ∞ (2.15)(for example, it is true if the coefficients of the SDE (2.13) satisfy the classical Lipschitz andsub-linear growth conditions, see [Gihman and Skorohod, 1972]). Furthermore, we assume theNovikov condition: E [ e ∫ T | µ ( t,Pt ) − Rσ ( t,Pt ) | dt ] < ∞ , (2.16)which implies the existence of a risk-neutral measure for { P t } t ∈ [0 ,T ] and ensures that the financialmarket does not admit arbitrage.We will denote with w t the total amount invested in the risky asset at time t ∈ [0 , T ], so that X t − w t will be the capital invested in the risk-free asset (now X t indicates the total wealth,but it will be defined more accurately below, see equation (2.18)). We also allow the insurer toshort-sell and to borrow/lend any infinitesimal amount, so that w t ∈ R .Finally, we only consider self-financing strategies: the insurer company only invests the surplusobtained with the core business, neither subtracting anything from the gains, nor adding some-thing from another business.The insurer’s wealth { X αt } t ∈ [0 ,T ] associated with a given strategy α t = ( u t , w t ) is described This is a classical assumption which implies that the financial market is independent on the insurance market.
6y the following SDE: dX αt = dR ut + w t dP t P t + ( X αt − w t ) dB t B t = [ c ( t, Y t ) − q ( t, Y t , u t ) ] dt + w t [ µ ( t, P t ) dt + σ ( t, P t ) dW ( P ) t ] + ( X αt − w t ) R dt − ∫ D (1 − u t ) z m ( dt, dz ) (2.17)with X α = R ∈ R + . Remember that { u t } t ∈ [0 ,T ] and { w t } t ∈ [0 ,T ] are, respectively, the proportionof reinsured claims and the total amount invested in the risky asset { P t } t ∈ [0 ,T ] . Remark 2.3.
It can be verified that the solution of the SDE (2.17) is given by the following: X αt = X α e Rt + ∫ t e R ( t − r ) [ c ( r, Y r ) − q ( r, Y r , u r ) ] dr + ∫ t e R ( t − r ) w r [ µ ( r, P r ) − R ] dr + ∫ t e R ( t − r ) w r σ ( r, P r ) dW ( P ) r − ∫ t ∫ D e R ( t − r ) (1 − u r ) z m ( dr, dz ) . (2.18)Now we are ready to formulate the optimization problem of an insurance company whichsubscribes a proportional reinsurance contract and invests its surplus in a financial market ac-cording with a strategy { α t = ( u t , w t ) } t ∈ [0 ,T ] in order to maximize the expected utility of itsterminal wealth: sup α ∈U E [ U ( X αT ) ] where U denotes a suitable class of admissible controls defined below (see Definition 2.2) and U : R → [0 , + ∞ ) is the utility function representing the insurer preferences. We focus on CARA( Constant Absolute Risk Aversion ) utility functions, whose general expression is given by U ( x ) = 1 − e − ηx x ∈ R where η > α s , for s ∈ [ t, T ], for the following optimization problem given theinformation available at the time t ∈ [0 , T ]:sup α ∈U t E [ U ( X αt,x ( T )) | F t ] t ∈ [0 , T ]where U t denotes the class of admissible controls in the time interval [ t, T ] (see Definition 2.2below). Here { X αt,x ( s ) } s ∈ [ t,T ] denotes the solution to equation (2.17) with initial condition X αt = x .For the sake of simplicity, we will reduce ourselves studying the function − e − ηx . Another possiblechoice is to study the corresponding minimizing problem for the function e − ηx , but the first choiceis usually preferred in the literature. Definition . We will denote with U the set of all admissible strategies, which are all the {F t } -predictable processes α t = ( u t , w t ), t ∈ [0 , T ], with values in [0 , × R , such that E [∫ T | w r || µ ( r, P r ) − R | dr ] < ∞ , E [∫ T w r σ ( r, P r ) dr ] < ∞ . When we want to restrict the controls to the time interval [ t, T ], we will use the notation U t .7rom now on we assume the following assumptions fulfilled. Assumption 2.1. E [ e ηZe RT ] < ∞ , E [ Ze ηZe RT ] < ∞ E [ Z e ηZe RT ] < ∞ (2.19) E [ e ( E [ e ηeRT Z ] − ∫ Tt λ s ds | F t ] < ∞ ⟨ P = 1 ⟩ ∀ t ∈ [0 , T ] . (2.20) Proposition 2.1.
Under the Assumption 2.1 the control (0 , is admissible and such that E [ e − ηX (0 , t,x ( T ) | F t ] < ∞ ⟨ P = 1 ⟩ ∀ ( t, x ) ∈ [0 , T ] × R . Proof.
See Appendix A.
Remark 2.4.
Let us observe that Proposition 2.1 implies that ess sup α ∈U t E [ U ( X αt,x ( T )) | F t ] > −∞ ⟨ P = 1 ⟩ t ∈ [0 , T ] and as a consequence that sup α ∈U E [ U ( X αT ) ] > −∞ . In order to solve this dynamic problem we introduce the value function associated with it v ( t, x, y, p ) = sup α ∈U t E [ − e − ηX αt,x ( T ) | Y t = y, P t = p ] (2.21)where the function v : V → R is defined in the domain V . = [0 , T ] × R × (0 , + ∞ ) . The following Lemma gives sufficient conditions to extend Proposition 2.1 to all constantstrategies.
Lemma 2.2.
Under the Assumption 2.1, let us suppose σ ( t, p ) and µ ( t, p ) are bounded for all ( t, p ) ∈ [0 , T ] × (0 , + ∞ ) . Then we have that all constant strategies α t = ( u, w ) with u ∈ [0 , and w ∈ R are admissible and such that E [ e − ηX αt,x ( T ) | F t ] < ∞ ⟨ P = 1 ⟩ ∀ ( t, x ) ∈ [0 , T ] × R . Proof.
See Appendix A.
3. Hamilton-Jacobi-Bellman equation
Let us consider the Hamilton-Jacobi-Bellman equation that the value function is expected tosolve if sufficiently regular ⎧⎨⎩ sup ( u,w ) ∈ [0 , × R L α v ( t, x, y, p ) = 0 v ( T, x, y, p ) = − e − ηx ∀ ( y, p ) ∈ R × (0 , + ∞ ) (3.1)where L α denotes the Markov generator of the triple ( X αt , Y t , P t ) associated with a constantcontrol α = ( u, w ). In what follows, we denote by C , b all bounded functions f ( t, x , . . . , x n ),with n ≥
1, with bounded first order derivatives ∂f∂t , ∂f∂x , . . . , ∂f∂x n and bounded second orderderivatives w.r.t. the spatial variables ∂ f∂x , . . . , ∂f∂x n .8 emma 3.1. Let f : V → R be a function in C , b . Then the Markov generator of the stochasticprocess ( X αt , Y t , P t ) for all constant strategies α = ( u, w ) ∈ [0 , × R is given by the followingexpression: L α f ( t, x, y, p ) = ∂f∂t ( t, x, y, p ) + ∂f∂x ( t, x, y, p ) [ Rx + c ( t, y ) − q ( t, y, u ) + w ( µ ( t, p ) − R ) ] + 12 w σ ( t, p ) ∂ f∂x ( t, x, y, p ) + b ( t, y ) ∂f∂y ( t, x, y, p ) + 12 γ ( t, y ) ∂ f∂y ( t, x, y, p )+ pµ ( t, p ) ∂f∂p ( t, x, y, p ) + 12 p σ ( t, p ) ∂ f∂p ( t, x, y, p ) + wσ ( t, p ) p ∂ f∂x∂p ( t, x, y, p )+ ∫ D [ f ( t, x − (1 − u ) z, y, p ) − f ( t, x, y, p ) ] λ ( t, y ) dF Z ( z ) . (3.2) Proof.
See Appendix A.Now let us introduce the following ansatz: v ( t, x, y, p ) = − e − ηxe R ( T − t ) φ ( t, y, p )where φ does not depend on x and it is a positive function . Then the original HJB problemgiven in (3.1) reduces to the simpler one given by − ∂φ∂t ( t, y, p ) − b ( t, y ) ∂φ∂y ( t, y, p ) − γ ( t, y ) ∂ φ∂y ( t, y, p ) + ηe R ( T − t ) c ( t, y ) φ ( t, y, p ) − pµ ( t, p ) ∂φ∂p ( t, y, p ) − σ ( t, p ) p ∂ φ∂p ( t, y, p )+ sup u ∈ [0 , Ψ u ( t, y ) φ ( t, y, p ) + sup w ∈ R Ψ w ( t, y, p ) = 0 (3.3)with final condition φ ( T, y, p ) = 1 for all ( y, p ) ∈ R × (0 , + ∞ ), definingΨ u ( t, y ) . = − ηe R ( T − t ) q ( t, y, u ) + λ ( t, y ) ∫ D [ − e η (1 − u ) ze R ( T − t ) ] dF Z ( z ) (3.4)and Ψ w ( t, y, p ) . = ηe R ( T − t ) ( ( µ ( t, p ) − R ) φ ( t, y, p ) + pσ ( t, p ) ∂φ∂p ( t, y, p ) ) w − σ ( t, p ) η e R ( T − t ) φ ( t, y, p ) w . (3.5)It should make it clear that we can split the optimal control research in two distinct problems:the optimization of Ψ u will give us the optimal level of reinsurance (see Section 4), while workingwith Ψ w we will find the optimal investment policy (see Section 5).
