Reduced-form setting under model uncertainty with non-linear affine processes
aa r X i v : . [ q -f i n . M F ] J un Reduced-form setting under model uncertainty withnon-linear affine intensities
Francesca Biagini ∗ Katharina Oberpriller † July 1, 2020
Abstract
In this paper we extend the reduced-form setting under model uncertainty introduced in[5] to include intensities following an affine process under parameter uncertainty, as definedin [15]. This framework allows to introduce a longevity bond under model uncertainty in aconsistent way with the classical case under one prior, and to compute its valuation numerically.Moreover, we are able to price a contingent claim with the sublinear conditional operator suchthat the extended market is still arbitrage-free in the sense of “No Arbitrage of the first kind”as in [6].
Keywords: sublinear expectation, reduced-from framework, non-linear affine processes, arbitrage-free pricing
Mathematics Subject Classification (2020):
JEL Classification:
C02, G22
Aim of this paper is to extend the reduced-form setting under model uncertainty as introducedin [5] to include non-linear affine intensities as defined in [15]. In this way we are able to introducea longevity bond under model uncertainty in an arbitrage-free way and numerically compute itsvalue process in several examples. Furthermore, we apply these results to the arbitrage-free pricingof a general contingent claim under model uncertainty.More precisely, in [5] the classical reduced-form framework as in [7] is extended under model un-certainty by defining a sublinear conditional operator with respect to a progressively enlargedfiltration G and a family of probability measures possibly mutually singular to each other, whichis an extension of the sublinear conditional operator with respect to F introduced in [38]. In thissetting no specific structure or assumptions are made for the intensity process. In the last fewyears several papers dealing with short rate modeling under model uncertainty have been pub-lished, e.g., [23], [22], [24], [16], [15]. A more general approach is treated in [15] by consideringaffine processes under parameter uncertainty, called non-linear affine processes, as an extension ofthe non-linear Lévy processes in [34]. More specifically, one-dimensional non-linear affine processesare defined as a family of semimartingale laws whose differential characteristics are bounded fromabove and below by affine functions of the current states. In financial applications, affine processesare not only relevant for short rate models but also for modeling the stochastic mortality/defaultintensity, e.g., in [12], [8], [44] and [30], as they allow analytically tractable models.Here we wish to provide the most general reduced-form setting under model uncertainty which al-lows numerical tractability or explicit computation for pricing insurance liabilities or credit deriva-tives. Hence we extend the results in [5] by representing the mortality intensity as non-linear affineprocesses in the sense of [15]. By doing so we are able to construct a general market model, where ∗ Department of Mathematics, Workgroup Financial and Insurance Mathematics, University of Munich (LMU),Theresienstraße 39, 80333 Munich, Germany. Email: [email protected] † Department of Mathematics of Natural, Social and Life Sciences, Gran Sasso Science Institute (GSSI), Viale F.Crispi 7, 67100 L’Aquila, Italy. Email: [email protected] F -martingales and the intensity process is a non-linear affine process underthe considered (time-dependent increasing) families of probability measures. The associated sub-linear conditional operator can then be used to evaluate insurance products by taking into accountthe (non-linear) affine structure of the mortality intensity. Furthermore, we give some examplesfor families of probability measures such that the market model satisfies the required assumptions.From a mathematical point of view the construction of the sublinear conditional operator requiressome regularity assumptions for the families of probability measures as in [38]. In our contextthe difficulty lies in constructing families of priors which satisfies these assumptions as well as thedesired properties concerning the market model and the affine structure of the intensity.In general, the mortality intensity is used to define the survival index and can be seen as buildingblock for mortality linked securities [10]. These kind of financial instruments which started appear-ing on the market around 2003 have the aim to reduce the mortality and longevity risk connectedto life insurance and pension products. One of the basic products of this type are longevity bondswhich pay the survivor index at the maturity and it is common to price them with the risk-neutralmeasure such that the extended market including the longevity bond is arbitrage-free [10]. In thiswork we are able to introduce the definition of a longevity bond under model uncertainty in aconsistent way with the classical setting under one prior. To this purpose, we use the sublinearconditional operator of [5]. As already mentioned in [5], the sublinear conditional operator is apriori not càdlàg which can lead to problems as càdlàg paths are a common standard assumption.This problem is solved in [5] by considering a fixed set of probability measures. Here this resultdoes not hold because we need to work with time-dependent increasing families of priors in orderto include non-linear affine intensities. Nevertheless, we are able to find conditions such that thereexists a càdlàg modification for the conditional sublinear operator. By generalizing the represen-tation of the sublinear expectation with Riccati equations from [15], we are able to numericallycompute the value of a longevity bond for some relevant examples. Moreover, numerical compu-tations can also be used for the valuation of general endowment contracts under an independenceassumption between the asset’s price process and the mortality intensity.Motivated by the valuation of the longevity bond, we examine if the sublinear conditional operatorin [5] can be used for pricing a contingent claim under model uncertainty such that the extendedmarket is arbitrage-free. To do so, we first need to choose an appropriate definition of arbitrage ina continuous time setting under model uncertainty. While for the discrete time setting there existsa broad literature about no arbitrage and related concepts under model uncertainty, e.g., [1], [4], [9]and [36], the situation is different for the continuous case. In [45] no-arbitrage is studied within asetting of volatility uncertainty. In [6] they introduce a robust version of arbitrage of the first kindand derive the fundamental theorem of asset pricing. By applying this definition to our setting,we show that the extended sublinear operator can be used to price a contingent claim such thatthe extended market allows no arbitrage of the first kind under model uncertainty as in [6]. Thisresult requires assumptions about the trading strategies which are however not restrictive in aninsurance setting. Moreover, we discuss the relation of this valuation to the superhedging price ofa contingent claim under model uncertainty given in [5].The paper is organized as follows. In Section 2 we outline the setting in [5] and extend the defi-nition of the sublinear conditional operator with respect to time-dependent increasing families ofprobability measures instead of a fixed set as in the original framework. In Section 3 we introducethe definition of a non-linear affine process defined as in [15]. Next, we define a market modelunder uncertainty combining the two settings and illustrate this with some examples in Section4. In Section 5 we give the definition of a longevity bond under model uncertainty and derive itsnumerical approximation via Riccati equations. Moreover, we show under which conditions it ispossible to find a càdlàg quasi-sure modification of the sublinear conditional operator and provethat these assumptions are satisfied in the examples given in Section 4. In Section 6 we introducethe definition of no arbitrage under first kind in our framework and study arbitrage-free pricing ofa contingent claim via the sublinear conditional operator.2 Reduced-form setting under model uncertainty
A reduced-form setting for credit and insurance markets under model uncertainty is introduced inthe paper [5] by defining a sublinear conditional operator with respect to a progressively enlargedfiltration and a family of probability measures possibly mutually singular to each other. This isnot a straight forward extension of the construction in [38], because the approach in [38] relies onspecial properties of the natural filtration generated by the canonical process which are not anylonger satisfied by the enlarged filtration. For example, the Galmarino’s test cannot be used inthis extended framework, as the assumptions under which it holds are not satisfied in the enlargedfiltration. In the following, we recall the approach in [5] in a more general version by taking intoaccount families of probability measures ( P ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω on a space Ω as in [38] instead of afixed set P on Ω as in [5].Fix T > and consider the space Ω = C ([0 , T ] , R ) of continuous functions ω = ( ω t ) t ∈ [0 ,T ] in R starting from zero, which is equipped with the topology of locally uniform convergence and istherefore a Polish space. The Borel σ -algebra on this space is given by F = B (Ω) and the setof probability measures on (Ω , F ) by P (Ω) . We assume that P (Ω) is endowed with the topologyof weak convergence. Furthermore, we denote by B := ( B t ) t ∈ [0 ,T ] the canonical process, i.e., B t ( ω ) = ω t , t ∈ [0 , T ] and its corresponding raw filtration by F := ( F t ) t ∈ [0 ,T ] with F = {∅ , Ω } and F T := W t ∈ [0 ,T ] F t = F . For every given P ∈ P (Ω) and t ∈ [0 , T ] , we define N Pt as thecollection of sets which are ( P, F t ) -null and we consider the following filtration F ∗ := ( F ∗ t ) t ∈ [0 ,T ] defined by F ∗ t := F t ∨ N ∗ t , N ∗ t := \ P ∈P (Ω) N Pt . (2.1)For a given family of probability measures P on Ω we define the σ -algebra F P T by F P T := F ∨ N P T , N P T := \ P ∈P N PT (2.2)and the filtration F ∗ , P := ( F ∗ , P t ) t ∈ [0 ,T ] is given by F ∗ , P t := F ∗ t ∨ N P T , t ∈ [0 , T ] , (2.3)where N P T is the collection of sets which are ( P, F T ) -null for all P ∈ P . We follow the approachof [38] for defining sublinear expectations and introduce the following notation. Let τ be a [0 , T ] -valued F -stopping time and ω ∈ Ω . For every ˜ ω ∈ Ω , the concatenation process ω ⊗ τ ˜ ω :=(( ω ⊗ τ ˜ ω ) t ) t ∈ [0 ,T ] of ( ω, ω ′ ) at τ is given by ( ω ⊗ τ ˜ ω ) t := ω t [0 ,τ ( ω )) ( t ) + ( ω τ ( ω ) + ˜ ω t − τ ( ω ) ) [ τ ( ω )) ,T ] ( t ) , t ∈ [0 , T ] . (2.4)Furthermore, for every function X on Ω we define the function X τ,ω on Ω by X τ,ω (˜ ω ) := X ( ω ⊗ τ ˜ ω ) , ˜ ω ∈ Ω . (2.5)Given a probability measure P ∈ P (Ω) and the regular conditional probability distribution P ωτ of P given F τ , we consider the probability measure P τ,ω ∈ P (Ω) given by P τ,ω ( A ) := P ωτ ( ω ⊗ τ A ) , A ∈ F , (2.6)with ω ⊗ τ A = { ω ⊗ τ ˜ ω : ˜ ω ∈ A } . Note that P ωτ is concentrated on the paths which coincide with ω up to time τ ( ω ) .For any ( s, ω ) ∈ [0 , T ] × Ω we fix the sets P ( s, ω ) ⊆ P (Ω) and assume that P ( s, ω ) = P ( s, ˜ ω ) if ω | [0 ,s ] = ˜ ω | [0 ,s ] . The set P (0 , ω ) is independent of ω and from now on denoted by P . For a stopping time σ we put P ( σ, ω ) := P ( σ ( ω ) , ω ) . Galmarino’s Test [42, Exercise 4.21]: Let
Ω = C ( R + , R ) , F the Borel σ -algebra with respect to the topologyof locally uniform convergence and F be the raw filtration generated by the canonical process B on Ω . Then a F -measurable function τ : Ω → R + is a F -stopping time if and only if τ ( ω ) ≤ t and ω | [0 ,t ] = ω ′ | [0 ,t ] imply τ ( ω ) = τ ( ω ′ ) .Furthermore, given a F -stopping time τ , and F -measurable function f is F τ -measurable if and only if f = f ◦ ι τ ,where ι τ : Ω → Ω is the stopping map ( ι τ ( ω )) t = ω t ∧ τ ( ω ) . ssumption 2.1. Let ( s, ω ) ∈ [0 , T ] × Ω , P ∈ P ( s, ω ) and τ be a stopping time such that T ≥ τ ≥ s .Set η := τ s,ω − s , then1. Measurability:
The graph { ( P ′ , ω ) : ω ∈ Ω , P ′ ∈ P ( τ, ω ) } ⊆ P (Ω) × Ω is analytic.2. Invariance: P η,ω ∈ P ( τ, ω ⊗ s ω ) for P -a.e. ω ∈ Ω .3. Stability under Pasting: If ν : Ω → P (Ω) is an F η -measurable kernel and ν ( ω ) ∈ P ( τ, ω ⊗ s ω ) for P -a.e. ω ∈ Ω , then the measure defined by P ( A ) = Z Z ( A ) η,ω ( ω ′ ) ν ( dω ′ ; ω ) P ( dω ) , A ∈ F , (2.7) is an element of P ( s, ω ) . The following proposition is the main result in [38, Theorem 2.3].
Proposition 2.2.