4. Optimal reinsurance strategy
In this section we discuss the problemsup u ∈ [0 , Ψ u ( t, y ) , ( t, y ) ∈ [0 , T ] × R (4.1)with Ψ u ( t, y ) given in (3.4).First, let us observe that Ψ u ( t, y ) is continuous w.r.t. u ∈ [0 , t, y ) ∈ [0 , T ] × R and admits continuous first and the second order derivatives w.r.t. u ∈ [0 , Intuitively, we note that X αt,x ( T ) = X αt, ( T ) + xe R ( T − t ) and we use the exponential form of the function v . Ψ u ( t, y ) ∂u = − ηe R ( T − t ) [ ∂q ( t, y, u ) ∂u − λ ( t, y ) ∫ D ze η (1 − u ) ze R ( T − t ) dF Z ( z ) ] ∂ Ψ u ( t, y ) ∂u = − ηe R ( T − t ) [ ∂ q ( t, y, u ) ∂u + ηe R ( T − t ) λ ( t, y ) ∫ D z e η (1 − u ) ze R ( T − t ) dF Z ( z ) ] . Notice that these derivatives are well defined thanks to (2.19).Now we are ready for the main result of this section.
Proposition 4.1.
Given Ψ u ( t, y ) in (3.4) , suppose that − ∂ q ( t, y, u ) ∂u < ηe R ( T − t ) λ ( t, y ) E [ Z e η (1 − u ) Ze R ( T − t ) ] ∀ ( t, y, u ) ∈ [0 , T ] × R × (0 , . (4.2) Then there exists a unique measurable function u ∗ ( t, y ) for all ( t, y ) ∈ [0 , T ] × R solution to (4.1) .Moreover, it is given by u ∗ ( t, y ) = ⎧⎪⎨⎪⎩ t, y ) ∈ A ˆ u ( t, y ) ( t, y ) ∈ ˆ A t, y ) ∈ A (4.3) where A . = { ( t, y ) ∈ [0 , T ] × R | λ ( t, y ) E [ Ze ηZe R ( T − t ) ] ≤ ∂q ( t, y, ∂u } ˆ A . = { ( t, y ) ∈ [0 , T ] × R | λ ( t, y ) E [ Ze ηZe R ( T − t ) ] > ∂q ( t, y, ∂u , ∂q ( t, y, ∂u > E [ Z ] λ ( t, y ) } A . = { ( t, y ) ∈ [0 , T ] × R | ∂q ( t, y, ∂u ≤ E [ Z ] λ ( t, y ) } and ˆ u ( t, y ) is the unique solution of the following equation: ∂q ( t, y, u ) ∂u = λ ( t, y ) ∫ D ze η (1 − u ) ze R ( T − t ) dF Z ( z ) . (4.4) Proof.
Since Ψ u ( t, y ) is continuous in u ∈ [0 ,
1] and ∂ Ψ u ( t,y ) ∂u < ∀ ( t, y, u ) ∈ [0 , T ] × R × (0 , u ( t, y ) is strictly concave in u ∈ [0 , u ∗ ( t, y ) of (4.1), whose measurability follows by classical selection theorems.Observe that A ∪ ˆ A ∪ A = [0 , T ] × R . In fact, let us define these subsets as follows: A = { ( t, y ) ∈ [0 , T ] × R | ∂ Ψ ( t, y ) ∂u ≤ } ˆ A = { ( t, y ) ∈ [0 , T ] × R | ∂ Ψ ( t, y ) ∂u > , ∂ Ψ ( t, y ) ∂u < } A ! = { ( t, y ) ∈ [0 , T ] × R | ∂ Ψ ( t, y ) ∂u ≥ } . Now, being ∂ Ψ u ( t,y ) ∂u strictly decreasing in u ∈ (0 , t, y ) ∈ ˆ A ∪ A we have that ∂ Ψ ( t, y ) ∂u > ∂ Ψ ( t, y ) ∂u ≥ ⇒ ( t, y ) ̸∈ A , T ] × R \ A = { ( t, y ) ∈ [0 , T ] × R | ∂ Ψ ( t, y ) ∂u > } = { ( t, y ) ∈ [0 , T ] × R | ∂ Ψ ( t, y ) ∂u > , ∂ Ψ ( t, y ) ∂u < } ˙ ∪ { ( t, y ) ∈ [0 , T ] × R | ∂ Ψ ( t, y ) ∂u > , ∂ Ψ ( t, y ) ∂u ≥ } = ˆ A ˙ ∪ A . Moreover, since ˆ A ∩ A = ∅ , then A ˙ ∪ ˆ A ˙ ∪ A = [0 , T ] × R .Let us recall that ∂ Ψ u ( t,y ) ∂u is continuous and strictly decreasing in u ∈ [0 , ∀ ( t, y ) ∈ [0 , T ] × R .If ( t, y ) ∈ A then Ψ u ( t, y ) is strictly decreasing in u ∈ [0 , u ∗ ( t, y ) = 0.If ( t, y ) ∈ ˆ A then there exists a unique u ∗ ( t, y ) ∈ (0 ,
1) such that ∂ Ψ u ( t,y ) ∂u = 0, and it is theunique solution to equation (4.4).Finally, if ( t, y ) ∈ A then Ψ u ( t, y ) is strictly increasing in u ∈ [0 , u ∗ ( t, y ) = 1. Remark 4.1.
We also observe for the sake of completeness that if λ ( t, y ) had been vanishedfor some ( t, y ) , then ∂ Ψ u ( t,y ) ∂u would have become strictly negative for all u , and in this case u ∗ ( t, y ) = 0 . In fact, the case of λ ( t, y ) = 0 corresponds to a degenerate situation: the riskpremia are paid, but there is no ”real” risk to be insured. From the economic point of view, we could say that if the reinsurance is not too muchexpensive (more precisely, if the price of an infinitesimal protection is below a certain dynamicthreshold) and if full reinsurance is not optimal, then the optimal strategy is provided by (4.4),i.e. by equating the marginal cost and the marginal gain; moreover, the following remark pointsout the relevance of the third case in (4.3).
Remark 4.2.
In the current literature full reinsurance is always considered sub-optimal, contraryto the result given by formula (4.3) . The main reason is that using premium calculation principlesmany authors force the reinsurance premium to have certain properties, such as the convexitywith respect to the protection level. In fact, it can be shown that if the reinsurance premium q ( t, y, u ) is convex w.r.t. u , full reinsurance is never optimal (see Remark 4.3). Nevertheless, itis reasonable that the insurer firm could regard full reinsurance as convenient for a limited periodand in some particular scenarios, because actually the objective is to maximize the expected utilityof the wealth at the end of the period.Moreover, from the reinsurer’s point of view, there is no reason to prevent the insurer frombuying a full protection, providing the cedant is ready to pay a fair price. At the same time, ifthe reinsurer is not able to sell a full reinsurance, then it is sufficient to choose q ( t, y, u ) suchthat A = ∅ . Now we provide some sufficient conditions in order to guarantee that condition (4.2) is fulfilled.
Lemma 4.1.
Suppose that at least one of the following condition holds:1. ∂q ( t,y, ∂u = 0 for all ( t, y ) ∈ [0 , T ] × R ;2. ∂ q ( t,y,u ) ∂u ≥ for all u ∈ (0 , and ( t, y ) ∈ [0 , T ] × R ;3. − ∂ q ( t,y,u ) ∂u < ηλ ( t, y ) E [ Z ] for all u ∈ (0 , and ( t, y ) ∈ [0 , T ] × R . hen the inequality (4.2) holds, which implies that the function Ψ u ( t, y ) is strictly concave in u ∈ (0 , .Proof. First, let us observe that 1 ⇒ ⇒
3. In fact, by the fundamental theorem of calculus wehave that ∂q ( t, y, u ) ∂u = ∂q ( t, y, ∂u + ∫ u ∂ q ( t, y, w ) ∂w dw and, being ∂q ( t,y,u ) ∂u ≥ ∂q ( t,y, ∂u = 0 implies that the integrand function must be nonnegative,that is 1 ⇒
2. The implication 2 ⇒ η > , λ ( t, y ) >
0. Now it is sufficient toshow that 3 implies (4.2); clearly − ∂ q ( t, y, u ) ∂u < ηλ ( t, y ) E [ Z ] < ηe R ( T − t ) λ ( t, y ) E [ Z e η (1 − u ) Ze R ( T − t ) ] and hence ∂ Ψ u ( t,y ) ∂u < u ∈ (0 , u ( t, y ) is strictlyconcave in u ∈ (0 , Remark 4.3.
Under the hypotheses that ∂ q ( t,y,u ) ∂u ≥ and c ( t, y ) > E [ Z ] λ ( t, y ) for all ( t, y, u ) ∈ [0 , T ] × R × (0 , , the full reinsurance is never optimal. In fact, for any arbitrary couple ( t, y ) we have that q ( t, y,
1) = q ( t, y,
0) + ∫ ∂q ( t, y, u ) ∂u du. Being q ( t, y,
0) = 0 and q ( t, y, > c ( t, y ) > E [ Z ] λ ( t, y ) (because the reinsurance is not cheapand using the net-profit condition for the insurance premium), we obtain that ∫ ∂q ( t, y, u ) ∂u du > E [ Z ] λ ( t, y ) . Since ∂q ( t,y,u ) ∂u is continuous in u ∈ [0 , by hypothesis, from the mean value theorem for integralswe know that there exists u ∈ (0 , such that ∂q ( t, y, u ) ∂u > E [ Z ] λ ( t, y ) . Under the hypothesis that ∂ q ( t,y,u ) ∂u ≥ for all u ∈ (0 , , ∂q ( t,y,u ) ∂u is an increasing function of u ,and this implies that ∂q ( t, y, ∂u ≥ ∂q ( t, y, u ) ∂u > E [ Z ] λ ( t, y ) . From this result we deduce that ∂ Ψ ( t, y ) ∂u = − ηe R ( T − t ) [ ∂q ( t, y, ∂u − E [ Z ] λ ( t, y ) ] < , ( t, y ) ∈ [0 , T ] × R which implies that A = ∅ , i.e. the full reinsurance is never optimal. Let us observe that the preceding Remark requires two special conditions. The first oneconcerns the concavity of the reinsurance premium and in Subsection 4.1 we will show that itis fulfilled by the most famous premium calculation principles. The second hypothesis is the socalled net-profit condition (e.g. see [Grandell, 1991]) and it is usually assumed in insurance riskmodels to ensure that the expected gross risk premium covers the expected losses.Now we investigate how Proposition 4.1 applies to a special case.12 xample 4.1. (Exponentially distributed claims)Let Z to be an exponential r.v. with parameter ζ > , then for any fixed ( t, y ) ∈ [0 , T ] × R equation (4.4) becomes λ ( t, y ) ∫ ∞ ze η (1 − u ) ze R ( T − t ) ζe − ζz dz = ∂q ( t, y, u ) ∂u . Taking k = η (1 − u ) e R ( T − t ) − ζ it can be written as λ ( t, y ) ∫ ∞ ze kz ζ dz = ∂q ( t, y, u ) ∂u and requiring that ζη > e RT (4.5) which implies that k < , finally equation (4.4) reads as λ ( t, y ) ζ ( η (1 − u ) e R ( T − t ) − ζ ) = ∂q ( t, y, u ) ∂u . (4.6) Summarizing, if Z is an exponential r.v. with parameter ζ > ηe RT , under the condition (4.2) wehave that expression (4.3) holds with A . = { ( t, y ) ∈ [0 , T ] × R | λ ( t, y ) ζ ( ηe R ( T − t ) − ζ ) ≤ ∂q ( t, y, ∂u } ˆ A . = { ( t, y ) ∈ [0 , T ] × R | λ ( t, y ) ζ ( ηe R ( T − t ) − ζ ) > ∂q ( t, y, ∂u , ∂q ( t, y, ∂u > λ ( t, y ) ζ } A . = { ( t, y ) ∈ [0 , T ] × R | ∂q ( t, y, ∂u ≤ λ ( t, y ) ζ } and with ˆ u ( t, y ) being the unique solution to equation (4.6) . Proposition 4.1 clarifies that the optimal reinsurance strategy crucially depends on the reinsur-ance premium. In this subsection we specialize that result using two of the most famous premiumcalculation principles: the expected value principle and the variance premium principle . We willshow that in both cases we loose the dependence of the optimal reinsurance strategy on thestochastic factor. Moreover, the optimal reinsurance strategy does not explicitly depend on theclaims intensity. These will be our motivations for introducing the intensity-adjusted variancepremium principle in Subsection 4.2.
Lemma 4.2.
Under the expected value principle, i.e. if the reinsurance premium admits thefollowing expression q ( t, y, u ) = (1 + θ r ) E [ Z ] λ ( t, y ) u ∀ ( t, y, u ) ∈ [0 , T ] × R × [0 ,
1] (4.7) for some constant θ r > (which is called the reinsurance safety loading), there exists a uniquemaximizer u ∗ ( t ) for all ( t, y ) ∈ [0 , T ] × R for the problem (4.1) . In particular, u ∗ ( t ) = { t ∈ A ˆ u ( t ) t ∈ [0 , T ] \ A (4.8) where A . = { t ∈ [0 , T ] | E [ Ze ηZe R ( T − t ) ] ≤ (1 + θ r ) E [ Z ] } and ˆ u ( t ) is the unique solution to the following equation: (1 + θ r ) E [ Z ] = ∫ D ze η (1 − u ) ze R ( T − t ) dF Z ( z ) . (4.9)13 roof. From (4.7) we get ∂q ( t, y, u ) ∂u = (1 + θ r ) E [ Z ] λ ( t, y ) , ∂ q ( t, y, u ) ∂u = 0 ∀ u ∈ (0 , u ( t, y ) is strictly concave in u ∈ (0 ,
1) thanks to Lemma 4.1. Moreover, bythe means of Remark 4.3 we know that the full reinsurance is always sub-optimal, in fact the set A in Proposition 4.1 is empty. Now we only have to apply Proposition 4.1.Note that we always have E [ Ze ηZe R ( T − t ) ] > E [ Z ] for each t ∈ [0 , T ], thus A could be anempty set when the reinsurer’s safety loading is close to 0. Example 4.2. (Exponentially distributed claims under the expected value principle)Let us come back to example 4.1. Under the expected value principle (4.7) the result for expo-nential claims is even more simplified, in fact we find the following explicit solution: u ∗ ( t ) = ⎧⎨⎩ − ζη ( − √ θ r ) e − R ( T − t ) t ∈ [0 , t ∧ T )0 t ∈ [ t ∧ T, T ] (4.10) where t = T − R log [ ζη ( − √ θ r )] . (4.11) The expression for t can be derived from the characterization of the set [0 , T ] × R \ A , whichin this case reads as follows: ζ − √ ζ (1+ θ r ) E [ Z ] η < e R ( T − t ) < ζ + √ ζ (1+ θ r ) E [ Z ] η where the second inequality is always fulfilled in view of (4.5) , hence we get t only from the firstinequality. Lemma 4.3.
Under the variance premium principle, i.e. if the reinsurance premium admits thefollowing expression q ( t, y, u ) = E [ Z ] λ ( t, y ) u + θ r E [ Z ] λ ( t, y ) u (4.12) for some constant reinsurance safety loading θ r > , the optimization problem (4.1) admits aunique maximizer u ∗ ( t ) ∈ (0 , for all ( t, y ) ∈ [0 , T ] × R , which is the solution to the followingequation: θ r E [ Z ] u = ∫ D ze η (1 − u ) ze R ( T − t ) dF Z ( z ) − E [ Z ] . (4.13) Proof.
Using the expression (4.12) we get that ∂q ( t, y, u ) ∂u = E [ Z ] λ ( t, y ) + 2 θ r E [ Z ] λ ( t, y ) u ∀ u ∈ (0 , ∂ q ( t, y, u ) ∂u = 2 θ r E [ Z ] λ ( t, y ) > ∀ u ∈ (0 , . By Lemma 4.1 Ψ u ( t, y ) is strictly concave w.r.t. u and the full reinsurance is never optimalbecause of Remark 4.3. Moreover, in order to apply Proposition 4.1 we notice that E [ Ze ηZe R ( T − t ) ] > E [ Z ] ⇒ A = ∅ thus the optimal strategy is unique and it belongs to (0 , Example 4.3. (Exponentially distributed claims under the variance premium principle)Under the variance premium principle (4.12) , suppose that the claims are exponentially dis-tributed with parameter ζ > ηe RT . Then it is easy to show that the optimal strategy is givenby u ∗ ( t ) = 1 − ζη ( − √ ζζ + 4 θ r ) e − R ( T − t ) t ∈ [0 , T ] . (4.14) We have shown that both the expected value principle (see Lemma 4.2) and the variance premiumprinciple (see Lemma 4.3) lead us to deterministic optimal reinsurance strategies, which do notdepend on the stochastic factor. This is a limiting factor, since the main objective of our paperis to solve the maximization problem under a stochastic factor model.In addition, in both cases the optimal reinsurance strategy does not explicitly depend on theclaims intensity. As a consequence, there is a paradox that we clarify with the following example.Let us consider two identical insurers (i.e. with the same risk-aversion, time horizon, and so on)who work in the same insurance business line, for example in automobile insurance, but in twodistinct territories with different riskiness. More precisely, let us assume that the two companiesinsure claims which have the same distribution F Z but occur with different probabilities. Henceit is a reasonable assumption that the claims arrival processes have two different intensities.Now let us suppose that both the insurers use Lemma 4.2 (or Lemma 4.3) in order to solve themaximization problem (4.1). Then they will obtain the same reinsurance strategy, but this is notwhat we expect. Hence the optimal reinsurance strategy should explicitly depend on the claimsintensity.In order to fix these two problems, in this subsection we introduce a new premium calculationprinciple, which will be referred as the intensity-adjusted variance premium principle .Let us first formalize that there exists a special class of premium calculation principles thatlead us to deterministic strategies which do not depend on the claims intensity. Remark 4.4.
For any reinsurance premium { q t } t ∈ [0 ,T ] admitting the following representation q ( t, y, u ) = λ ( t, y ) Q ( t, u ) (4.15) for a suitable function Q : [0 , T ] × [0 , → [0 , + ∞ ) , the optimal reinsurance strategy u ∗ t = u ∗ ( t, Y t ) given in Proposition 4.1 turns out to be deterministic. Moreover, it does not explicitlydepend on the claims intensity. For example, the expected value principle and the variancepremium principle admit the factorization (4.15) with, respectively, Q ( t, u ) = (1 + θ r ) E [ Z ] u and Q ( t, u ) = E [ Z ] u + θ r E [ Z ] u . Now the basic idea is to find a reinsurance premium { q t } t ∈ [0 ,T ] (see Definition 2.1) such that E [∫ t q ( s, Y s , u s ) ds ] = E [∫ t u s dC s ] + θ r var [∫ t u s dC s ] ∀ t ∈ [0 , T ] (4.16)for a given reinsurance safety loading θ r in order to dynamically satisfy the original formulationof the variance premium principle . For this purpose, we give the following result. Lemma 4.4.