Let Assumption 2.1 hold true, σ ≤ τ ≤ T be F -stopping times and X : Ω → R be an upper semianalytic function on Ω . Then the function E τ ( X ) defined by E τ ( X )( ω ) := sup P ∈P ( τ,ω ) E P [ X τ,ω ] , ω ∈ Ω , (2.8) is F ∗ τ -measurable and upper semianalytic. Moreover E σ ( X )( ω ) = E σ ( E τ ( X ))( ω ) for all ω ∈ Ω . (2.9) Furthermore, the following consistency condition is fulfilled, i.e., E τ ( X ) = ess sup P P ′ ∈P ( τ ; P ) E P ′ [ X |F τ ] P -a.s. for all P ∈ P , (2.10) where P ( τ ; P ) = { P ′ ∈ P : P ′ = P on F τ } . The family of sublinear conditional expectations ( E t ) t ∈ [0 ,T ] is called ( P , F ) -conditional expectation.We now enlarge the underlying space to introduce a random time ˜ τ , which is not an F -stoppingtime but has an F -progressively measurable intensity process µ to represent a totally unexpecteddefault or decease time under model uncertainty. Let ˆΩ be another Polish space equipped with itsBorel σ -algebra B ( ˆΩ) . On the product space ( ˜Ω , G ) := (Ω × ˆΩ , B (Ω) ⊗ B ( ˆΩ)) we adopt the followingconventions. For every function or process X on (Ω , B (Ω)) we denote its natural immersion intothe product space by X (˜ ω ) := X ( ω ) for all ω ∈ Ω and similarly for processes on ( ˜Ω , B ( ˜Ω)) .Furthermore, for every sub- σ -algebra A of B (Ω) , the natural extension as a sub- σ -algebra of G on ( ˜Ω , G ) is given by A ⊗ {∅ , ˜Ω } , similarly for sub- σ -algebras of B ( ˆΩ) .We fix a probability measure ˆ P on ( ˆΩ , B ( ˆΩ)) such that ( ˆΩ , B ( ˆΩ) , ˆ P ) is an atomless probabilityspace, i.e., there exists a random variable with an absolutely continuous distribution. Moreover,let ξ be a Borel-measurable surjective random variable ξ : ( ˆΩ , B ( ˆΩ) , ˆ P ) → ([0 , , B ([0 , with uniform distribution, i.e., ξ ∈ U ([0 , . Without loss of generality we assume B ( ˆΩ) = σ ( ξ ) .The family of all probability measures on ( ˜Ω , G ) is denoted by P ( ˜Ω) . In particular we are interestedin the following families of probability measures ( ˜ P ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω with ˜ P ( t, ω ) := { ˜ P ∈ P ( ˜Ω) : ˜ P = P ⊗ ˆ P , P ∈ P ( t, ω ) } (2.11)for ( t, ω ) ∈ [0 , T ] × Ω . As P (0 , ω ) does not depend on ω this also holds for ˜ P (0 , ω ) which isdenoted by ˜ P . Moreover, we consider an R -valued, F -adapted, continuous and increasing process Γ := (Γ t ) t ≥ on (Ω , B (Ω)) with Γ := 0 and Γ ∞ := + ∞ such that Γ t := Z t µ s ds, t ≥ , for all t ≥ , for all ω ∈ Ω , (2.12)4here µ = ( µ t ) t ≥ is a nonnegative F -progressively measurable stochastic process with R t µ s ( ω ) ds < ∞ for all t ≥ , ω ∈ Ω . On ˜Ω = Ω × ˆΩ we define the stopping time ˜ τ by ˜ τ = inf { t ≥ e − Γ t ≤ ξ } = inf { t ≥ t ≥ − ln ξ } , (2.13)where we use the convention inf ∅ = ∞ .We define the filtration H := ( H t ) t ∈ [0 ,T ] on ˜Ω which is generated by the process H := ( H t ) t ∈ [0 ,T ] with H t := { ˜ τ ≤ t } , t ∈ [0 , T ] , (2.14)and consider the enlarged filtration G := ( G t ) t ∈ [0 ,T ] with G t := F t ∨ H t , t ∈ [0 , T ] . With thisconstruction it holds G = F T ⊗ σ ( ξ ) = H T ∨ F T = σ (˜ τ ) ∨ F T . As in (2 . we denote by G ∗ the corresponding universally completed filtration. Moreover, let G P := G ∨ N PT for P ∈ P and G P := G ∨ N P T with N P T defined in (2 . . In addition, we define L ( ˜Ω) as the space of all R -valued G P -measurable functions, where we use the following convention. For every ˜ P ∈ P ( ˜Ω) , we set E ˜ P [ X ] := E ˜ P [ X + ] − E ˜ P [ X − ] if E ˜ P [ X + ] or E ˜ P [ X − ] is finite and E ˜ P [ X ] := −∞ if E ˜ P [ X + ] = E ˜ P [ X − ] = + ∞ . Furthermore, we introduce the set L ( ˜Ω) := { ˜ X | ˜ X : ( ˜Ω , G P ) → ( R , B ( R )) measurable function such that ˜ E ( | ˜ X | ) < ∞} . Here ˜ E denotes the upper expectation associated to ˜ P defined as ˜ E ( ˜ X ) := sup ˜ P ∈ ˜ P E ˜ P [ ˜ X ] , ˜ X ∈ L ( ˜Ω) . One important step for the main result of [5] is Proposition 2.13 in [5]. A similar result has alsobeen derived in Proposition 2.2 in [11].
Proposition 2.3.
Let t ∈ [0 , T ] . If ˜ X is a real-valued σ (˜ τ ) ∨ F t -measurable function on ˜Ω , thenthere exists a unique measurable function ϕ : ( R + × Ω , B ( R + ) ⊗ F t ) → ( R , B ( R )) , such that ˜ X ( ω, ˆ ω ) = ϕ (˜ τ ( ω, ˆ ω ) , ω ) , ( ω, ˆ ω ) ∈ ˜Ω . (2.15)The existence of such a measurable function ϕ does not depend on the structure of the consideredfamily of probability measures on ˜Ω , as the proof is based on a monotone class argument. Moreover,the other crucial point to extend the sublinear operator to the enlarged space is Proposition 2.2.Thus, we are able to state a generalized version of Theorem 2.18, Proposition 2.21 in [5]. Proposition 2.4.
Let Assumption 2.1 hold for ( P ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω and consider an upper semi-analytic function ˜ X on ˜Ω such that ˜ X ∈ L ( ˜Ω) or ˜ X is G P -measurable and nonnegative. If t ∈ [0 , T ] , then the following function ˜ E t ( ˜ X ) := { ˜ τ ≤ t } E t ( ϕ ( x, · )) | x =˜ τ + { ˜ τ>t } E t ( e Γ t E ˆ P [ { ˜ τ>t } ˜ X ]) (2.16) is well-defined, where ϕ is the measurable function ϕ : ( R + × Ω , B ( R + ) ⊗ F P T ) → ( R , B ( R )) , (2.17) such that ˜ X ( ω, ˆ ω ) = ϕ (˜ τ ( ω, ˆ ω ) , ω ) , ( ω, ˆ ω ) ∈ ˜Ω . (2.18) Furthermore, for every t ∈ [0 , T ] the function ˜ E t ( ˜ X ) is upper semianalytic and measurable withrespect to G ∗ t and G P and satisfies the following consistency condition, i.e., for every t ∈ [0 , T ]˜ E t ( ˜ X ) = ess sup ˜ P ˜ P ′ ∈ ˜ P ( t ; ˜ P ) E ˜ P ′ [ ˜ X |G t ] ˜ P -a.s. for all ˜ P ∈ ˜ P , (2.19) where ˜ P ( t ; ˜ P ) = { ˜ P ′ ∈ ˜ P : ˜ P ′ = ˜ P on G t } . ( ˜ E t ) t ∈ [0 ,T ] is called ( ˜ P , G ) -conditional expectation.In general, the ( ˜ P , G ) -conditional expectation does not satisfy a strong tower property as the ( P , F ) -conditional expectation does in (2.9). However, it is shown in Theorem 2.22 in [5] that ( ˜ E t ( ˜ X )) t ∈ [0 ,T ] fulfills a weak form of time-consistency, which means ˜ E s ( ˜ E t ( ˜ X )) ≥ ˜ E s ( ˜ X ) for all ≤ s ≤ t ≤ T ˜ P -a.s. for all ˜ P ∈ ˜ P . Moreover, for the fundamental building blocks of life insurance liabilities, namely term insurance,annuity and pure endowment contract, it is proved in Proposition 2.31 in [5] that the strong towerproperty is satisfied.
Remark 2.5.
We note that the results in [5] are also valid if we replace Ω by C x ( R + , R d ) orthe space D x ( R + , R d ) of càdlàg functions in R d starting in x ∈ R d equipped with the topologyof locally uniform convergence or with the Skorokhod topology, respectively. In the last case ageneralized version of Proposition 2.2 is derived in Theorem 4.29 in [21], where the definition ofthe concatenation process in (2 . needs to be adapted in the following way ( ω ⊗ τ ˜ ω ) t := ω t [0 ,τ ( ω )) ( t ) + ( ω τ ( ω ) + ˜ ω t − τ ( ω ) − x ) [ τ ( ω )) , ∞ ) ( t ) , t ≥ with ω, ˜ ω ∈ Ω x and τ stopping time. Remark 2.6.
In general, we do not need to assume the existence of the intensity process µ =( µ t ) t ≥ as in [5] in order to define ( ˜ P , G ) -conditional expectations, see (2 . . However, with therepresentation of Γ as in (2 . we get more tractable results and µ will also be necessary for usingthe framework of [15].The above construction of the product space ˜Ω via the stopping time ˜ τ is a special case of theCox model, see e.g., Remark 2.24 (a) in [3], which was suggested for modeling credit risk the firsttime in [29]. However, the construction of the stopping time ˜ τ in (2 . can be generalized toinclude other possible distributions for ˜ τ . Note that this will change the definition of ( ˜ E t ) t ∈ [0 ,T ] in (2 . . The Cox model can also be generalized by considering a process Γ with càdlàg instead ofcontinuous paths as done in [20]. Unfortunately, we cannot transfer this case to the setting of [5]as the construction of the operator ( ˜ E t ) t ∈ [0 ,T ] requires the continuity of Γ , see the proofs of Lemma2.10 and Proposition 2.11 in [5].In the framework of insurance modeling we now wish to apply the above results to the valuationof insurance products under model uncertainty. In [5] it is shown that the conditional sublinearoperator ( ˜ E t ) t ∈ [0 ,T ] can be used as pricing operator for life insurance liabilities. For simplicitywe focus on endowments, i.e., contracts with payoff { ˜ τ>T } Y , where Y is an F ∗ , P T -measurablenonnegative upper semianalytic function on Ω such that E ( Y ) := sup P ∈P E P [ Y ] < ∞ . In thiscase the payment is made at the maturity of the contract only if the default event does not occurbefore the maturity date. For this contract the following valuation formula is deduced in Lemma2.26 in [5]. Lemma 2.7.
Let Y = Y ( ω ) , ω ∈ Ω , be an F ∗ , P T -measurable upper semianalytic function such that E ( | Y | ) < ∞ . Then for every t ∈ [0 , T ] , { ˜ τ>T } and Y e − R Tt µ u du are upper semianalytic functions and belong to L ( ˜Ω) . Furthermore, if ( P ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω satis-fies Assumption 2.1, the following holds pathwisely for every t ∈ [0 , T ]˜ E t ( Y { ˜ τ>T } ) = { ˜ τ>t } E t ( Y e − R Tt µ u du ) . (2.20)In order to evaluate (2 . we wish to use the results on affine processes under parameter un-certainty from the paper [15]. However, we need first to embed the framework of [15] into oursetting. 6 Affine processes under parameter uncertainty
We now briefly recall the framework of [15]. Consider a probability measure P ∈ P (Ω) such that thecanonical process B is a continuous ( P, F ) -semimartingale such that B = B + M P + A P , where A P is a stochastic process with continuous paths of finite variation P -a.s., M P is a continuous ( P, F ) -local martingale and A P = M P = 0 . The ( P, F ) -characteristics of the semimartingale B with such a decomposition are then given by the pair ( A P , C ) with C = h M P i . From now on weonly consider semimartingales with absolutely continuous (a.c.) characteristics ( β P , α ) , i.e., withpredictable processes β P and α ≥ such that for all t ∈ [0 , T ] A Pt = Z t β Ps ds, C t = Z t α s ds. We define the set P acsem = { P ∈ P (Ω) | B is a ( P, F )-semimartingale with a.c. characteristics } . To consider model risk a parameter vector θ = ( b , b , a , a ) is introduced. For b i < b i , i = 0 , , and a i < a i , i = 0 , , we define the compact set Θ := [ b , b ] × [ b , b ] | {z } := B × [ a , a ] × [ a , a ] | {z } := A ⊂ R × R ≥ . (3.1)Moreover, we define for x ∈ R the following set-valued functions for A and Bb ∗ ( x ) := { b + b x : b ∈ B } ,a ∗ ( x ) := { a + a x + : a ∈ A } , (3.2)where a := ( a , a ) , b := ( b , b ) ∈ R , and ( · ) + := max {· , } . As Θ is an interval, b ∗ ( x ) and a ∗ ( x ) are intervals and can be described by b ∗ ( x ) = [ b + ( b { x ≥ } + b { x< } ) x, b + ( b { x ≥ } + b { x< } ) x ] ,a ∗ ( x ) = [ a + a x + , a + a x + ] . Definition 3.1. [15, Definition 2.1] Let Θ be a set as in (3 . with associated a ∗ , b ∗ as in (3 . .Consider t ∈ [0 , T ] and P ∈ P acsem be a semimartingale law. We say that P is affine-dominated on ( t, T ] by Θ , if ( β P , α ) satisfy β Ps ∈ b ∗ ( B s ) , α s ∈ a ∗ ( B s ) , for dP ⊗ dt -almost all ( ω, s ) ∈ Ω × ( t, T ] . If t = 0 , we say that P is affine-dominated by Θ .Let O be the considered state space, i.e., either R , R ≥ or R > . Definition 3.2. [15, Definition 2.2] Let Θ be a set as in (3 . with associated a ∗ , b ∗ as in (3 . .A family of semimartingale laws P ∈ P acsem such that1. P ( B = x ) = 1 ,2. P is affine dominated by Θ is called affine process under parameter uncertainty starting at x ∈ O . If this holds for P , then weuse the notation P ∈ A ( x, Θ) . Furthermore, for t ∈ [0 , T ] we say that P ∈ A ( t, x, Θ) if P ∈ P acsem and1. P ( B t = x ) = 1 ,2. P is affine dominated on ( t, T ] by Θ .The state space O and the parameter space Θ cannot be chosen to be completely independent.Otherwise, it can happen that the set A ( x, Θ) is empty. To avoid this problem, we introduce thedefinition of proper families of affine processes under parameter uncertainty. Definition 3.3.