For any {F Yt } t ∈ [0 ,T ] -predictable reinsurance strategy { u t } t ∈ [0 ,T ] we have that forany t ∈ [0 , T ] var [∫ t u s dC s ] = E [ Z ] E [∫ t u s λ s ds ] + E [ Z ] var [∫ t u s λ s ds ] . (4.17) I.e. Q is such that q fulfills the Definition 2.1. See e.g. [Young, 2006]. roof. Let us denote with { M ut } t ∈ [0 ,T ] the following {F t } -martingale: M ut = ∫ D ∫ t u s z ( m ( ds, dz ) − dF Z ( z ) λ s ds ) . Recalling that { C t } t ∈ [0 ,T ] is defined in (2.6), the variance of the reinsurer’s cumulative losses atthe time t ∈ [0 , T ] is given byvar [∫ t u s dC s ] = E [(∫ t u s dC s ) ] − E [∫ t u s dC s ] = E [ | M ut | + (∫ t u s λ s E [ Z ] ds ) + 2 M ut ∫ t u s λ s E [ Z ] ds ] − E [∫ t u s dC s ] . Denoting with ⟨ M u ⟩ t the predictable covariance process of M ut , using Remark 2.1, we find thatvar [∫ t u s dC s ] = E [ ⟨ M u ⟩ t ] + E [ Z ] E [(∫ t u s λ s ds ) ] − E [ Z ] E [∫ t u s λ s ds ] = E [ Z ] E [∫ t u s λ s ds ] + E [ Z ] var [∫ t u s λ s ds ] ∀ t ∈ [0 , T ] . (4.18)Here we have used that E [ M ut ∫ t u s λ s E [ Z ] ds ] = 0. In fact we notice that E [ M ut ∫ t u s λ s E [ Z ] ds ] = E [ E [ M ut ∫ t u s λ s E [ Z ] ds | F YT ]] = E [ E [ M ut | F YT ] ∫ t u s λ s E [ Z ] ds ] and being G = F ∨ F YT ⊇ F YT (see Remark 2.2) we have that E [ M ut | F YT ] = E [ E [ M ut | G ] F YT ] = E [ M u | F YT ] = 0and the proof is complete. Remark 4.5.
We highlight that Lemma 4.4 applies to {F Yt } t ∈ [0 ,T ] -predictable reinsurance strate-gies, but this is not restrictive. In fact, from Lemma 4.1 we know that the optimal strategy belongsto the class of {F Yt } t ∈ [0 ,T ] -predictable processes. Remark 4.6.
In the classical Cram´er-Lundberg model, i.e. λ ( t, y ) = λ , for any deterministicstrategy u t = u ( t ) var [∫ t u s λ ds ] = 0 , thus in this case we choose expression (4.12) and the equation (4.16) is satisfied. Under any risk model with stochastic intensity the formula (4.12) neglects the term E [ Z ] var [∫ t u s λ s ds ] in the equation (4.17). In order to capture the effect of this term, we can find the followingestimate: var [∫ t u s λ s ds ] ≤ E [(∫ t u s λ s ds ) ] ≤ E [ T ∫ t u s λ s ds ] .
16s a consequence, we can choose as premium calculation rule q ( t, y, u ) = E [ Z ] λ ( t, y ) u + θ r E [ Z ] [ λ ( t, y ) + T λ ( t, y ) ] u (4.19)which will be called intensity-adjusted variance principle in this work; using this formula, weensure that E [∫ t q ( s, Y s , u s ) ds ] ≥ E [∫ t u s dC s ] + θ r var [∫ t u s dC s ] ∀ t ∈ [0 , T ]for all {F Yt } t ∈ [0 ,T ] -predictable reinsurance strategies and for any arbitrary level of reinsurancesafety loading θ r > Lemma 4.5.
Under the intensity-adjusted variance premium principle (4.19) , the optimizationproblem (4.1) admits a unique maximizer u ∗ ( t, y ) ∈ (0 , for all ( t, y ) ∈ [0 , T ] × R , which is thesolution to the following equation: θ r E [ Z ] [ T λ ( t, y ) ] u = ∫ D ze η (1 − u ) ze R ( T − t ) dF Z ( z ) − E [ Z ] . (4.20) Proof.
From the expression (4.19) we get ∂q ( t, y, u ) ∂u = E [ Z ] λ ( t, y ) + 2 θ r E [ Z ] [ λ ( t, y ) + T λ ( t, y ) ] u ∀ u ∈ (0 , ∂ q ( t, y, u ) ∂u = 2 θ r E [ Z ] [ λ ( t, y ) + T λ ( t, y ) ] > ∀ u ∈ (0 , . By Lemma 4.1 Ψ u ( t, y ) is strictly concave w.r.t. u and full reinsurance is never optimal becauseof Remark 4.3. Moreover, we notice that A = ∅ as in Lemma 4.3, thus the optimal strategy isunique and it belongs to (0 , intensity-adjustedvariance premium principle leads to optimal strategies which are consistent with the desiredproperties obtained under the other premium calculation principles. Moreover, the reinsurancestrategies under the intensity-adjusted variance premium principle are not deterministic and ex-plicitly depend on the (stochastic) intensity. Hence the problems described in the beginning ofthis subsection are fixed.Using the result given in Example 4.3, it is easy to specialize Lemma 4.5 to the case ofexponentially distributed claims. Example 4.4. (Exponentially distributed claims under the intensity-adjusted variance premiumprinciple)Under the intensity-adjusted variance premium principle (4.19) , suppose that the claims areexponentially distributed with parameter ζ > ηe RT . Then the optimal strategy u ∗ ( t, y ) ∈ (0 , isgiven by u ∗ ( t, y ) = 1 − ζη ( − √ ζζ + 4 θ r [ T λ ( t, y ) ] ) e − R ( T − t ) ( t, y ) ∈ [0 , T ] × R . (4.21) Remark 4.7.
In [Liang and Yuen, 2016] and [Yuen et al., 2015] the authors used, respectively,the variance premium and the expected value principles to obtain optimal reinsurance strategiesin a risk model with multiple dependent classes of insurance business. In those papers the optimalstrategies explicitly depend on the claims intensities, but it is due to the presence of more thanone business line, hence our arguments are not valid there. Nevertheless, in [Yuen et al., 2015]the authors realized that in the diffusion approximation of the classical risk model the variancepremium principle lead to optimal strategies which do not depend on the claims intensities. Infact, this was the main motivation of their work. Their observation confirms our perplexities ofstrategies independent on the claims intensity. . Optimal investment policy Lemma 5.1.
The problem sup w ( t,y,p ) ∈ R Ψ w ( t, y, p ) where Ψ w ( t, y, p ) is defined in (3.5) , admits a unique solution w ∗ ( t, y, p ) for all ( t, y, p ) ∈ [0 , T ] × R × (0 , + ∞ ) given by w ∗ ( t, y, p ) = µ ( t, p ) − Rησ ( t, p ) e R ( T − t ) + pηe R ( T − t ) ∂φ∂p ( t, y, p ) φ ( t, y, p ) . (5.1) Proof.
Since φ ( t, y, p ) >
0, Ψ w ( t, y, p ) is strictly concave w.r.t. w and the result follows from thefirst order condition.We emphasize that the optimal w ∗ is the sum of the classical solution plus an adjustmentterm due to the dependence of the risky asset price coefficients on the stochastic process { P t } . Remark 5.1. If µ, σ are continuous function and φ ∈ C , , then w ∗ is a continuous functionw.r.t. ( t, y, p ) . Corollary 5.1.