The families of non-linear affine process laws ( A ( x, Θ)) x ∈O with state space O are called proper if either a > or a = a = 0 and b ≥ a > holds.7 Reduced form setting under model uncertainty with non-linear affine intensities
In the sequel, we include affine processes under parameter uncertainty in the setting of Section 2to obtain analytically tractable models for credit risk/insurance markets under model uncertainty. Ω x We consider the space Ω x := C x ([0 , T ] , R ) of continuous functions with values in R starting ata fixed point x ∈ R . An element of this space is denoted by ω := ( ω S , ω µ ) and the canonicalprocess B := ( B S , B µ ) given by B t ( ω ) = ( B St ( ω ) , B µt ( ω )) = ( ω St , ω µt ) , t ∈ [0 , T ] . We assume to bein the setting introducd in Section 2 applied to Ω x and keep the notation as there. On (Ω x , F ) we consider a financial market model consisting of a riskfree asset S ≡ and of a risky asset S = ( S t ) t ∈ [0 ,T ] driven by B S . The mortality intensity µ in (2 . is given by B µ . As the mortalityintensity is nonnegative, we assume that B µ is nonnegative Z -q.s. A sufficient condition that thisassumption holds for a non-linear affine process defined as in Definition 3.2 is given in Proposition2.3 in [15]. Remark 4.1.
Without loss of generality it is possible to consider a financial market consist-ing of d risky assets by setting Ω x := C x ([0 , T ] , R d +1 ) with the canonical process B t ( ω ) :=( B St ( ω ) , B µt ( ω )) := ( ω S t , ..., ω S d t , ω µt ) for ω ∈ Ω dx , x ∈ R d +1 , t ∈ [0 , T ] and d ∈ N . In this casethe d assets S , ..., S d are driven by the d -dimensional process B S .We define the families of probability measures ( Z ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x on Ω x by Z ( t, ω ) := P S ∩ A µ ( t, ω µt , Θ µ ) (4.1)for ( t, ω ) ∈ [0 , T ] × Ω x , where the set A µ ( t, ω µt , Θ µ ) is introduced next. Definition 4.2.
Let ( t, ω ) ∈ [0 , T ] × Ω x , Θ µ as in (3 . , b ∗ ( B µ · ) and a ∗ ( B µ · ) as in (3 . . Given P µ ∈ P (Ω x ) we have that P µ ∈ A µ ( t, ω µt , Θ µ ) if1. the process B µ is a one-dimensional ( P µ , F ) -semimartingale with a.c. characteristics withthe corresponding predictable processes β P µ and α ≥ ,2. P µ ( B µt = ω µt ) = P µ ( { ω ∈ Ω x : B µt ( ω ) = ω µt } ) = 1 ,3. P µ is affine-dominated by ( t, T ] by Θ µ , i.e., β P µ s ∈ b ∗ ( B µs ) and α s ∈ a ∗ ( B µs ) for dP µ ⊗ dt -almost all ( ω, s ) ∈ Ω x × ( t, T ] .In addition, the following assumptions hold for the set P S ⊆ P (Ω x ) in (4 . . Assumption 4.3. P S satisfies Assumption 2.1, where we define the families ( P S ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x by P S ( t, ω ) := P S for all ( t, ω ) ∈ [0 , T ] × Ω x .2. For all P ∈ P S the process S = ( S t ) t ∈ [0 ,T ] is a ( P, F ) -local martingale. Note that by definition the set Z (0 , ω ) is independent of ω , which is crucial for Proposition 2.2.From now on, set Z := Z (0 , ω ) . Remark 4.4.
By the definition of the families of probability measures ( Z ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x in (4 . the mortality intensity B µ is a non-linear affine process and the risky asset S is modeled inan arbitrage-free way under model uncertainty. In a more general setting one could work with aset P S of semimartingale measures for S and follow the approach of [6]. However, in this casethe existence of an equivalent local martingale measure can only be guaranteed by consideringan additional cemetery state at which the paths jump at the stopping time ξ . To avoid furthertechnicalities and to be consistent with classical reduced form models, we directly assume that P S are local martingale measures for S . Moreover, for our purpose it is sufficient to consider a setof probability measures P S instead of the families ( P S ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x . However, our resultscan be easily extended for families by requiring that the local martingale property holds for all P ∈ P S (0 , ω ) . 8e now show that Assumption 2.1 holds for ( Z ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x , so that we can define thesublinear operator ( E t ) t ∈ [0 ,T ] on Ω x as in Proposition 2.2 with respect to ( Z ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x in (4 . . Proposition 4.5.
Let ( Q ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x and ( ˜ Q ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x be two families satisfyingAssumption 2.1. Then the families ( P ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x defined by P ( t, ω ) := Q ( t, ω ) ∩ ˜ Q ( t, ω ) , ( t, ω ) ∈ [0 , T ] × Ω x also fulfill Assumption 2.1.Proof. Let ( s, ω ) ∈ [0 , T ] × Ω x , P ∈ P ( s, ω ) and τ be a stopping times such that s ≤ τ ≤ T . Set η := τ s,ω − s .1) Measurability: As Q ( s, ω ) and ˜ Q ( s, ω ) satisfy Assumption 2.1, we know that the sets { ( P ′ , ω ) : ω ∈ Ω , P ′ ∈ Q ( τ, ω ) } and { ( P ′ , ω ) : ω ∈ Ω , P ′ ∈ ˜ Q ( τ, ω ) } are analytic. As the countable intersec-tion of analytic sets is again analytic, the property of measurability also holds for ( P ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x .2) Invariance: As P ∈ Q ( s, ω ) ∩ ˜ Q ( s, ω ) we get P η,ω ∈ Q ( τ, ω ⊗ s ω ) for P -a.e. ω ∈ Ω x and P η,ω ∈ ˜ Q ( τ, ω ⊗ s ω ) for P -a.e. ω ∈ Ω x and we can conclude P η,ω ∈ Q ( τ, ω ⊗ s ω ) ∩ ˜ Q ( τ, ω ⊗ s ω ) | {z } = P ( τ,ω ⊗ s ω ) for P -a.e. ω ∈ Ω x . Stability under Pasting:
Let κ : Ω x → P (Ω x ) be an F η -measurable kernel and κ ( ω ) ∈ P ( τ, ω ⊗ s ω ) for P -a.e. ω ∈ Ω x . Due to the fact that P ∈ Q ( s, ω ) ∩ ˜ Q ( s, ω ) , we get for A ∈ F P ( A ) = Z Z ( A ) η,ω ( ω ′ ) κ ( dω ′ , ω ) P ( dω ) ∈ Q ( s, ω ) and P ( A ) = Z Z ( A ) η,ω ( ω ′ ) κ ( dω ′ , ω ) P ( dω ) ∈ ˜ Q ( s, ω ) . So it follows P ( A ) ∈ P ( s, ω ) .Next we show that ( A µ ( t, ω µt , Θ µ )) ( t,ω ) ∈ [0 ,T ] × Ω x satisfies Assumption 2.1. To do so, we first usethe results of Lemma 3.1 and 3.2 in [15] for the set A ( t, ω t , Θ) with ( t, ω ) ∈ [0 , T ] × Ω x defined inDefinition 3.2. Note, here we work with the one-dimensional path space Ω x = C x ([0 , T ] , R ) for afixed x ∈ R . Then we prove that the families ( A µ ( t, ω µt , Θ µ )) ( t,ω ) ∈ [0 ,T ] × Ω x in Definition 4.2 satisfyAssumptions 2.1. This requires to transfer the results in [33] to a two-dimensional setting. Proposition 4.6.
Let ( A µ ( s, ω µs , Θ µ )) ( s,ω ) ∈ [0 ,T ] × Ω x be the families as in Definition 4.2 for a fixedset Θ µ as in (3 . . Fix s ∈ [0 , T ] , ω ∈ Ω x and τ be a stopping time taking values in [ s, T ] . Moreover,let P ∈ A µ ( s, ω µ ( s ) , Θ µ ) , then1. Measurability:
The graph { ( P ′ , ω ) : ω ∈ Ω x , P ′ ∈ A µ ( τ ( ω ) , ω µτ ( ω ) , Θ µ ) } ⊆ P (Ω x ) × Ω x isanalytic.2. Invariance:
There exists a family of conditional probabilities ( P τ,ω ) ω ∈ Ω x with respect to F τ such that P τ,ω ∈ A µ ( τ ( ω ) , ω µτ ( ω ) , Θ µ ) for P -a.e. ω ∈ Ω x .3. Stability under Pasting:
Assume that there exists a family of probability measures ( Q ω ) ω ∈ Ω x such that Q ω ∈ A µ ( τ ( ω ) , ω µτ ( ω ) , Θ µ ) for P -a.e. ω ∈ Ω x and the map ω → Q ω is F τ measur-able. Then the probability measure P ⊗ Q defined by P ⊗ Q ( · ) = Z Ω x Q ω ( · ) P ( dω ) is an element of A µ ( s, ω µ ( s ) , Θ µ ) . roof. See Appendix.The statements in Proposition 4.6 do not correspond directly to Assumption 2.1 but are in line withthe ones used in [13], [14]. In these papers families of probability measures ( P ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω with support on D ( ω,t ) := { ˜ ω : ˜ ω | [0 ,t ] = ω | [0 ,s ] } are considered. As a consequence the concatenationis defined as ( ω ⊗ t ˜ ω ) s := ω s { s ≤ t } + ˜ ω s { s ≥ t } (4.2)for ω, ˜ ω ∈ Ω , such that ω t = ˜ ω t . Note, this is equivalent to (2 . if ω t = ˜ ω t , i.e., (2 . extends (4 . to the case where ω t = ˜ ω t . Therefore, the equivalence between Assumption 2.1 and the formulationin Proposition 4.6 is clear by taking into account the two different conventions in (2 . and (4 . ,respectively. Remark 4.7.
In [15] it is stated that the invariance and stability under pasting properties of thefamilies ( A ( t, ω t , Θ)) ( t,ω ) ∈ [0 ,T ] × Ω x in Lemma 3.1 and 3.2 in [15] follow directly by Theorem 2.1in [33]. The latter paper deals with a special case of our setting, as the differential characteristicsare assumed to be in a fixed set Θ ⊂ R × S d + × L , where S d + is the family of all positive definitesymmetric real-valued ( d × d ) -matrices and L the set of Lévy measures. It is shown in Theorem2.1 in [33] that P Θ = { P ∈ P acsem : ( β P , α P , F P ) ∈ Θ , P ⊗ dt -a.e. } , (4.3)where ∅ 6 = Θ ⊆ R d × S d + × L is a Borel measurable subset, satisfies Assumption 2.1. The maindifference with respect to our setting is that Θ is fixed in [33] and does not depend on the stateof the canonical process B, as in the affine case. In addition, [33] takes only into account a family P independent of ( s, ω ) ∈ R + × Ω . So, we cannot directly conclude that Assumption 2.1 holds forthe families ( A ( t, ω t , Θ)) ( t,ω ) ∈ [0 ,T ] × Ω x in our case.In the following we present some examples for the families ( Z ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x in (4 . . Example 4.8.
The following sets are considered M = { P ∈ P (Ω x ) : B is a local P -martingale }M a = { P ∈ M : h B i P is absolutely continuous P -a.s. } , where B is the canonical process and h B i P is the R × -valued quadratic variation process of B under P . For the sake of simplicity, we here assume S = B S . Now, we define a set-valued process D : Ω x × [0 , T ] → R d × d as in [35] and consider the following families of probability measures P D ( s, ω ) for all ( s, ω ) ∈ [0 , T ] × Ω x given by P D ( s, ω ) := (cid:26) P ∈ M a : d h B i Pu du ( ω ) ∈ D u + s ( ω ⊗ s ω ) for du × P -a.e. ( u, ω ) ∈ [0 , T ] × Ω x (cid:27) (4.4)Thereby, it is assumed that the set D satisfies the following regularity assumption. Assumption 4.9.