Suppose that there exist two functions f ( t, y ) : [0 , T ] × R → (0 , + ∞ ) and g ( t, p ) :[0 , T ] × (0 , + ∞ ) → R such that φ ( t, y, p ) = f ( t, y ) e g ( t,p ) for all ( t, y, p ) ∈ [0 , T ] × R × (0 , + ∞ ) ,with f ( t, y ) > . Then the optimal investment strategy (5.1) reads as follows: w ∗ ( t, p ) = µ ( t, p ) − Rησ ( t, p ) e R ( T − t ) + pηe R ( T − t ) ∂g∂p ( t, p ) . (5.2)
6. Verification Theorem
Now we conjecture a solution to equation (3.3) of the form φ ( t, y, p ) = f ( t, y ) e g ( t,p ) , with f ( t, y ) >
0. Using Lemma 5.1, replacing all the derivatives and performing some calculations,the equation (3.3) reads as follows ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ − ∂f∂t ( t, y ) − b ( t, y ) ∂f∂y ( t, y ) − γ ( t, y ) ∂ f∂y ( t, y ) + [ ηe R ( T − t ) c ( t, y ) + max u ( t,y ) ∈ [0 , Ψ u ( t, y ) ] f ( t, y )+ f ( t, y ) [ − ∂g∂t ( t, p ) − pR ∂g∂p ( t, p ) − p σ ( t, p ) ∂ g∂p ( t, p ) + 12 ( µ ( t, p ) − R ) σ ( t, p ) ] = 0 f ( T, y ) e g ( T,p ) = 1 ∀ ( y, p ) ∈ R × (0 , + ∞ ) (6.1)It is easy to show that if f, g are two solutions of the following Cauchy problems ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ − ∂f∂t ( t, y ) − b ( t, y ) ∂f∂y ( t, y ) − γ ( t, y ) ∂ f∂y ( t, y )+ [ ηe R ( T − t ) c ( t, y ) + max u ( t,y ) ∈ [0 , Ψ u ( t, y ) ] f ( t, y ) = 0 f ( T, y ) = 1 (6.2) ⎧⎪⎨⎪⎩ − ∂g∂t ( t, p ) − pR ∂g∂p ( t, p ) − p σ ( t, p ) ∂ g∂p ( t, p ) + 12 ( µ ( t, p ) − R ) σ ( t, p ) = 0 g ( T, p ) = 0 (6.3)then they solve the Cauchy problem (6.1) and v ( t, x, y, p ) = − e − ηxe R ( T − t ) f ( t, y ) e g ( t,p ) solves theoriginal HJB equation given in (3.1).Before we prove a verification theorem, we must show that our proposed optimal controls areadmissible strategies. See e.g. [Merton, 1969]. emma 6.1. Suppose that (6.2) and (6.3) admit classical solutions with ∂g∂p satisfying the fol-lowing growth condition: ⏐⏐⏐⏐ ∂g∂p ( t, p ) ⏐⏐⏐⏐ ≤ C (1 + | p | β ) ∀ ( t, p ) ∈ [0 , T ] × (0 , + ∞ ) (6.4) for some constants β > and C > . Moreover, assume that E [∫ T | µ ( t, P t ) | P β +1 t dt + ∫ T σ ( t, P t ) P β +2 t dt ] < ∞ . (6.5) Let be u ∗ ( t, y ) as given in Proposition 4.1 and w ∗ ( t, p ) in Lemma 5.1. Let us define the processes u ∗ t . = u ∗ ( t, Y t ) and w ∗ t . = w ∗ ( t, P t ) ; then the pair ( u ∗ t , w ∗ t ) is an admissible strategy, i.e. ( u ∗ t , w ∗ t ) ∈U .Proof. First let us observe that both u ∗ t , w ∗ t are {F t } -predictable processes since u ∗ ( t, u ) and w ∗ ( t, p ) are measurable functions of their arguments and Y is adapted. Moreover, they takevalues, respectively, in [0 ,
1] and in R . Furthermore, using the expression (5.2) we have that E [∫ T | w ∗ t || µ ( t, P t ) − R | dt ] ≤ E [∫ T ( µ ( t, P t ) − R ) ησ ( t, P t ) e R ( T − t ) dt ] + E [∫ T | µ ( t, P t ) − R | P t ηe R ( T − t ) ⏐⏐⏐⏐ ∂g∂p ( t, P t ) ⏐⏐⏐⏐ dt ] ≤ E [∫ T ( µ ( t, P t ) − R ) ησ ( t, P t ) e R ( T − t ) dt ] + C E [∫ T | µ ( t, P t ) | (1 + P βt ) P t dt ] < ∞ and E [∫ T ( w ∗ t σ ( t, P t )) dt ] = E [∫ T ( µ ( t, p ) − R ) η σ ( t, p ) e R ( T − t ) dt ] + E [∫ T σ ( t, P t ) P t η e R ( T − t ) ( ∂g∂p ( t, P t ) ) dt ] + 2 E [∫ T ( µ ( t, p ) − R ) P t η e R ( T − t ) ∂g∂p ( t, P t ) dt ] ≤ E [∫ T ( µ ( t, p ) − R ) η σ ( t, p ) e R ( T − t ) dt ] + C E [∫ T σ ( t, P t ) P t η e R ( T − t ) ( P βt ) dt ] + C E [∫ T | µ ( t, p ) − R | P t η e R ( T − t ) (1 + P βt ) dt ] < ∞ where C denotes any positive constant and the expectations are finite because of the Novikovcondition (2.16) together with (6.4) and (6.5).Now we are ready for the verification argument. Theorem 6.1 (Verification Theorem) . Suppose that (6.2) and (6.3) admit bounded classicalsolutions, respectively f ∈ C , ((0 , T ) × R )) ∩ C ([0 , T ] × R )) and g ∈ C , ((0 , T ) × (0 , + ∞ )) ∩C ([0 , T ] × (0 , + ∞ )) .Let us assume that the conditions (6.4) and (6.5) hold and suppose that ⏐⏐⏐⏐ ∂f∂y ( t, y ) ⏐⏐⏐⏐ ≤ ˜ C (1 + | y | β ) ∀ ( t, y ) ∈ [0 , T ] × R (6.6)19 or some constants β > and ˜ C > . As an alternative, the conditions (6.4) , (6.5) and (6.6) may be replaced by the boundedness of ∂g∂p and ∂f∂y .Then the function v : V → R defined by the following v ( t, x, y, p ) = − e − ηxe R ( T − t ) f ( t, y ) e g ( t,p ) (6.7) is the value function of the reinsurance-investment problem and α ∗ ( t, Y t , P t ) = ( u ∗ ( t, Y t ) , w ∗ ( t, P t )) with u ∗ ( t, y ) given in Proposition 4.1 and w ∗ ( t, p ) in (5.2) is an optimal control.Proof. Let f ( t, y ) : [0 , T ] × R → (0 , + ∞ ) and g ( t, p ) : [0 , T ] × (0 , + ∞ ) → R be functions satisfyingthe assumptions required by Theorem 6.1 and suppose that they are solutions of the Cauchy prob-lems (6.2) and (6.3). Now consider the function φ ( t, y, p ) = f ( t, y ) e g ( t,p ) . As already observed,it satisfies equation (3.3), i.e. it is a solution of the problem ⎧⎨⎩ sup ( u,w ) ∈ [0 , × R H α φ ( t, y, p ) = 0 φ ( t, y, p ) = 1 ∀ ( y, p ) ∈ R × (0 , + ∞ ) . (6.8)Now, taking v ( t, x, y, p ) = − e − ηxe R ( T − t ) φ ( t, y, p ), we have that v is a solution of the Cauchyproblem (3.1). This implies that, for any ( t, x, y, p ) ∈ [0 , T ] × R × R × (0 , + ∞ ) L α v ( s, X αt,x ( s ) , Y t,y ( s ) , P t,p ( s )) ≤ ∀ s ∈ [ t, T ]for all α ∈ U t , where { Y t,y ( s ) } s ∈ [ t,T ] denotes the solution to equation (2.1) with initial condition Y t = y and, similarly, { P t,p ( s ) } s ∈ [ t,T ] denotes the solution to equation (2.13) with initial condition P t = p .Now, from Itˆo’s formula we have that v ( T, X αt,x ( T ) , Y t,y ( T ) , P t,p ( T )) − v ( t, x, y, p ) = ∫ Tt L α v ( s, X αt,x ( s ) , Y t,y ( s ) , P t,p ( s )) ds + M T (6.9)where { M r } r ∈ [ t,T ] is the following stochastic process: M r = ∫ rt w s σ ( s, P s ) ∂v∂x ( s, X αs , Y s , P s ) dW ( P ) s + ∫ rt P s σ ( s, P s ) ∂v∂p ( s, X αs , Y s , P s ) dW ( P ) s + ∫ rt γ ( s, Y s ) ∂v∂y ( s, X αs , Y s , P s ) dW ( Y ) s + ∫ D ∫ rt [ v ( s, X αs − (1 − u ) z, Y s , P s ) − v ( s, X αs , Y s , P s ) ]( m ( ds, dz ) − λ ( s, Y s ) dF Z ( z ) ) . (6.10)Now we prove that { M r } r ∈ [ t,T ] is an {F r } -local martingale. Since the jump term is a realmartingale because v is bounded, we only need to show that E [∫ T ∧ τ n t ( w s σ ( s, P s ) ∂v∂x ( s, X αs , Y s , P s ) ) ds ] < ∞ E [∫ T ∧ τ n t ( P s σ ( s, P s ) ∂v∂p ( s, X αs , Y s , P s ) ) ds ] < ∞ E [∫ T ∧ τ n t ( γ ( s, Y s ) ∂v∂y ( s, X αs , Y s , P s ) ) ds ] < ∞ { τ n } n =1 ,... such that lim n → + ∞ τ n = + ∞ .Taking into account the expression (6.7), we note that ∂v∂x ( t, x, y, p ) = ηe R ( T − t ) e − ηxe R ( T − t ) f ( t, y ) e g ( t,p ) ∂v∂y ( t, x, y, p ) = − e − ηxe R ( T − t ) e g ( t,p ) ∂f∂y ( t, y ) ∂v∂p ( t, x, y, p ) = − e − ηxe R ( T − t ) f ( t, y ) e g ( t,p ) ∂g∂p ( t, p ) . Let us define a sequence of random times { τ n } n =1 ,... as follows: τ n . = inf { s ∈ [ t, T ] | X αs < − n ∨ | Y s | > n } n = 1 , . . . In the sequel of the proof we denote with C n any constant depending on n = 1 , . . . .Then we have that E [∫ T ∧ τ n ( w s σ ( s, P s ) ∂v∂x ( s, X αs , Y s , P s ) ) ds ] = E [∫ T ∧ τ n ( w s σ ( s, P s ) ηe R ( T − s ) e − ηX αs e R ( T − s ) f ( s, Y s ) e g ( s,P s ) ) ds ] ≤ C n E [∫ T ( w s σ ( s, P s ) ) ds ] < ∞ ∀ n = 1 , . . . because w t is admissible and f and g are bounded by hypothesis. Moreover,we have that E [∫ T ∧ τ n ( γ ( s, Y s ) ∂v∂y ( s, X αs , Y s , P s ) ) ds ] = E [∫ T ∧ τ n ( γ ( s, Y s ) e − ηX αs e R ( T − s ) e g ( s,P s ) ∂f∂y ( s, Y s ) ) ds ] ≤ ˜ C E [∫ T ∧ τ n ( γ ( s, Y s ) e − ηX αs e R ( T − s ) e g ( s,P s ) ) (1 + | Y s | β ) ds ] ≤ C n E [∫ T γ ( s, Y s ) ds ] < ∞ ∀ n = 1 , . . . because g is bounded and using the assumptions (2.2) and (6.6). Finally, we obtain that E [∫ T ∧ τ n ( P s σ ( s, P s ) ∂v∂p ( s, X αs , Y s , P s ) ) ds ] = E [∫ T ∧ τ n P s σ ( s, P s ) ( e − ηX αs e R ( T − s ) f ( s, Y s ) e g ( s,P s ) ∂g∂p ( s, P s ) ) ds ] ≤ C E [∫ T ∧ τ n P s σ ( s, P s ) ( e − ηX αs e R ( T − s ) f ( s, Y s ) e g ( s,P s ) ) (1 + | P s | β ) ds ] ≤ C n E [∫ T σ ( s, P s ) ( P s + P β +2 s ) ds ] < ∞ ∀ n = 1 , . . . because f and g are bounded by hypothesis and using conditions (2.14), (6.4) and (6.5). Thus { M r } r ∈ [ t,T ] is an {F r } -local martingale and { τ n } n =1 ,... is a localizing sequence for { M r } r ∈ [ t,T ] .Taking the expected value of both sides of (6.9) with T replaced by T ∧ τ n , we obtain that E [ v ( T ∧ τ n , X αt,x ( T ∧ τ n ) , Y t,y ( T ∧ τ n ) , P t,p ( T ∧ τ n )) | F t ] ≤ v ( t, x, y, p )21or any α ∈ U t , t ∈ [0 , T ∧ τ n ] , n ≥
1. Now notice that E [ v ( T ∧ τ n , X αt,x ( T ∧ τ n ) , Y t,y ( T ∧ τ n ) , P t,p ( T ∧ τ n )) ]= E [ e − ηX αt,x ( T ∧ τ n ) e R ( T ∧ τn − t ) f ( T ∧ τ n , Y T ∧ τ n ) e g ( T ∧ τ n ,P T ∧ τn ) ] ≤ C e − ηne R ( T ∧ τn ) ≤ C thus { v ( T ∧ τ n , X αt,x ( T ∧ τ n ) , Y t,y ( T ∧ τ n ) , P t,p ( T ∧ τ n )) } n =1 ,... is a family of uniformly integrablerandom variables. Hence it converges almost surely. Observing that { τ n } n =1 ,... is a bounded andnon-decreasing sequence, since P [ | X αt | < + ∞ ] = 1 (see (2.18)) and using (2.3) and (2.15), takingthe limit for n → + ∞ , we conclude that E [ v ( T, X αt,x ( T ) , Y t,y ( T ) , P t,p ( T )) | F t ]= lim n → + ∞ E [ v ( T ∧ τ n , X αt,x ( T ∧ τ n ) , Y t,y ( T ∧ τ n ) , P t,p ( T ∧ τ n )) | F t ] ≤ v ( t, x, y, p ) ∀ α ∈ U t , t ∈ [0 , T ] . (6.11)To be precise, we have thatlim n → + ∞ X αt,x ( T ∧ τ n ) = X αt,x ( T − ) = X αt,x ( T ) P -a.s.since the jump of { N t } t ∈ [0 ,T ] occurs at time T with probability zero. Using the final conditionof the HJB equation (3.1), from (6.11) we get E [ U ( X αt,x ( T ))] ≤ v ( t, x, y, p ) ∀ α ∈ U t , t ∈ [0 , T ] . Now note that α ∗ ( t, y, p ) was calculated in order to obtain L α ∗ v ( t, x, y, p ) = 0; replicating thecalculations above, replacing L α with L α ∗ , we find the equality:sup α ∈U t E [ U ( X αt,x ( T )) | Y t = y, P t = p ] = v ( t, x, y, p )thus α ∗ ( t, Y t , P t ) is an optimal control.After the characterization of the value function, we provide a probabilistic representationby means of the Feynman-Kac formula. In preparation for this result, let us introduce a newprobability measure Q ≪ P . Novikov condition (2.16) implies that the process { L t } t ∈ [0 ,T ] definedby L t = e − ( ∫ t | µ ( s,Ps ) − Rσ ( s,Ps ) | ds + ∫ t µ ( s,Ps ) − Rσ ( s,Ps ) dW ( P ) s ) is an {F t } -martingale and we can introduce the following probability measure Q : d Q d P ⏐⏐⏐⏐ F t = L t t ∈ [0 , T ] . (6.12)By Girsanov theorem we know that ˜ W ( P ) t = W ( P ) t + ∫ t µ ( s,P s ) − Rσ ( s,P s ) ds is a Q -Brownian motion andwe can rewrite the risky asset dynamic as d ˜ P t = P t [ R dt + σ ( t, P t ) d ˜ W ( P ) t ] . (6.13)Since the discounted price { ˜ P t = P t e − Rt } t ∈ [0 ,T ] turns out to be an {F t } -martingale, then Q is amartingale or risk-neutral measure for { P t } . We will denote by E Q the conditional expectationwith respect to the martingale measure Q . Let us observe that under Q the dynamics of { Y t } and { R t } do not change. roposition 6.1. Suppose that (6.2) and (6.3) admit classical solutions f ∈ C , ((0 , T ) × R )) ∩C ([0 , T ] × R )) and g ∈ C , ((0 , T ) × (0 , + ∞ )) ∩ C ([0 , T ] × (0 , + ∞ )) , respectively, both bounded with ∂f∂y and ∂g∂p satisfying the growth conditions (6.6) and (6.4) . Then f and g admit the followingFeynman-Kac representations: f ( t, y ) = E [ e − ∫ Tt ( ηe R ( T − s ) c ( s,Y s )+Ψ u ∗ ( s,Y s ) ) ds | Y t = y ] (6.14) g ( t, p ) = − E Q [∫ Tt ( µ ( s, P s ) − R ) σ ( s, P s ) ds | P t = p ] (6.15) where Ψ u ∗ ( t, y ) is the function defined by (3.4) , replacing u with u ∗ ( t, y ) , and Q is the probabilitymeasure introduced in (6.12) .Proof. The result is a simple consequence of the Feynman-Kac theorem.In Section 8 we will provide sufficient conditions which ensure that the functions f and g given in (6.14) and (6.15) are, respectively, C , ((0 , T ) × R ) and C , ((0 , T ) × (0 , + ∞ )) solutionsto the Cauchy problems (6.2) and (6.3).
7. Simulations and numerical results
Here we illustrate some numerical results based on the theoretical framework developed in theprevious sections. In particular, we perform sensitivity analysis of the optimal reinsurance-investment strategy in order to study the effect of the model parameters on the insurer’s decision.
First, we compare the optimal reinsurance strategy under the expected value principle (seeLemma 4.2) and the intensity-adjusted variance premium principle (see Lemma 4.5). In thissubsection the first one will be shortly referred as EVP, while the second one as IAVP. Themain difference is that under EVP we loose the dependence on the stochastic factor, while underIAVP we keep this dependence; moreover, IAVP also depends on the second moment of the r.v. Z introduced in Section 2.In what follows we assume that { Z i } i =1 ,... is a sequence of i.i.d. positive random variables Paretodistributed with shape parameter 1 . . b = 0 . , γ = 0 . Y = 1. For the sake of simplicity, we assume that λ ( t, y ) = λ e y , that is { λ t = λ ( t, Y t ) } t ∈ [0 ,T ] solves dλ t = λ t dY t λ = 0 . , which guarantees that the intensity is positive. Finally, we consider the model parameters inTable 1, using the notation introduced in Section 2. Table 1:
Simulation parameters
Parameter Value T η . θ r . R igure 1: The effect of the risk-aversion parameter η on the optimal initial strategy any variation of the risk-aversion.In Figure 2 we notice that any increase in the reinsurance safety loading leads to a decreaseof the reinsured risks. It is a simple consequence of the well-known law of demand: the higherthe price, the lower the quantity demanded. It is worth noting that under our assumptions thestrategy under IAVP is more sensitive than under EVP. Figure 2:
The effect of the reinsurance safety loading θ r on the optimal initial strategy Finally, in Figure 3 we can see that the insurer increases the protection when the time horizonis higher. Again, the strategy under EVP turns out to be more sensitive to any change of thetime horizon. It is interesting that over 15 years EVP leads to more conservative strategies.We conclude this subsection investigating the dynamical properties of the reinsurance strate-gies under EVP and IAVP . Figure 4 shows that the mean behavior of the optimal reinsurance Under a practical point of view, we simulated the stochastic processes using the classical Euler’s approxima-tion method, with dt = T . igure 3: The effect of the time horizon T on the optimal initial strategy strategy is decreasing over the time interval; nevertheless, under IAVP the strategy cruciallydepends on the stochastic factor, hence the insurer will react to any movement of the claimsintensity, while under EVP she will follow a deterministic strategy. Figure 4:
Dynamical reinsurance strategies. The dashed line represents the optimal (deterministic)strategy under EVP.
Summarizing the main results of our numerical simulations, we can conclude that any varia-tion of the model parameters has the same effect on the optimal strategy under EVP and IAVP,at least from a qualitative point of view. It is important noting that any quantitative comparisonis affected by the parameters initial choice. Nevertheless, we can state that using our model pa-rameters under EVP the strategy is more sensitive with respect to the model parameters, exceptfor the safety loading, but it is dynamically more stable during the time interval [0 , T ], becauseit does not take into account any variation of the claims intensity.25 .2. Investment strategy
Now we illustrate a sensitivity analysis for the investment strategy based on the Corollary 5.1.In our simulations we assumed that the risky asset follows a CEV model, that is dP t = P t [ µ dt + σP βt dW ( P ) t ] P = 1with µ = 0 . , σ = 0 . , β = 0 .
5, while the risk-free interest rate is R = 5% as in the previoussubsection. Let us observe that this model corresponds to (2.13) assuming that µ ( t, p ) = µ and σ ( t, p ) = σp β , with constant µ, σ >
0. The numerical computation of the function g ( t, p )and its partial derivative ∂g∂p ( t, p ) is required by the equation (5.2); for this purpose we usedthe Feynman-Kac representation given in (6.15) evaluated through the standard Monte Carlomethod.In figure 5 we show that the higher is the insurer’s risk aversion, the lower is the total amountinvested in the risky asset. Figure 5:
The effect of the risk-aversion parameter η on the optimal initial strategy Figure 6 illustrates that if the volatility increases, then an increasing portion of the insurer’swealth is invested in the risk-free asset.Finally, if the risk-free interest rate grows up, then the insurer will find it more convenientto invest its surplus in the risk-free asset, as shown in figure 7.Similar results can be found in [Sheng et al., 2014]. In particular, figure 6 confirms the resultobtained in Figure 3a of that paper; in addition, figures 5 and 7 completes the sensitivity analysesperformed there. 26 igure 6:
The effect of the volatility parameter σ on the optimal initial strategy Figure 7:
The effect of the risk-free interest rate R on the optimal initial strategy . Existence and uniqueness of classical solutions In this section we are interested in providing sufficient conditions for existence and uniquenessof the solutions to the PDEs involved in the reinsurance-investment problem, see the Cauchyproblems (6.2) and (6.3) and as a consequence of a classical solution to HJB equation associatedwith our problem.First, let us consider (6.3). The following Lemma prepares the main result.