For every t ∈ [0 , T ] , { ( s, ω, M ) ∈ [0 , t ] × Ω x × R d × d : M ∈ D s ( ω ) } ∈ B ([0 , t ]) ⊗ F t ⊗ B ( R d × d ) . Using this condition it is shown in Theorem 4.3 in [38] that P D ( τ, ω ) satisfy Assumption 2.1, where τ is a (finite) stopping time and ω ∈ Ω .For every ( s, ω ) ∈ [0 , T ] × Ω x set Z ( s, ω ) := P D ( s, ω ) and define the process D = ( D t ) t ∈ [0 ,T ] withvalues in a subset of R d × d such that D t ( ω ) = D t ( ω S , ω µ ) ∈ (cid:26) (cid:18) σ S a µ, + a µ, ω µt (cid:19) : σ S ∈ [ σ S , σ S ] , a µ, ∈ [ a , a ] , a µ, ∈ [ a , a ] (cid:27) . We first verify that the set P D ( s, ω ) in (4 . with our choice of the process D contains the proba-bility measures we are interested in. To do so, fix ( s, ω ) ∈ [0 , T ] × Ω x and consider P ∈ P D ( s, ω ) .Then, for du × P -a.e. ( u, ω ) ∈ [0 , T ] × Ω x we have d h B i Pu du ( ω ) ∈ D u + s ( ω ⊗ s ω ) for σ S ∈ [ σ S , σ S ] , a µ, ∈ [ a , a ] , a µ, ∈ [ a , a ]
10f and only if d h B S , B µ i Pu du ( ω ) = 0 , d h B S i Pu du ( ω ) ∈ [ σ S , σ S ] ,α P,µu ( ω ) = d h B µ i Pu du ( ω ) ∈ [ a + a ( ω µ ⊗ s ω µ ) u + s , a + a ( ω µ ⊗ s ω µ ) u + s ] = a ∗ (( ω µ ⊗ s ω µ ) u + s ) . (4.5)Note, the statement in (4 . is equivalent to d h B S , B µ i Pu du ( ω ) = 0 , d h B S i Pu du ( ω ) ∈ [ σ S , σ S ] ,α P ,µu ( ω ) ∈ a ∗ ( ω µu ) for du × P -a.e. ( u, ω ) ∈ [ s, T ] × Ω x , P ∈ P D ( s, ω ) , P ( B µs = ω µs ) = 1 , for du × P -a.e. ( u, ω ) ∈ [0 , T ] × Ω x with P ∈ P D ( s, ω ) , i.e., P is affine-dominated in the senseof Definition 4.2. Assumption 4.9 is obviously satisfied as the set-valued function D is defined byelementary operations. In the outlined setting B µ is an affine process under parameter uncertaintywith Θ = { }×{ }× [ a , a ] × [ a , a ] and the process B S is a local martingale for all P ∈ P D ( s, ω ) .In this case the set Z is saturated.By defining the set-valued process D as a constant process with values in a nonempty, convex andcompact set of matrices D ⊆ R × , we are in the case of the classical G -setting. The definition ofthe families of probability measures ( P D ( s, ω )) ( s,ω ) ∈ [0 ,T ] × Ω x in (4 . reduces to P D = { P ∈ M a : d h B i Pt /dt ∈ D P × dt -a.e. } for all ( s, ω ) ∈ [0 , T ] × Ω x . It is shown in Proposition 3.1 in [38] that P D satisfies Assumption2.1. Assume D contains matrices of the form ˜ D = ( ˜ d i,j ) ≤ i,j ≤ with ˜ d , = ˜ d , = 0 , ˜ d , = σ S and ˜ d , = σ µ with σ S ∈ [ σ S , σ S ] and σ µ ∈ [ σ µ , σ µ ] for < σ S ≤ σ S , < σ µ ≤ σ µ . Then thiscorresponds to the affine structure of the semimartingale components of B µ with Θ µ = { } × { } × [ σ µ , σ µ ] × { } and volatility uncertainty for B S . Note, D is convex and also compact as the set ofdiagonalizable matrices with bounded eigenvalues is compact. Example 4.10.
The last example can be generalized by considering an affine structure on thedrift of B µ , i.e., Θ = [ b , b ] × [ b , b ] × [ a , a ] × [ a , a ] . Therefore, define the set-valued process L : Ω x × [0 , T ] → R by L t ( ω S , ω µ ) ∈ (cid:26) (cid:18) b µ, + b µ, ω µt (cid:19) : b µ, ∈ [ b , b ] , b µ, ∈ [ b , b ] (cid:27) . For all ( s, ω ) ∈ [0 , T ] × Ω x we set Z ( s, ω ) := (cid:26) P ∈ P acsem : d [ B ] Pu du ( ω ) ∈ D u + s ( ω ⊗ s ω ) , dA Pu du ( ω ) ∈ L u + s ( ω ⊗ s ω ) (4.6)for du × P -a.e. ( u, ω ) ∈ [0 , T ] × Ω x (cid:27) , (4.7)where A P denotes the finite variation part of the semimartingale decomposition of B . In Proposi-tion 4.3 in [21] it is shown that Assumption 2.1 is satisfied under the following condition. Assumption 4.11.
For every t ∈ [0 , T ] { ( s, ω, M, N ) ∈ [0 , t ] × Ω x × R × × R : M ∈ D s ( ω ) , N ∈ L s ( ω ) } ∈ B ([0 , t ]) ⊗ F t ⊗ B ( R × × R ) . In this example the local martingale property of B S is satisfied on [ t, T ] for all P ∈ Z ( t, ω ) .11 .2 Extended market model on ˜Ω x Given the market model on (Ω x , F , F ) , a final time horizon T > and the families of probabilitymeasures ( Z ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x in (4 . , we now use the construction described in Section todefine an extended market model on ( ˜Ω x , G , G ) . As in (2 . the families of probability measures ( ˜ Z ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x are given by ˜ Z ( t, ω ) := { ˜ P ∈ P ( ˜Ω x ) : ˜ P = P ⊗ ˆ P , P ∈ Z ( t, ω ) } for ( t, ω ) ∈ [0 , T ] × Ω x with ˜ Z := ˜ Z (0 , ω ) .Note, by Assumption 4.3 for all ( t, ω ) ∈ [0 , T ] × Ω x and P ∈ Z ( t, ω ) the process S is a local ( P, F ) -martingale. Let ˜ P ∈ ˜ Z ( t, ω ) , ≤ s ≤ t ≤ T and ( τ n ) n ∈ N a suitable sequence of stoppingtimes, then E ˜ P [ S t ∧ τ n |F s ] = E P ⊗ ˆ P [ S t ∧ τ n |F s ] = E P [ S t ∧ τ n |F s ] = S s ∧ τ n , which means that S is also a ( ˜ P , F ) -local martingale for every ˜ P ∈ ˜ Z ( t, ω ) . By construction (seeSection 6.5 in [7]) we have that in our setting the immersion property holds, i.e., every F -martingaleis also a G -martingale. Note that the usual hypotheses on the filtrations are not necessary for theimmersion property to hold. Thus, S is also a local ( ˜ P , G ) -martingale for every ˜ P ∈ ˜ Z ( t, ω ) , ( t, ω ) ∈ [0 , T ] × Ω x . Assumption 4.3 implies that S is also a local martingale for all ˜ P ∈ ˜ Z ( t, ω ) with ( t, ω ) ∈ [0 , T ] × Ω x . In the following we introduce a longevity bond with price process S L := ( S Lt ) t ∈ [0 ,T ] and maturity T > by means of the survivor index as in Section 2.1.2 in [10] on the financial market ( ˜Ω x , G ) .The survivor index S sur = ( S sur t ) t ∈ [0 ,T ] is defined by S sur t = exp (cid:18) − Z t B µs ds (cid:19) , t ∈ [0 , T ] , (5.1)where B µ = ( B µs ) s ∈ [0 ,T ] represents the mortality intensity of a fixed age cohort. Definition 5.1.
A longevity bond with maturity T is a bond paying the amount S sur T at time T .In the sequel, our aim is to introduce the price process S L := ( S Lt ) t ∈ [0 ,T ] for a longevity bond withmaturity T under model uncertainty in a way that the resulting extended market ( S , S, S L ) isarbitrage-free in the sense of Definition 6.1.In [15] the upper bond prices under the non-linear affine term structure model A ( t, x, Θ) , x ∈ O isdefined as p ( t, T, x ) := sup P ∈A ( t,x, Θ) E P [ e − R Tt B µs ds | B µt = x ] , ≤ t ≤ T. (5.2)This bond price is then given as the solution of generalized Riccati equations in Proposition 6.2in [15] and in some important special case leads to a closed-form solution. Note, that p ( t, T, x ) = sup P ∈A ( t,x, Θ) E P [ e − R Tt B µs ds | B µt = x ] = sup P ∈A ( t,x, Θ) E P [ e − R Tt B µs ds { B µt = x } ] , where we have used that P ( B µt = x ) = 1 for all P ∈ A ( t, x, Θ) . Hence p ( t, T, x ) represents theworst case estimation for the bond price given the class of models A ( t, x, Θ) . However, it will notin general coincide with the superreplication price for the defaultable contingent claim H = { τ>t } .We now discuss a more general definition for the longevity bond in our setting.The first candidate for S L is the process Y = ( Y t ) t ∈ [0 ,T ] given by Y t := ˜ E t ( e − R T B µs ds ) = ess sup ˜ P ˜ P ′ ∈ ˜ Z ( t ; P ) E ˜ P ′ [ e − R T B µs ds |G t ] ˜ P -a.s. for all ˜ P ∈ ˜ Z , (5.3)where ˜ Z ( t ; ˜ P ) := { ˜ P ′ ∈ ˜ Z : ˜ P = ˜ P ′ on G t } and ( ˜ E t ) t ∈ [0 ,T ] is the conditional sublinear operatorintroduced in (2 . . By Proposition 2.4 the process Y is G ∗ -adapted and well-defined as e − R T B µs ds
12s a nonnegative Borel-measurable function. Moreover, ˜ E t ( e − R T B µs ds ) coincides with E t ( e − R T B µs ds ) by Remark 2.19 1. in [5], which means Y is F ∗ -measurable by Proposition 2.2.Unfortunately, the process Y has no càdlàg paths which is often necessary for standard results infinancial mathematics. Motivated by the proof of Theorem 3.2 in [37] we define the value of thelongevity bond as càdlàg process as follows. Definition 5.2.
The value process of the longevity bond S L := ( S Lt ) t ∈ [0 ,T ] is given by S L := Y ′ N c , where N belongs to the family N ˜ Z T of ( ˜ P , G T ) -null sets for all ˜ P ∈ ˜ Z denoted by N ˜ Z T and Y ′ is given by Y ′ t := lim sup r ↓ t,r ∈ Q Y r = lim sup r ↓ t,r ∈ Q ˜ E r ( e − R T B µs ds ) = lim sup r ↓ t,r ∈ Q E r ( e − R T B µs ds ) for t < T (5.4) Y ′ T := Y T = ˜ E T ( e − R T B µs ds ) = E T ( e − R T B µs ds ) . (5.5)Under the assumption sup P ∈Z E P [ e − R T B µs ds ] < ∞ , the process S L is a càdlàg ( P, F ∗ , Z + ) -supermartingale for all P ∈ Z by the proof of Theorem 3.2 in [37]. By the immersion property S L is also a ( ˜ P , G ∗ , ˜ Z + ) -supermartingale for all ˜ P ∈ ˜ Z . Here, the filtration G ∗ , ˜ Z := ( G ∗ , ˜ Z t ) t ∈ [0 ,T ] (respectively F ∗ , Z ) is defined similar as in (2 . , i.e., G ∗ , ˜ Z t := G ∗ t ∨ N ˜ Z T , t ∈ [0 , T ] . (5.6)It is quite standard to use the filtration G ∗ , ˜ Z in the framework of model uncertainty, see e.g., [37],[31]. By considering the right-continuous version G ∗ , ˜ Z + of this filtration, we intuitively give theagent a “little bit” more information than the one available up to time t at the market. For thisreason G ∗ , ˜ Z + is not the natural filtration for financial applications. Hence, we discuss here furtherconditions to achieve more regularity on the paths of ( ˜ E t ) t ∈ [0 ,T ] . Proposition 5.3.
Let Assumption 2.1 hold for the families ( P ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x and X be anonnegative upper semianalytic function on Ω x . Furthermore, assume that for all P ∈ P , t ∈ [0 , T ] the set Φ P,Xt := { E Q [ X |F t ] : Q ∈ P ( t ; P ) } is upward directed, i.e., for all E Q [ X |F t ] , E Q [ X |F t ] ∈ Φ P,Xt there exists E Q [ X |F t ] ∈ Φ P,Xt such that E Q [ X |F t ] = E Q [ X |F t ] ∨ E Q [ X |F t ] P -a.s. Thenthe process ( E t ( X )) t ∈ [0 ,T ] has a càdlàg P -modification. As E ( X ) is a ( P, F ∗ ) -supermartingale for all P ∈ P , the existence of a càdlàg P -modification isequivalent to the fact that t E P [ E t ( X )] is right-continuous for all P ∈ P [39, p. 2037]. Note,this is a generalization of Theorem 7 in [41] which states that a supermartingale Y has a uniquecàdlàg modification with respect to P if and only if ( E P [ Y t ]) t ∈ [0 ,T ] is right-continuous. Proof.
Fix P ∈ P and t ∈ [0 , T ] . Choose a sequence of ( t n ) n ∈ N with t n > t and t n ↓ t for n → ∞ .We show that t E P [ E t ( X )] is right-continuous for all P ∈ P . The proof follows the idea ofProposition 4.3 in [28].1) In a first step we show that E P (cid:20) ess sup P P ′ ∈P ( t ; P ) E P ′ [ X |F t ] (cid:21) = sup P ′ ∈P ( t ; P ) E P (cid:2) E P ′ [ X |F t ] (cid:3) . (5.7)As we assumed that the set Φ P,Xt is upward directed it follows by Theorem A.33 in [19] that there ex-ists an increasing sequence ( E Q n [ X |F t ]) n with Q n ∈ P ( t ; P ) in Φ P,Xt such that lim n →∞ E Q n [ X |F t ] =ess sup P P ′ ∈P ( t ; P ) E P ′ [ X |F t ] . Then by Fatou’s Lemma we get E P (cid:20) ess sup P P ′ ∈P ( t ; P ) E P ′ [ X |F t ] (cid:21) = E P (cid:20) lim n →∞ E Q n [ X |F t ] (cid:21) ≤ lim inf n →∞ E P (cid:20) E Q n [ X |F t ] (cid:21) ≤ sup P ′ ∈P ( t ; P ) E P (cid:2) E P ′ [ X |F t ] (cid:3) . Note, the assumption sup P ∈Z E P [ e − R T B µs ds ] < ∞ is always satisfied for B µ > which is the case for amortality intensity.
13s it holds E P (cid:20) ess sup P P ′ ∈P ( t ; P ) E P ′ [ X |F t ] (cid:21) ≥ sup P ′ ∈P ( t ; P ) E P (cid:2) E P ′ [ X |F t ] (cid:3) , we obtain the claim in (5 . .2) As P ′ ∈ P ( t ; P ) it holds for all A ∈ F t that E P [ A ] = E P ′ [ A ] . Furthermore, every nonnegative F t -measurable random variable Y can be approximated by simple functions by Sombrero’s Lemma.So it follows E P [ Y ] = E P ′ [ Y ] by monotone convergence.Let ǫ > . As P n ∈ P ( t n ; P ) ⊆ P ( t ; P ) for t n > t , we have by (5 . that E P (cid:20) ess sup P P ′ ∈P ( t ; P ) E P ′ [ X |F t ] (cid:21) < E P (cid:2) E P n [ X |F t ] (cid:3) + ǫ = E P n (cid:2) E P n [ X |F t ] (cid:3) + ǫ = E P n [ X ] + ǫ = E P n (cid:2) E P n [ X |F t n ] (cid:3) + ǫ = E P (cid:2) E P n [ X |F t n ] (cid:3) + ǫ ≤ sup P ′ ∈P ( t n ; P ) E P (cid:2) E P ′ [ X |F t n ] (cid:3) + ǫ = E P (cid:20) ess sup P P ′ ∈P ( t n ; P ) E P ′ [ X |F t n ] (cid:21) + ǫ.
3) Due to the supermartingale property of E ( X ) the expectation is decreasing such that E P (cid:20) ess sup P P ′ ∈P ( t ; P ) E P ′ [ X |F t ] (cid:21) ≥ lim n →∞ E P (cid:20) ess sup P P ′ ∈P ( t n ; P ) E P ′ [ X |F t n ] (cid:21) , which finishes the proof.In the proof of Lemma 3.4 in [39] it is used that the set Φ P,Xt is upward directed for all t ∈ [0 , T ] , P ∈ P and X F T -measurable with sup P ∈P E P [ | X | ] < ∞ if P is stable under pasting, i.e.,for all P ∈ P , τ F -stopping time, Λ ∈ F τ , P , P ∈ P ( F τ ; P ) the measure P ( A ) =: E P [ P ( A |F τ ) Λ + P ( A |F τ ) Λ c ] , A ∈ F (5.8)is again an element of P . Note that condition (5 . is not the same concept as the stability underpasting of Assumption 2.1. Remark 5.4.
In Proposition 5.3 we first choose the contingent claim X in which we are interestedin and then assume that only for this fixed X the set Φ P,Xt is upward directed. However, if P satisfies the property of stability under pasting in the sense of (5 . , it follows that Φ X,Pt is upwarddirected for all P ∈ P and any F T -measurable random variables X with sup P ∈P E P [ | X | ] < ∞ .Thus for X nonnegative, upper semianaliytic and F T -measurable the assumptions in Proposition5.3 are always satisfied if P is stable under pasting in the sense of (5 . .We now provide an example of families of priors for which (5 . is satisfied. Proposition 5.5.
Consider the setting of Example . with the families ( P D ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x defined as in (4 . . Furthermore, let Assumption 4.9 hold and the process D = ( D t ) t ∈ [0 ,T ] be givenfor fixed a , a , a , a . Then the set P D satisfies (5 . and thus the set Φ P,Xt is upward directed forall t ∈ [0 , T ] , P ∈ P D and X F T -measurable.Proof. We prove this result in several steps. From now on let P ∈ P D , τ ∈ [0 , T ] a F -stoppingtime, Λ ∈ F τ and P , P ∈ P ( τ ; P ) . In addition, consider P defined as in (5 . .1) We first show that P = P on F τ . (5.9)Let A ∈ F τ , then P ( A ) = E P [ P ( A |F τ ) Λ + P ( A |F τ ) Λ c ] = E P [ A Λ + A Λ c ] = P ( A ) . Moreover, for A ∈ F we have P ( A ) = E P [ P ( A |F τ ) Λ + P ( A |F τ ) Λ c ] = E P [ E P [ A ∩ Λ |F τ ]] + E P [ E P [ A ∩ Λ c |F τ ]]= E P [ A ∩ Λ ] + E P [ A ∩ Λ c ] = P ( A ∩ Λ) + P ( A ∩ Λ c ) . (5.10)14) By using (5 . for a random variable X on Ω x and t ∈ [0 , T ] we have E P [ X Λ |F t ] = E P [ X Λ |F t ] and E P [ X Λ c |F t ] = E P [ X Λ c |F t ] . (5.11)Next we show that B S , B µ are local ( P , F ) -martingales. We prove it for a (local) ( Q, F ) -martingale X = ( X t ) t ∈ [0 ,T ] for Q ∈ { P, P , P } . We have to distinguish two cases. Combining the twoequations in (5 . we get for ≤ τ ≤ t ≤ T and t ≤ sE P [ X s |F t ] = E P [ X s Λ |F t ] + E P [ X s Λ c |F t ] = E P [ X s Λ |F t ] + E P [ X s Λ c |F t ]= E P [ X s |F t ] Λ + E P [ X s |F t ] Λ c = X t . (5.12)For the case ≤ t ≤ τ ≤ T and t ≤ s it holds E P [ X s Λ |F t ] = E P [ X s Λ |F t ] = E P [ E P [ X s Λ |F τ ] |F t ] = E P [ Λ E P [ X s |F τ ] |F t ]= E P [ Λ X τ |F t ] = E P [ X τ Λ |F t ] . (5.13)Here, we used in the last step that for t ≤ τ and X ∈ F τ we have E P [ X Λ |F t ] = E P [ X Λ |F t ] .In the same way as in (5 . we can derive E P [ X s Λ c |F t ] = E P [ X τ Λ c |F t ] , which implies with (5 . and (5 . that E P [ X s |F t ] = E P [ X s Λ |F t ] + E P [ X s Λ c |F t ] = E P [ X s Λ |F t ] + E P [ X s Λ c |F t ] = X t . Thus, B µ and B S are local ( P , F ) -martingales.3) We show α Pt ( ω ) := d h B i Pt /dt ( ω ) = [0 ,τ ] ( t ) α Pt ( ω ) + ] τ,T ] ( t )( α P t ( ω ) Λ ( ω ) + α P t ( ω ) Λ c ( ω )) (5.14)with α Qt := d h B i Qt /dt for Q ∈ { P, P , P } . First, we prove h B i Pt ( ω ) = h B i P t ( ω ) Λ ( ω ) + h B i P t ( ω ) Λ c ( ω ) . (5.15)Consider a partition π : 0 = t < t < ... < t n = t of the interval [0 , t ] with mesh size k π k :=max {| t k − t k − | : k = 1 , ..., n } . Set ∆ t k := ( B t k − B t k − ) for k ∈ N . Then it holds P (cid:0) { ω ∈ Ω : | n X k =0 ∆ t k ( ω ) − h B i P t ( ω ) Λ ( ω ) − h B i P t ( ω ) Λ c ( ω ) | > ǫ } (cid:1) = P (cid:0) { ω ∈ Λ : | n X k =0 ∆ t k ( ω ) − h B i P t ( ω ) | > ǫ } (cid:1) + P (cid:0) { ω ∈ Λ c : | n X k =0 ∆ t k ( ω ) − h B i P t ( ω ) | > ǫ } (cid:1) = P (cid:0) { ω ∈ Λ : | n X k =0 ∆ t k ( ω ) − h B i P t ( ω ) | > ǫ } (cid:1) + P (cid:0) { ω ∈ Λ c : | n X k =0 ∆ t k ( ω ) − h B i P t ( ω ) | > ǫ } (cid:1) ≤ P (cid:0) { ω ∈ Ω : | n X k =0 ∆ t k ( ω ) − h B i P t ( ω ) | > ǫ } (cid:1) + P (cid:0) { ω ∈ Ω : | n X k =0 ∆ t k − h B i P t ( ω ) | > ǫ } (cid:1) −→ k π k→ , where we used (5 . . Thus (5 . follows as the limit of convergence in probability is almost surelyunique. If the quadratic variation for the process X with respect to P exists, then X has the samequadratic variation with respect to all probability measures Q ∼ P [43, p. 15]. As P = P on F τ ,it follows from step 1) that h B i Pt = h B i Pt for t ∈ [0 , τ ] . By putting all these facts together we canconclude that (5 . holds. Furthermore, as α P , α P i , i = 1 , take values in D , it follows that also α P take values in this set, i.e., P ∈ P D .Next, we show that the property of Φ P,Xt being upward directed, which is a property on (Ω x , F ) ,can be transferred to the extended space ( ˜Ω x , G ) .15 roposition 5.6. Let Assumption 2.1 hold for the families ( P ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x . Assume thatfor every nonnegative upper semianalytic function X on Ω x , t ∈ [0 , T ] and P ∈ P the set Φ P,Xt := { E Q [ X |F t ] : Q ∈ P ( t ; P ) } is upward directed. Then for every nonnegative upper semianalyticfunction ˜ X on ˜Ω x which is G P T -measurable, t ∈ [0 , T ] and ˜ P ∈ ˜ P the set ˜Φ ˜ P ,Xt := { E ˜ Q [ ˜ X |G t ] : ˜ Q ∈ ˜ P ( t ; ˜ P ) }} is upward directed with ˜ P := P ⊗ ˆ P .Proof. Let P ∈ P , t ∈ [0 , T ] and X nonnegative upper semianalytic such that the correspondingset Φ P,Xt is upward directed. Consider ˜ P = P ⊗ ˆ P ∈ ˜ P and ˜ Q ∈ ˜ P ( t ; ˜ P ) . As ˜ Q ∈ ˜ P there exists Q ∈ P such that ˜ Q = Q ⊗ ˆ P . Next, we prove the following statement ˜ Q ∈ ˜ P ( t ; ˜ P ) if and only if Q ∈ P ( t ; P ) , (5.16)is equivalent to show that ˜ Q ( ˜ A ) = ˜ P ( ˜ A ) ∀ ˜ A ∈ G t ⇐⇒ Q ( A ) = P ( A ) ∀ A ∈ F t . Let A ∈ F t ⊆ G t then we have Q ( A ) = Q ⊗ ˆ P ( A ) = ˜ Q ( A ) = ˜ P ( A ) = P ⊗ ˆ P ( A ) = P ( A ) , whichshows the first implication. For the other direction take ˜ A ∈ G t and use Lemma 2.12 in [5] suchthat we have ˜ Q ( ˜ A ) = E ˜ Q [ ˜ A ] = E Q [ E ˆ P [ ˜ A ]] = E P [ E ˆ P [ ˜ A ]] = E ˜ P [ ˜ A ] = ˜ P ( ˜ A ) , which proves the claim in (5 . . Now consider also ˜ Q ∈ P ( t ; ˜ P ) , i.e., ˜ Q = Q ⊗ ˆ P . As Φ P,Xt isupward directed we know that there exists Q ∈ P ( t ; P ) such that E Q [ X |F t ] = E Q [ X |F t ] ∨ E Q [ X |F t ] P -a.s. (5.17)for any X nonnegative and upper semianalytic. Set ˜ Q := Q ⊗ ˆ P . We now show that ˜ Q ∈ ˜ P ( t ; ˜ P ) and E ˜ Q [ ˜ X |G t ] = E ˜ Q [ ˜ X |G t ] ∨ E ˜ Q [ ˜ X |G t ] with ˜ Q i := Q i ⊗ ˆ P for i = 1 , . The first property followsby (5 . . By Proposition 2.16 in [5] we know that for t ≥ , ˜ Q = Q ⊗ ˆ P and ˜ X nonnegative and G P T -measurable E ˜ Q [ ˜ X |G t ] = { ˜ τ ≤ t } E Q [ ϕ ( x, · ) |F t ] (cid:12)(cid:12) x =˜ τ + { ˜ τ>t } e Γ t E Q [ E ˆ P [ { ˜ τ>t } ˜ X ] |F t ]= { ˜ τ ≤ t } (cid:0) E Q [ ϕ ( x, · ) |F t ] ∨ E Q [ ϕ ( x, · ) |F t ] (cid:12)(cid:12) x =˜ τ (cid:1) ++ { ˜ τ>t } e Γ t (cid:0) E Q [ E ˆ P [ { ˜ τ>t } ˜ X ] |F t ] ∨ E Q [ E ˆ P [ { ˜ τ>t } ˜ X ] |F t ] (cid:1) = E ˜ Q [ ˜ X |G t ] ∨ E ˜ Q [ ˜ X |G t ] ˜ Q -a.s.with ϕ as in (2 . . Here, we used (5 . which is possible as E ˆ P [ { ˜ τ>t } ˜ X ] is nonnegative. Lemma 5.7.