Lemma 8.1.
Let us define the set D n . = ( n , n ) for n = 1 , . . . and assume that the functions µ ( t, p ) , σ ( t, p ) are Lipschitz-continuous in p ∈ D n , uniformly in t ∈ [0 , T ] . Moreover, assumethat σ ( t, p ) is bounded from below, i.e. there exists a constant δ σ > such that σ ( t, p ) ≥ δ σ forall ( t, p ) ∈ [0 , T ] × (0 , + ∞ ) .Then for each n = 1 , . . . the function k : [0 , T ] × (0 , + ∞ ) → R defined by k ( t, p ) = ( µ ( t, p ) − R ) σ ( t, p ) (8.1) is uniformly Lipschitz-continuous on [0 , T ] × D n .Proof. Firstly, using the Lipschitz-continuity of the parabolic function on the bounded domain D n we have that | k ( t, p ) − k ( t ′ , p ′ ) | = ⏐⏐⏐⏐⏐( µ ( t, p ) − Rσ ( t, p ) ) − ( µ ( t ′ , p ′ ) − Rσ ( t ′ , p ′ ) ) ⏐⏐⏐⏐⏐ ≤ K n ⏐⏐⏐⏐ µ ( t, p ) − Rσ ( t, p ) − µ ( t ′ , p ′ ) − Rσ ( t ′ , p ′ ) ⏐⏐⏐⏐ = K n ⏐⏐⏐⏐ σ ( t ′ , p ′ )[ µ ( t, p ) − R ] − σ ( t, p )[ µ ( t ′ , p ′ ) − R ] σ ( t, p ) σ ( t ′ , p ′ ) ⏐⏐⏐⏐ for a positive constant K n > n . Now, being σ ( t, p ) bounded from below,setting ˜ K n = K n δ σ we have that | k ( t, p ) − k ( t ′ , p ′ ) | ≤ ˜ K n | σ ( t ′ , p ′ )[ µ ( t, p ) − R ] − σ ( t, p )[ µ ( t ′ , p ′ ) − R ] |≤ ˜ K n R | σ ( t, p ) − σ ( t ′ , p ′ ) | + ˜ K n | σ ( t ′ , p ′ ) µ ( t, p ) − σ ( t, p ) µ ( t ′ , p ′ ) |≤ ˜ K n R | σ ( t, p ) − σ ( t ′ , p ′ ) | + ˜ K n | σ ( t ′ , p ′ ) µ ( t, p ) − σ ( t ′ , p ′ ) µ ( t ′ , p ′ ) | + ˜ K n µ ( t ′ , p ′ ) | σ ( t ′ , p ′ ) − σ ( t, p ) | and, observing that any Lipschitz-continuous function on a bounded domain is also bounded,the result is a consequence of our hypotheses. Theorem 8.1.
Suppose that the following conditions are satisfied:1. µ ( t, p ) and σ ( t, p ) are locally Lipschitz-continuous in p , uniformly in t ∈ [0 , T ] , i.e. foreach n = 1 , . . . there exists a positive constant K n such that | µ ( t, p ) − µ ( t, p ′ ) | + | σ ( t, p ) − σ ( t, p ′ ) | ≤ K n | p − p ′ | ∀ p, p ′ ∈ [ n , n ] , t ∈ [0 , T ];
2. for all couple ( t, p ) ∈ [0 , T ] × (0 , + ∞ ) the solution { P t,p ( s ) } s ∈ [ t,T ] does not explode, i.e. P [ sup s ∈ [ t,T ] P t,p ( s ) < ∞ ] = 1; for instance, it is true if we assume the sub-linear growth for σ ( t, p ) : | σ ( t, p ) | ≤ K σ (1 + p ) ∀ p ∈ (0 , + ∞ ) , t ∈ [0 , T ] together with the other hypotheses of this theorem; See [Pascucci, 2011], Theorem 9.11, p. 281. . σ ( t, p ) is bounded from below, i.e. there exists a constant δ σ > such that σ ( t, p ) ≥ δ σ forall ( t, p ) ∈ [0 , T ] × (0 , + ∞ ) ;4. µ ( t, p ) is bounded, i.e. there exists a constant δ µ > such that | µ ( t, p ) | ≤ δ µ for all ( t, p ) ∈ [0 , T ] × (0 , + ∞ ) .Then the function g ( t, p ) given in (6.15) satisfies the Cauchy problem (6.3) and there exists aunique classical solution to (6.3) . Moreover, we have that g ∈ C , ((0 , T ) × (0 , + ∞ )) .Proof. The proof is a consequence of Theorem 1 and Lemma 2 in [Heath and Schweizer, 2000],toghether with Lemma 8.1. We highlight that in order to use those results, we take D n . = ( n , n )for n = 1 , . . . as bounded domains such that (0 , + ∞ ) = ⋃ ∞ n =1 D n . Moreover, we observe that thefunction k defined in (8.1) is bounded, as requested in Lemma 2 of [Heath and Schweizer, 2000],because µ ( t, p ) is bounded and σ ( t, p ) is bounded from below. Remark 8.1.
In [Sheng et al., 2014] the authors found an explicit solution of the Cauchy prob-lem (6.3) in the particular case of the CEV model, i.e. when µ ( t, p ) = µ and σ ( t, p ) = kp β . Now we turn the attention to the second PDE involved in the reinsurance-investment problem,see the Cauchy problem (6.2). Before proving the existence theorem, let us state some preliminaryresults.
Lemma 8.2.
Given a compact set K ∈ R let us assume that H ( t, y, u ) : [0 , T ] × R × K isH¨older-continuous in ( t, y ) ∈ [0 , T ] × R uniformly in u ∈ K with exponent < ξ ≤ . Then max u ∈ K H ( t, y, u ) is H¨older-continuous in ( t, y ) ∈ [0 , T ] × R with exponent < ξ ≤ .Proof. Given t, t ′ ∈ [0 , T ] and y, y ′ ∈ R , let us define h ( u ) = H ( t, y, u ) h ( u ) = H ( t ′ , y ′ , u ) . Then we have that | max u ∈ K h ( u ) − max u ∈ K h ( u ) | ≤ max u ∈ K | h ( u ) − h ( u ) | . (8.2)In fact, observing that | max u ∈ K h ( u ) − max u ∈ K h ( u ) | = { max u ∈ K h ( u ) − max u ∈ K h ( u ) if max u ∈ K h ( u ) ≥ max u ∈ K h ( u )max u ∈ K h ( u ) − max u ∈ K h ( u ) if max u ∈ K h ( u ) < max u ∈ K h ( u )we notice that in the first casemax u ∈ K h ( u ) − max u ∈ K h ( u ) = max u ∈ K [ h ( u ) − h ( u ) + h ( u )] − max u ∈ K h ( u ) ≤ max u ∈ K [ h ( u ) − h ( u )] ≤ max u ∈ K | h ( u ) − h ( u ) | , and in the second case we have thatmax u ∈ K h ( u ) − max u ∈ K h ( u ) ≤ max u ∈ K [ h ( u ) − h ( u )] ≤ max u ∈ K | h ( u ) − h ( u ) | . Now, using inequality (8.2), we have that | max u ∈ K H ( t, y, u ) − max u ∈ K H ( t ′ , y ′ , u ) | ≤ max u ∈ K | H ( t, y, u ) − H ( t ′ , y ′ , u ) |≤ L ( | t − t ′ | ξ + | y − y ′ | ξ )and this completes the proof. 29 orollary 8.1. Let us assume that the following hypotheses hold: • q ( t, y, u ) is bounded and H¨older-continuous in ( t, y ) ∈ [0 , T ] × R uniformly in u ∈ [0 , withexponent < ξ ≤ ; • λ ( t, y ) is bounded and H¨older-continuous in ( t, y ) ∈ [0 , T ] × R with exponent < ξ ≤ .Then max u ( t,y ) ∈ [0 , Ψ u ( t, y ) is H¨older-continuous in ( t, y ) ∈ [0 , T ] × R with exponent < ξ ≤ .Proof. In view of Lemma 8.2, it is sufficient to show that Ψ u ( t, y ) is H¨older-continuous in ( t, y ) ∈ [0 , T ] × R uniformly in u ∈ [0 ,
1] with exponent 0 < ξ ≤
1. Let us recall equation (3.4):Ψ u ( t, y ) = − ηe R ( T − t ) q ( t, y, u ) + λ ( t, y ) ∫ D [ − e η (1 − u ) ze R ( T − t ) ] dF Z ( z )Since e R ( T − t ) is differentiable and bounded on t ∈ [0 , T ], our first hypothesis ensures that thefirst term ηe R ( T − t ) q ( t, y, u ) is H¨older-continuous in ( t, y ) ∈ [0 , T ] × R uniformly in u ∈ [0 ,
1] withexponent 0 < ξ ≤
1. For the second term we notice that it is a product of two bounded andH¨older-continuous functions, in fact ⏐⏐⏐⏐⏐∫ D e η (1 − u ) ze R ( T − t ) dF Z ( z ) − ∫ D e η (1 − u ) ze R ( T − t ′ ) dF Z ( z ) ⏐⏐⏐⏐⏐ ≤ E [⏐⏐⏐ e η (1 − u ) Ze R ( T − t ) − e η (1 − u ) Ze R ( T − t ′ ) ⏐⏐⏐] . Using Lagrange’s theorem, there exists ¯ t ∈ [0 , T ] such that E [⏐⏐⏐ e η (1 − u ) Ze R ( T − t ) − e η (1 − u ) Ze R ( T − t ′ ) ⏐⏐⏐] ≤ E [⏐⏐⏐ Rη (1 − u ) Ze R ( T − ¯ t ) e η (1 − u ) Ze R ( T − ¯ t ) ⏐⏐⏐] | t − t ′ |≤ Rηe RT E [ Ze ηZe RT ] | t − t ′ | and the proof is complete.The following theorem is based on the main result of [Heath and Schweizer, 2000]. Theorem 8.2.