Let Assumption 2.1 hold for the families ( P ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω x and ˜ X be a non-negative upper semianalytic function on ˜Ω x which is G P T -measurable. Furthermore, assume thatfor every nonnegative upper semianalytic function X on Ω x , t ∈ [0 , T ] and P ∈ P the set Φ P,Xt := { E Q [ X |F t ] : Q ∈ P ( t ; P ) } is upward directed. Then the process ( ˜ E t ( X )) t ∈ [0 ,T ] has acàdlàg ˜ P -modification.Proof. Proposition . allows us to transfer the property of upward directed from the set Φ P,Xt to ˜Φ ˜ P , ˜ Xt . Then we can use exactly the same arguments in Proposition 5.3 which is possible as thesublinear conditional operator ˜ E admits the representation as essential supremum in (2 . underAssumption 2.1.The results in Proposition 5.3, 5.6 and 5.7 are also valid by replacing Ω x defined as in Subsection4.1 by Ω = C ( R + , R d ) or Ω = D ( R + , R d ) for d ∈ N .One advantage by working with the càdlàg P -modification compared with the Definition 5.2 isthat we get the path regularity without being forced to consider a process adapted to the right-continuous version of a filtration. Nevertheless, if P does not allow the existence of càdlàg P -modification, the approach in Definition 5.2 always guarantees càdlàg paths.16 .1 Numerical valuation In the sequel we derive a numerical representation of the longevity bond S L introduced in Definition5.2 by using the affine structure with parameter uncertainty of the underlying intensity. This ispossible by generalizing Theorem 6.2 in [15].We define the upper bounds for the intervals a ∗ ( x ) and b ∗ ( x ) in (3 . which are given by a ( x ) = a + a x + and b ( x ) = b + b { x< } + b { x ≥ } | {z } := B ,x . (5.18) Proposition 5.8.
Assume that for all P ∈ Z β Pt ≤ b + B ,x B µt ,dP ⊗ dt -almost everywhere for ≤ t ≤ T . Moreover, assume either that a = a = 0 or that forall P ∈ Z , B µt ≥ P ⊗ dt -a.e. Furthermore, there exists P ∈ Z and a one-dimensional ( P , F ) -Brownian motion W such that the componentwise canonical process B µ under P is the uniquestrong solution of dB µt = ( b + B ,x B µt ) dt + q a ( B µt ) dW t , B µ = ω µ . (5.19) Then, for all u ≥ and ≤ t ≤ T E t ( e − R Tt B µs ds ) = ess sup PP ′ ∈Z ( t ; P ) E P ′ (cid:2) e − R Tt B µs ds (cid:12)(cid:12) F t ] = exp( φ ( T − t,
0) + ψ ( T − t, B µt ) P -a.s. , (5.20) where φ and ψ solve the Riccati equations ∂ t φ ( t, u ) = 12 a ψ ( t, u ) + b ψ ( t, u ) φ (0 , u ) = 0 ∂ t ψ ( t, u ) = 12 a φ ( t, u ) + B ,x φ ( t, u ) − ψ (0 , u ) = u. Proof.
1) Let P ∈ Z . With the same arguments as in Proposition 6.2 in [15] we have E P [ e − R t B µs ds ] ≤ E P [ e − R t B µs ds ] , where P is given by the assumptions of the Proposition. As P ∈ Z is arbitrary and P ∈ Z itfollows E P [ e − R T B µs ds ] = sup P ∈Z E P [ e − R T B µs ds ] . (5.21)2) We now show that E P [ e − R T B µs ds |F t ] = ess sup PP ′ ∈Z ( t ; P ) E P ′ (cid:2) e − R T B µs ds (cid:12)(cid:12) F t ] P -a.s. (5.22)As P ∈ Z ( t ; P ) ⊆ Z , the inequality E P [ e − R T B µs ds |F t ] ≤ ess sup P P ′ ∈Z ( t,P ) E P ′ (cid:2) e − R Tt B µs ds (cid:12)(cid:12) F t ] follows directly. For the other direction we show that for all P ′ ∈ Z ( t ; P ) E P [ E P [ e − R T B µs ds |F t ]] ≥ E P [ E P ′ [ e − R T B µs ds |F t ]] . Fix P ′ ∈ Z ( t ; P ) , then by (5 . we have E P [ E P [ e − R T B µs ds |F t ]] = E P [ e − R T B µs ds ] = sup P ∈Z E P [ e − R T B µs ds ] ≥ sup P ′ ∈Z ( t ; P ) E P ′ [ e − R T B µs ds ] ≥ E P ′ [ e − R T B µs ds ] = E P ′ [ E P ′ [ e − R T B µs ds |F t ]] = E P [ E P ′ [ e − R T B µs ds |F t ]] , where we use in the last equality that P ′ = P on F t .3) As B µ is an affine process in the classical sense, we get by Theorem 10.14 in [17] and (5 . therepresentation via Riccati equations as in (5 . .17y Definition 5.2 the value process of the longevity bond S Lt = lim sup r ↓ t,r ∈ Q Y r N c , t ∈ [0 , T ) canbe rewritten by (5 . and the corresponding Riccati equations as Y r = ˜ E r ( e − R T B µs ds ) = E r ( e − R T B µs ds ) = ess sup P P ′ ∈Z ( r ; P ) E P ′ (cid:2) e − R T B µs ds (cid:12)(cid:12) F r ]=( e − R r B µs ds ) ess sup P P ′ ∈Z ( r ; P ) E P ′ (cid:2) e − R Tr B µs ds (cid:12)(cid:12) F r ]=( e − R r B µs ds ) exp( φ ( T − r,
0) + ψ ( T − r, B µr ) . (5.23)As already mentioned in Remark 6.3 in [15] there are two important cases in which the assumptionsof Proposition 5.8 are satisfied.1. Non-linear Vasicek model with state space O = R , i.e., Θ = [ b , b ] × { b } × [ a , a ] × { } with b = b .2. Non-linear CIR model with state space O = R > , i.e., Θ = [ b , b ] × [ b , b ] × { } × [ a , a ] with b ≥ a / . Remark 5.9.
In [30] non-mean reverting processes are suggested as they better fit observed data onmortality intensity. As the Vasicek and the CIR model also belong to classes of Ornstein-Uhlenbeckand Feller processes, we can include this non-mean reverting property to the correspondent non-linear cases by setting b = b = 0 .We now consider the valuation of a contingent claim f ( S T ) in a setting for given families ofprobability measures ( ˜ A ( t, y )) ( t,y ) ∈ [0 ,T ] × R ⊆ P (Ω x ) with y = ( y µ , y S ) and a Lipschitz function f : O S → R + , O S ⊆ R . We want to find a way to numerically compute the following valuefunction v : [0 , T ] × O → R , O ⊆ R v ( t, y ) := sup P ∈ ˜ A ( t,y ) E P [ e − R Tt B µs ds f ( S T ) | B µt = y µ , S t = y S ] . (5.24)We construct an example for a space Ω x and families of probability measures ( ˜ A ( t, y )) ( t,y ) ∈ [0 ,T ] × R such that the value function v ( t, y ) in (5 . can be explicitly computed via generalized Riccatiequations and PDEs. More generalized cases are studied in [2]. Example 5.10.
Set (Ω x , F ) := (Ω µ × Ω S , F µ ⊗ F S ) with x := ( x µ , x S ) ∈ R , Ω µ := C x µ ([0 , T ] , R ) and Ω S := C x S ([0 , T ] , R ) equipped with the Borel σ -algebra F µ := B (Ω µ ) and F S := B (Ω S ) respectively. The canonical processes on Ω µ and Ω S are denoted by B µ and B S , respectively. For t ∈ [0 , T ] , y = ( y µ , y S ) ∈ O µ × R , we consider on Ω x the following family of probability measures ˜ A ( t, y ) := { P = P µ ⊗ P S : P µ ∈ A ( t, y µ , Θ µ ) , P S ∈ P S } ⊆ P (Ω x ) , (5.25)where P S is the weakly compact set of probability measures representing the G -expectation as anupper expectation on Ω S as in Theorem 2.5, Chapter VI in [40]. Let ( A ( t, y µ , Θ µ )) ( t,y µ ) ∈ [0 ,T ] ×O µ be proper families of probability measures on Ω µ with state space O µ ⊆ R as in Definition 3.2.Moreover, assume that the asset price S = ( S s ) s ∈ [ t,T ] on Ω S satisfies the following SDE dS s = b ( S s ) d s + h ( S s ) d h B S i s + σ ( S s ) dB S , s ∈ [ t, T ] S t = y S , where the canonical process B S is a one-dimensional G -Brownian motion on Ω S due to the defi-nition of P S and b, h, σ : R → R are Lipschitz continuous functions. Then v S ( t, y S ) is the uniqueviscosity solution of the following PDE ∂ t v S + F ( D v S , Dv S , v S , y S ) = 0 v S ( T, y S ) = f ( y S ) with F ( D v S , Dv S , v S , y S ) = G ( σ ( y S ) D v S + h ( y S ) Dv S ) + b ( y S ) Dv S
18y Theorem 3.7, Chapter V in [40]. By construction the canonical processes B S and B µ areindependent under all P ∈ ˜ A ( t, y ) , ( t, y ) ∈ [0 , T ] × R . We have v ( t, y ) = sup P ∈ ˜ A ( t,y ) E P [ e − R Tt B µs ds f ( S T ) { B µt = y µ } { S t = y S } ] P ( B µt = y µ , S t = y S )= sup P ∈ ˜ A ( t,y ) E P [ e − R Tt B µs ds { B µt = y µ } ] E P [ f ( S T ) { S t = y S } ] P ( B µt = y µ ) P ( S t = y S ) (5.26) = sup P µ ∈A ( t,y µ , Θ µ ) E P µ [ e − R Tt B µs ds { B µt = y µ } ] P ( B µt = y µ ) sup P S ∈P S E P S [ f ( S T ) { S t = y S } ] P ( S t = y S ) (5.27) = sup P µ ∈A ( t,y µ , Θ µ ) E P µ [ e − R Tt B µs ds { B µt = y µ } ] sup P S ∈P S E P S [ f ( S T ) { S t = y S } ] P ( S t = y S ) (5.28) =: v µ ( t, y µ ) v S ( t, y S ) . In (5 . we used the independence of B S and B µ . Moreover, by Definition 3.2 of A ( t, y µ , Θ µ ) itholds P µ ( B µt = y µ ) = 1 for all P µ ∈ A ( t, y µ , Θ µ ) which implies (5 . and (5 . . If A µ ( y µ , Θ x ) satisfies the assumptions in Proposition 6.2 in [15] (which corresponds to conditions in Proposition5.8 in a one-dimensional setting), then the function v µ : [0 , T ] × O µ → R can be expressed viageneralized Riccati equations in Proposition 5.8. We now wish to show how the extended market model on ˜Ω x introduced in Subsection 4.2 con-taining the riskfree asset S , the risky asset S and the longevity bond S L is arbitrage-free. Morein general, we allow the trading of a contingent claim represented by a G Z T -measurable randomvariable Y . We price this contingent claim with the sublinear conditional operator ( ˜ E t ) t ∈ [0 ,T ] in-troduced in Proposition 2.4, i.e., we set S Yt := ˜ E t ( Y ) for t ∈ [0 , T ] . To guarantee that S Y iswell-defined we assume from now on that Y is upper semianalytic on ˜Ω x and nonnegative. Wethen show that the extended market model ( S , S, S Y ) is arbitrage-free. Setting Y := e − R T B µs ds we obtain the desired result for the market model extended with the longevity bond.We now consider the concept of “absence of arbitrage of the first kind” NA ( ˜ Z ) under model un-certainty introduced in [6] and directly apply it to our market model on ˜Ω x . For a σ -field A ⊆ G the set of all [0 , ∞ ] -valued, G -measurable random variables that are ˜ Z -q.s. finite is denoted by L ( A , ˜ Z ) . A trading strategy H is given by a simple predictable processes H = P ni =1 h i ] τ i − ,τ i ] ,where h i = ( h ji ) , j ∈ { S, Y } is G τ i − -measurable for all i ≤ n and ( τ i ) i ≤ n is a nondecreasingsequence of G -stopping times with τ = 0 . The set of possible strategies for a given initial wealth x ∈ R + is given by H simp ( x ) = { H : simple predictable process such that X x,H ≥ Z -q.s. } , (6.1)where X x,H is the associated wealth process of the form X x,Ht = x + n X i =1 h Si ( S τ i ∧ t − S τ i − ∧ t ) | {z } := X x,H, t + n X i =1 h Yi ( S Yτ i ∧ t − S Yτ i − ∧ t ) | {z } := X x,H, t . (6.2)A simple strategy H is in H simp ( x ) if X x,H stays nonnegative ˜ Z -q.s. We introduce the set X simp = { X x,H : x ∈ R + , H ∈ H simp ( x ) } . (6.3)For T ∈ R + and f ∈ L ( G T , ˜ Z ) the superhedging price of the claim f is defined by ν simp ( T, f ) := inf { x ∈ R + : ∃ H ∈ H simp ( x ) with X x,HT ≥ f ˜ Z -q.s. } . (6.4)19 efinition 6.1. [6, Definition 2.1] The market model ( S, S Y ) on ˜Ω x presents no arbitrage of firstkind with respect to ˜ Z , (NA ( ˜ Z ) ) if ∀ s ∈ [0 , T ] and f ∈ L ( G s , ˜ Z ) , ν simp ( s, f ) = 0 = ⇒ f = 0 ˜ Z -q.s., (6.5)where the wealth process X x,H given as in (6 . and ˜ Z := ˜ Z (0 , ω ) defined in (4 . .The arbitrage condition in Definition 6.1 takes only into account the set ˜ Z and not the fami-lies ( ˜ Z ( t, ω )) ( t,ω ) ∈ (0 ,T ] × Ω x which is in line with the assumptions in Theorem 3.2 in [37]. This ismotivated by the fact that ˜ Z is the set of probability measures we are really interested in andthe families ( ˜ Z ( t, ω )) ( t,ω ) ∈ (0 ,T ] × Ω x are auxiliary constructions. In addition, in our setting the set ˜ Z ( t, ω ) intuitively considers the market on the interval [ t, T ] instead from time zero. Remark 6.2.
In contrast to [6] we do not assume the asset S to have ˜ Z -q.s. continuous pathswhich is crucial for proving the fundamental theorem of asset pricing in Theorem 3.4 in [6]. Here,we only require paths to be càdlàg as in the classical case, e.g. [27], or without any assumptionsregarding regularity. Another difference to [6] is that the simple predictable strategies H aredefined with respect to the filtration G and not with respect to the right-continuous filtration G + .As already mentioned in [6] this is not a problem as the set of predictable processes on ( ˜Ω , G + ) coincides with the class of predictable processes on ( ˜Ω , G ) . Furthermore, the set of local martingalemeasures in Definition 3.3 in [6] is also defined by the local martingale property with respect to G + .By Proposition 2.2 in [32] it holds that for any right-continuous G -adapted process it is equivalentto be a ( ˜ P , G ) -semimartingale or a ( ˜ P , G P + ) -semimartingale or a ( ˜ P , G + ) -semimartingale and thesemimartingale characteristics are the same. Thus, it is also possible to consider local ( ˜ P , G ) -martingales instead of local ( ˜ P , G + ) -martingales.We now introduce the weaker notion NA ( ˜ P ) := NA ( { ˜ P } ) for ˜ P ∈ ˜ Z which is used in the proofof [6, Theorem 3.4]. This condition means that ∀ T ∈ R + and f ∈ L ( G T , ˜ P ) , ν simp , ˜ P ( T, f ) = 0 = ⇒ f = 0 ˜ P -a.s. , where ν simp , ˜ P ( T, f ) := inf { x ∈ R + : ∃ H ∈ H simp , ˜ P ( x ) with X x,HT ≥ f ˜ P -a.s. } and H simp , ˜ P ( x ) is the class of all simple predictable processes such that X x,H is non-negative ˜ P -a.s.We have the following useful relation between NA ( ˜ P ) and NA ( ˜ P ) . Proposition 6.3.
Assume that S has ˜ Z -q.s. continuous paths. ThenNA ( ˜ Z ) holds if and only if NA ( ˜ P ) holds for all ˜ P ∈ ˜ Z . (6.6) If S has càdlàg paths, thenNA ( ˜ P ) holds for all ˜ P ∈ ˜ Z implies NA ( ˜ Z ) . (6.7) Proof.
Equivalence (6 . follows by Theorem 3.4 in [6]. As H simp ( x ) ⊆ H simp ,P ( x ) we have that (6 . holds. Remark 6.4.
The other direction in (6 . relies on the property of ˜ Z -q.s. continuous paths of S and does not hold in general.The following lemma shows that the original market model ( S , S ) on (Ω x , F T ) satisfies NA ( Z ) in the sense of Definition . by considering Z and the filtration F instead of ˜ Z and G respectively.Moreover, the wealth process X x,H defined in (6 . consists only of X x,H, t , i.e., S Y ≡ . Lemma 6.5.
Under Assumption 4.3 the condition NA ( Z ) is satisfied for the market model ( S , S ) on (Ω x , F T ) defined in Subsection 4.1.Proof. Assumption 4.3 ensures that for every Q ∈ Z the NFLVR-condition holds for the Q -market.This implies that NA ( Q ) holds for all Q ∈ Z and by Proposition 6.3 we can conclude the NA ( Z ) holds. Here, we used that in the classical case, i.e., when the set of priors consists only of onesingle probability measure, it holds that NFLVR implies NA by Lemma A.2 in [26].20n the sequel, we prove that no arbitrage of first kind under model uncertainty also holds forextended models ( S , S, S Y ) on ( ˜Ω x , G T ) . Assumption 6.6.
Let ˜ P ∈ ˜ Z and t ∈ [0 , T ] . Then for all X x,H ∈ X simp we have E ˜ P [ X x,Ht ] ≤ E ˜ P [ X x,H ] . (6.8) Proposition 6.7.
Let Y be an upper semianalytic, G Z T -measurable and nonnegative random vari-able. Set S Yt := ˜ E t ( Y ) for t ∈ [0 , T ] . Under Assumption 6.6 the extended market model ( S , S, S Y ) on ˜Ω x satisfies NA ( ˜ Z ) .Proof. This follows by the arguments in the proof of Theorem 3.5 in [6].Assumption 6.6 may appear restrictive. However, it is satisfied in many cases, as we now showbelow.
Lemma 6.8.
If one of the following properties holds for every X x,H ∈ X simp , then condition (6 . is satisfied.1. X x,H is a ˜ P -supermartingale for all ˜ P ∈ ˜ Z .2. X x,H, and X x,H, are ˜ P -supermartingales for all ˜ P ∈ ˜ Z .3. X x,H, ≥ Z -q.s. and E ˜ P [ X x,H, t ] ≤ for all ˜ P ∈ ˜ Z and t ∈ [0 , T ] .4. X x,H, ≥ Z -q.s. and we do not allow short-selling for S Y , i.e., h Yi ≥ for i = 1 , .., n .5. S ≥ Z -q.s. and we do not allow short-selling for S, S Y , i.e., h ji ≥ for i = 1 , .., n and j ∈ { S, Y } .Clearly, . ⇒ . and . ⇒ . ⇒ . .Proof. Since (6 . obviously holds under conditions 1. and 2., we start with 3. Let X x,H ∈ X simp .As for any ˜ P ∈ ˜ Z the asset S is a ( ˜ P , G ) -local martingale, there exists an increasing sequence (˜ τ n ) n ∈ N of G -stopping times with ˜ τ n ↑ ∞ ˜ P -a.s. such that ( S ˜ τ n ∧ t ) t ≥ is a ˜ P -martingale for all ˜ P ∈ ˜ Z , n ∈ N . It follows that for x ∈ R + , H ∈ H simp ( x ) , n ∈ N X x,H, ·∧ ˜ τ n is a local ( ˜ P , G ) -martingale for all ˜ P ∈ ˜ Z . (6.9)By X x,H, > Z -q.s., X x,H, ·∧ ˜ τ n is a ˜ P -supermartingale for all ˜ P ∈ ˜ Z and n ∈ N . Thus, by Fatou’sLemma we get for ≤ s ≤ t and ˜ P ∈ ˜ Z E ˜ P [ X x,H, t |G s ] = E ˜ P (cid:2) lim n →∞ X x,H, t ∧ ˜ τ n |G s (cid:3) ≤ lim inf n →∞ E ˜ P (cid:2) X x,H, t ∧ ˜ τ n |G s (cid:3) ≤ lim inf n →∞ X x,H, s ∧ ˜ τ n = X x,H, s , i.e., X x,H, is a ˜ P -supermartingale for all ˜ P ∈ ˜ Z . As E ˜ P [ X x,H, t ] ≤ for all ˜ P ∈ ˜ Z and t ∈ [0 , T ] it follows for ˜ P ∈ ˜ Z E ˜ P [ X x,Ht ] = E ˜ P [ X x,H, t ] + E ˜ P [ X x,H, t ] ≤ E ˜ P [ X x,H, ]= E ˜ P (cid:20) X x,H, + n X i =1 h Yi ( S Yτ i ∧ − S Yτ i − ∧ ) | {z } =0 (cid:21) = E ˜ P [ X x,H ] . For condition 4. it is enough to observe that E ˜ P [ X x,H, t ] = n X i =1 h Yi (cid:0) E ˜ P [ S Yτ i ∧ t ] − E ˜ P [ S Yτ i − ∧ t ] (cid:1)| {z } ≤ ≤ , by using the no short-sale constraint and the fact that S Y is a supermartingale for all ˜ P ∈ ˜ Z .Furthermore, it is obvious that the nonnegativity of S and the additional short-sale constraintguarantee that X x,H, ≥ Z -q.s.. 21 emark 6.9. In condition 1. and 2. in Lemma 6.8 the chosen filtrations play no role as (6 . involves only the expectation.The results in Lemma 6.8 show that the sublinear conditional operator ( ˜ E t ) t ∈ [0 ,T ] allows to pricea European contingent claim in a way that the extended market is arbitrage-free way under someadditional assumptions. These supplementary constraints can be regarded as the price we pay forconsidering a setting under model uncertainty. On the one hand, allowing only strategies H suchthat the wealth process X x,H is a supermartingale for all ˜ P ∈ ˜ Z is in line with the definitionof admissible strategies under model uncertainty in [37, p. 4450] with the difference that therenot only simple strategies are considered. On the other hand, the supermartingale assumptionseems too strong due to Assumption 6.6 which only requires decreasing expectation. Conditions3. and 4. in Lemma 6.8 could be regarded as restrictive in an economical sense. However, in aninsurance context constraints as no short-selling or a positive wealth-process are often required bythe regulatory framework.For the next result we consider general families of probability measures ( P ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω on Ω satisfying Assumption 2.1 in the setting of Section 2. Lemma 6.10.