Suppose that the following conditions are satisfied:1. b ( t, y ) and γ ( t, y ) are locally Lipschitz-continuous in y , uniformly in t ∈ [0 , T ] , i.e. for each n = 1 , . . . there exists a positive constant K n such that | b ( t, y ) − b ( t, y ′ ) | + | γ ( t, y ) − γ ( t, y ′ ) | ≤ K n | y − y ′ | ∀ y, y ′ ∈ [ − n, n ] , t ∈ [0 , T ];
2. for all couple ( t, y ) ∈ [0 , T ] × R the solution { Y t,y ( s ) } s ∈ [ t,T ] does not explode, i.e. P [ sup s ∈ [ t,T ] Y t,y ( s ) < ∞ ] = 1; for instance, it is true when we assume that for some positive constant K | b ( t, y ) | + | γ ( t, y ) | ≤ K (1 + | y | ) ∀ t ∈ [0 , T ] , y ∈ R together with the previous assumption;
3. there exists a constant δ γ > such that γ ( t, y ) ≥ δ γ ; See [Pascucci, 2011], Theorem 9.11, p. 281. . the intensity function λ ( t, y ) is bounded and H¨older-continuous in ( t, y ) ∈ [0 , T ] × R withexponent < ξ ≤ ;5. the reinsurance premium q ( t, y, u ) is bounded and H¨older-continuous in ( t, y ) ∈ [0 , T ] × R uniformly in u ∈ [0 , with exponent < ξ ≤ .Then the function f ( t, y ) defined in (6.14) satisfies the Cauchy problem (6.2) and there exists aunique classical solution to (6.2) . Moreover, we have that f ∈ C , ((0 , T ) × R ) .Proof. The proof is a consequence of Theorem 1 and Lemma 2 in [Heath and Schweizer, 2000]together with Corollary 8.1, observing that under our assumptions max u ( t,y ) ∈ [0 , Ψ u ( t, y ) is con-tinuous and bounded from above. A. Appendix
Proof of Lemma 2.1.
First, let us start considering all the [0 , D ]-indexed processes { H ( t, z ) } t ∈ [0 ,T ] of this type: H ( t, z ) = ˜ H t A ( z ) t ∈ [0 , T ] , A ∈ [0 , D ]where { ˜ H t } t ∈ [0 ,T ] is a nonnegative and {F t } -predictable process. Using the independence be-tween { N t } t ∈ [0 ,T ] and { Z n } n ≥ we have that E [∫ T ∫ D H ( t, z ) m ( dt, dz ) ] = E [∑ n ≥ ˜ H T n A ( Z n ) { T n ≤ T } ] = ∑ n ≥ P [ Z n ∈ A ] E [ ˜ H T n { T n ≤ T } ] = P [ Z ∈ A ] E [∑ n ≥ ˜ H T n { T n ≤ T } ] = P [ Z ∈ A ] E [∫ T ˜ H t λ t dt ] = E [∫ D ∫ T H ( t, z ) dF Z ( z ) λ t dt ] Using [Br´emaud, 1981, App. A1, T4 Theorem, p.263] this result can be extended to all non-negative, {F t } -predictable and [0 , D ]-indexed process { H ( t, z ) } t ∈ [0 ,T ] and this completes theproof. Proof of Proposition 2.1.
For any constant strategy α t = ( u, w ) with u ∈ [0 ,
1] and w ∈ R wehave that E [ e − ηX αt,x ( T ) | F t ] == e − ηxe R ( T − t ) E [ e − η ∫ Tt e R ( T − s ) [ c ( s,Y s ) − q ( s,Y s ,u )] ds e η ∫ Tt ∫ D e R ( T − r ) (1 − u ) z m ( dr,dz ) | F t ] ×× E [ e − η ∫ Tt e R ( T − s ) w [ µ ( s,P s ) − R ] ds e − η ∫ Tt e R ( T − s ) wσ ( s,P s ) dW ( P ) s | F t ] (A.1)because of the independence between the financial and the insurance markets. In particular, forthe null strategy α t = (0 , E [ e − ηX (0 , t,x ( T ) | F t ] ≤≤ e − ηxe R ( T − t ) e η KR ( e R ( T − t ) − E [ e η ∫ Tt ∫ D e R ( T − r ) z m ( dr,dz ) | F t ] . E [ e ηe RT ∫ Tt ∫ D z m ( dr,dz ) | F t ] = E [ e ηe RT ∑ NTi = Nt Z i | F t ]= ∑ n ≥ N t E [ e ηe RT ∑ ni = Nt Z i | F t ] P [ N T = n | F t ]= ∑ n ≥ N t E [ n ∏ i = N t e ηe RT Z i | F t ] P [ N T = n | F t ]= ∑ n ≥ N t E [ e ηe RT Z | F t ] ( n − N t ) P [ N T = n | F t ]= ∑ n ≥ E [ e ηe RT Z ] n P [ N T − N t = n | F t ]= ∑ n ≥ E [ e ηe RT Z ] n E [ (∫ Tt λ s ds ) n n ! e − ∫ Tt λ s ds | F t ] = E [ e ( E [ e ηeRT Z ] − ∫ Tt λ s ds | F t ] < ∞ ⟨ P = 1 ⟩ because of the Assumption 2.1. Proof of Lemma 2.2.
Assume that there exists a positive constant K ′ such that | µ ( t, p ) | + σ ( t, p ) ≤ K ′ ∀ ( t, p ) ∈ [0 , T ] × (0 , + ∞ ) . From the proof of the Proposition 2.1 (see above), we know that for any constant strategy α t = ( u, w ) with u ∈ [0 ,
1] and w ∈ R the equation (A.1) holds. Now, using the inequality (2.10),we have that E [ e − ηX αt,x ( T ) | F t ] ≤≤ e − ηxe R ( T − t ) e η KR ( e R ( T − t ) − E [ e η ∫ Tt ∫ D e R ( T − r ) (1 − u ) z m ( dr,dz ) | F t ] ×× E [ e − η ∫ Tt e R ( T − s ) w [ µ ( s,P s ) − R ] ds e − η ∫ Tt e R ( T − s ) wσ ( s,P s ) dW ( P ) s | F t ] ≤ C e − ηx e η KR ( e R ( T − t ) − E [ e ηe RT ∫ Tt ∫ D z m ( dr,dz ) | F t ] E [ e − ηwe RT ∫ Tt σ ( s,P s ) dW ( P ) s | F t ]where C is a positive constant and the first expectation is finite because of the proof of theProposition 2.1.Now let us define the stochastic process { h t } t ∈ [0 ,T ] as h t = ηwe RT σ ( t, P t )and set L t = e − ∫ t h s dW ( P ) s − ∫ t h s ds Since h t is bounded, the Novikov condition is satisfied: E [ e ∫ T h s ds ] < ∞ . This allows us to introduce a new probability measure ˆ P using the change of measure given by L t = d ˆ P d P ⏐⏐⏐⏐ F t . Using the Kallianpur-Striebel formula, we obtain that E [ e − ηwe RT ∫ Tt σ ( s,P s ) dW ( P ) s | F t ] = E [ e − ∫ Tt h s dW ( P ) s | F t ]= E [ L T e ∫ Tt h s ds | F t ] L t ≤ E ˆ P [ e ∫ Tt h s ds | F t ] < ∞ and the proof is complete. 32 roof of Lemma 3.1. Looking at (2.17), we apply Itˆo’s formula to the stochastic process f ( t, X αt , Y t , P t ): f ( t, X αt , Y t , P t ) = f (0 , X α , Y , P ) + ∫ t L α f ( s, X αs , Y s , P s ) ds + m t where m t = ∫ t w s σ ( s, P s ) ∂f∂x ( s, X αs , Y s , P s ) dW ( P ) s + ∫ t P s σ ( s, P s ) ∂f∂p ( s, X αs , Y s , P s ) dW ( P ) s + ∫ t γ ( s, Y s ) ∂f∂y ( s, X αs , Y s , P s ) dW ( Y ) s + ∫ D ∫ t [ f ( s, X αs − (1 − u ) z, Y s , P s ) − f ( s, X αs , Y s , P s ) ]( m ( ds, dz ) − λ ( s, Y s ) dF Z ( z ) ) . (A.2)We only need to prove that this is an {F t } -martingale.Let us observe that E [∫ T ( w s σ ( s, P s ) ∂f∂x ( s, X αs , Y s , P s ) ) ds ] < ∞ E [∫ T ( P s σ ( s, P s ) ∂f∂p ( s, X αs , Y s , P s ) ) ds ] < ∞ E [∫ T ( γ ( s, Y s ) ∂f∂y ( s, X αs , Y s , P s ) ) ds ] < ∞ because all the partial derivatives are bounded and using, respectively, the definition of the set U , (2.14) and (2.2).Thus the first three integrals in (A.2) are well defined and, according to the Itˆo integral theory,they are martingales. Finally, the jump term in (A.2) is a martingale too, being the function f bounded. Acknowledgements
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