Let Assumption 2.1 hold for ( P ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω and Y be an upper semianalytic, G P T -measurable and nonnegative function on ˜Ω . Set S Yt := N c lim sup r ↓ t,r ∈ Q ˜ E r ( Y ) for t ∈ [0 , T ) and S YT := ˜ E T ( Y ) with N ∈ N ˜ PT . Let S be an G ∗ , ˜ P -adapted continuous ( ˜ P , G ∗ , ˜ P ) -semimartingale for all ˜ P ∈ ˜ P . Assume that ˜ P is a non-empty saturated set of sigma-martingale measures for S .Under the assumption h Yi ≥ , i = 1 , ...n, the extended market model ( S , S, S Y ) satisfies N A ( ˜ P ) for ˜ Z = ˜ P in (6 . .Proof. By applying Theorem 2.4 in [37], a version of the optional decomposition theorem undermodel uncertainty, for every ˜ P ∈ ˜ P there exists a G ∗ , ˜ P + -predictable process ˜ H which is S -integrablefor all ˜ P ∈ ˜ P such that D t := S Yt − Z t, ( ˜ P )0 HdS is nonincreasing ˜ P -a.s. for all ˜ P ∈ ˜ P . (6.10)In this case R t, ( ˜ P )0 HdS is the ItÃť-integral under the fixed measure ˜ P ∈ ˜ P . As S is a continuouslocal ( ˜ P , G ∗ , ˜ P + ) -martingale for all ˜ P ∈ ˜ P and H is integrable, it follows that also R t, ( ˜ P )0 HdS is acontinuous local ( ˜
P , G ∗ , ˜ P + ) -martingale for all ˜ P ∈ ˜ P . Here, we use that a sigma-martingale withcontinuous paths is a local martingale by Theorem 91 (IV.9) in [41]. Consider now X x,H ∈ X simp .By (6 . we get ≤ X x,Ht = X x,H, t + n X i =1 h Yi ( S Yτ i ∧ t − S Yτ i − ∧ t )= X x,H, t + n X i =1 h Yi (cid:18) D τ i ∧ t + Z τ i ∧ t, ( ˜ P )0 HdS − D τ i − ∧ t − Z τ i − ∧ t, ( ˜ P )0 HdS (cid:19) = X x,H, t + n X i =1 h Yi ( D τ i ∧ t − D τ i − ∧ t ) | {z } ≤ + n X i =1 h Yi (cid:18) Z τ i ∧ t, ( ˜ P ) τ i − ∧ t HdS (cid:19) ≤ X x,H, t + n X i =1 h Yi (cid:18) Z τ i ∧ t, ( ˜ P ) τ i − ∧ t HdS (cid:19) := ˜ X x,Ht . Note, ˜ X x,H ·∧ ˜ τ m is a ( ˜ P , G ∗ , ˜ P + ) -supermartingale for all ˜ P ∈ ˜ P , m ∈ N . It follows by Fatou’s Lemma By the same arguments regarding the filtration as in Remark 6.2 S is also ( ˜ P , G ∗ , ˜ P + ) -semimartingale for all ˜ P ∈ ˜ P . The sigma-martingale property holds with respect to the filtration G ∗ , ˜ P + for all ˜ P ∈ ˜ P . ˜ X x,H is a ( ˜ P , G ∗ , ˜ P + ) -supermartingale for all ˜ P ∈ ˜ P . Thus, for all t ∈ [0 , T ] E ˜ P [ X x,Ht ] ≤ E ˜ P [ ˜ X x,Ht ] ≤ E ˜ P [ ˜ X x,H ] = E ˜ P (cid:2) X x,H + n X i =1 h Yi ( D τ i ∧ − D τ i − ∧ ) | {z } =0 (cid:3) = E ˜ P [ X x,H ] for ˜ P ∈ ˜ P which implies Assumption 6.6.Here, we only assume a no short-selling constraint for the strategies. The price we pay for this aremore assumptions on P . An example for a set of probability measures satisfying these conditionsis given in Lemma 4.2 in [37]. However, the set ˜ Z does not satisfy these assumptions as alreadya set of affine processes is not saturated even under one single prior. In general, the optionaldecomposition theorem under model uncertainty in [37] also requires that S has non-dominatingdiffusions under each ˜ P ∈ ˜ P . However, this property is always satisfied if S is continuous, seeExample 2.3 ii) in [37].As already mentioned the set Z is in general not saturated due to the affine structure, as it isoutlined in the following. Let P ∈ Z such that S is a positive local ( P, F ∗ , Z + ) -martingale. Consider P ′ ∈ P (Ω x ) such that P ∼ P ′ and B S is a local ( P ′ , F ∗ , Z + ) -martingale. By the definition of Z , B S is F -adapted and thus a local ( P ′ , F ) -martingale by Theorem 10 in [18]. Furthermore, as B µ is a ( P, F ) -semimartingale, it follows that B µ is a ( P ′ , F ) -semimartingale due to P ∼ P ′ by TheoremIII.3.13 in [25]. By applying Girsanov’s theorem for semimartingales in Proposition III.3.24 in [25]to B µ , there exists a predictable process b satisfying Z | α s b s | ds < ∞ and Z b s α s ds < ∞ P ′ -a.s. for t ∈ [0 , T ] and such that a version of the characteristics of B µ relative to P ′ is given by A P ′ = A P + Z α s b s ds = Z (cid:0) β Ps + α s b s (cid:1)| {z } := β P ′ s ds, C P ′ = C (6.11)up to a P ′ -null set. By (6 . we can see why the saturation property is not satisfied for anarbitrary affine structure in Definition 3.2, as we can not guarantee β P ′ s ∈ b ∗ ( B µs ) for dP ′ ⊗ dt -almost all ( s, ω ) ∈ Ω x × [ t, T ] . However, by considering only an affine structure on the volatility ofthe mortality intensity, as it is the case in Example 4.8, the set Z is saturated.We now compare the price process ( ˜ E t ( Y )) t ∈ [0 ,T ] of the contingent claim Y with its correspondingsuperhedging price. In this setting Theorem 3.11 and 3.12 in [5] can be reformulated if ˜ P satisfiesAssumption 3.1 in [5]. Assumption 6.11. ˜ P is a set of sigma martingale measures for S , i.e., S is a ( ˜ P , G ∗ , ˜ P + ) -sigma-martingale forall ˜ P ∈ ˜ P ;2. ˜ P is saturated: all equivalent sigma-martingale measures of its elements still belong to ˜ P ;3. S has dominating diffusion under every ˜ P ∈ ˜ P . Here, S is assumed to be a d -dimensional G ∗ , ˜ P -adapted process with càdłàg paths such that S is a ( ˜ P , G ∗ , ˜ P ) -semimartingale for every ˜ P ∈ ˜ P . Furthermore, the set of d -dimensional G ∗ , ˜ P -predictableprocesses which are S -integrable for all ˜ P ∈ ˜ P is denoted by ˜ L ( S, ˜ P ) and the admissible strategieson ˜Ω are given by ˜ △ := (cid:26) ˜ δ ∈ ˜ L ( S, ˜ P ) : Z ( ˜ P ) ˜ δdS is a ( ˜ P , G ∗ , ˜ P + ) -supermartingale for all ˜ P ∈ ˜ P (cid:27) . In this case the notation R ( ˜ P ) ˜ δdS := ( R ( ˜ P ) ,t ˜ δdS ) t ∈ [0 ,T ] is the usual Itô integral under ˜ P . Werecall [5, Theorem 3.11]. 23 heorem 6.12. Let Assumption 2.1 hold for ( P ( t, ω )) ( t,ω ) ∈ [0 ,T ] × Ω and Assumption 6.11 for ˜ P ,respectively. Consider Y to be an upper semianalytic, G P T -measurable and nonnegative contingentclaim such that ˜ E t ( Y ) ∈ L ( ˜Ω) for all t ∈ [0 , T ] . If t ∈ [0 , T ] and there exists a G ∗ , ˜ P -adaptedprocess ˜ X = ( ˜ X s ) s ∈ [0 ,T ] with càdlàg paths, such that for s ∈ [0 , T ]˜ X s = ˜ E s ( Y ) ˜ P -a.s. for all ˜ P ∈ ˜ P , and if the tower property holds for Y , i.e., for all r, s ∈ [0 , t ] with r ≤ s , ˜ E t ( Y ) = ˜ E r ( ˜ E s ( Y )) ˜ P -a.s. for all ˜ P ∈ ˜ P , then we have the following equivalent dualities for all ˜ P ∈ ˜ P and s ∈ [0 , T ]˜ E s ( Y ) = ess inf ˜ P { ˜ v is G ∗ , ˜ P s -measurable : ∃ ˜ δ ∈ ˜ △ such that ˜ v + Z ( ˜ P ′ ) ,Ts ˜ δ u dS u ≥ Y ˜ P ′ -a.s.for all ˜ P ′ ∈ ˜ P} =: ess inf ˜ P { D s } ˜ P -a.s. (6.12) = ess inf ˜ P { ˜ v is G ∗ , ˜ P s -measurable : ∃ ˜ δ ∈ ˜ △ such that ˜ v + Z ( ˜ P ′ ) ,Ts ˜ δ u dS u ≥ Y ˜ P ′ -a.s.for all ˜ P ′ ∈ ˜ P ( s ; ˜ P ) } =: ess inf ˜ P { D ( ˜ P ) s } ˜ P -a.s. (6.13)If ˜ P does not satisfy Assumption 6.11, the superhedging dualities (6 . , (6 . do not hold ingeneral. However, by having a look at the proof of Theorem 5.2.21 in [46] one of the two inequalitiesis still valid, i.e., for s ∈ [0 , T ] and ˜ P ∈ ˜ P ˜ E s ( Y ) ≤ ess inf ˜ P { D ( ˜ P ) s } ≤ ess inf ˜ P { D s } ˜ P -a.s. (6.14)for D ( ˜ P ) s given in (6 . and D s in (6 . . Note, in this context ess inf ˜ P { D ( ˜ P ) s } corresponds to the ˜ P -superhedging price and ess inf ˜ P { D s } to the ˜ P ( s ; ˜ P ) -superhedging price. The inverse inequalityis not valid as an optional decomposition result for semimartingales given by Theorem 2.4 in [37]is used which requires Assumption 6.11 to hold. Nevertheless, even if the price of the contingentclaim Y is lower than the superhedging price, the extended market ( S , S, ˜ E ( Y )) on ˜Ω is stillarbitrage-free under reasonable assumptions, as shown in Proposition 6.7. Moreover, even underone single prior the superhedging price of a contingent claim is often criticized as being too high.In the case of considering several priors, we see in (6 . that the quasi-sure ˜ P ( s ; ˜ P ) -superhedgingprice is even more conservative than the one under a single prior. Remark 6.13.
In Definition 6.1 of NA ( P ) we only allow for simple trading strategies. However,in (6 . , (6 . the strategy ˜ δ ∈ ˜∆ does not need to be simple. The same situation occurs in thesuperreplication result in Theorem 5.1 in [6]. However, in the setting analyzed in Section 6 it holds ˜∆ simp := { H ∈ H simp : X ,H is a ( ˜ P , G ∗ , ˜ P + ) -supermartingale for all ˜ P ∈ ˜ P} ⊆ ˜∆ , (6.15)where X ,H is defined as X ,H, in (6 . . Thus, it follows D simp s ⊆ D s with D simp s := { ˜ v is G ∗ , ˜ P s -measurable : ∃ ˜ δ ∈ ˜ △ such that ˜ v + Z ( ˜ P ′ ) ,Ts ˜ δ u dS u ≥ Y ˜ P ′ -a.s. for all ˜ P ′ ∈ ˜ P ( s ; ˜ P ) } . This together with (6 . implies that for all s ∈ [0 , T ] and ˜ P ∈ ˜ P ˜ E s ( Y ) ≤ ess inf ˜ P { D s } ≤ ess inf ˜ P { D simp s } ˜ P -a.s. . In this paper we were able to define an extended market model within a reduced-form frameworkunder model uncertainty where the mortality intensity follows a non-linear affine price process.This allows both to introduce the definition of a longevity bond under model uncertainty as wellas to compute it by explicit formulas or by numerical methods. We are also able to guarantee theexistence of a càdlàg modification for the longevity bond’s value process. Furthermore, we showhow the resulting market model extended with the longevity bond is arbitrage-free. These resultscan be used for further research on hedging under model uncertainty.24
Appendix
Proof.
Proposition 4.6The proof consists in verifying that the arguments in [32] are also valid in our setting. We show thisexplicitly how this is done for one property. The other properties follow with similar arguments.
Step 1:
The families ( A ( t, ω t , Θ)) ( t,ω ) ∈ [0 ,T ] × Ω x defined in Definition 3.2 satisfy Assumption 2.1.1) Measurability:
From [15, Lemma 3.1] we know that the set { ( ω, t, P ) ∈ Ω x × [0 , T ] × P (Ω x ) | P ∈ A ( t, ω t , Θ) } (8.1)is Borel which implies that the set is analytic.2) Invariance:
Let ( s, ω ) ∈ [0 , T ] × Ω x , P ∈ A ( s, ω s , Θ) and τ be a stopping time taking values in [ s, T ] . It is clear that for every ω ∈ Ω x we can define the conditional probability P τ,ω with respectto F τ as in (2 . . We have to prove that P τ,ω ∈ A ( τ, ω τ ( ω ) , Θ) for P -a.e. ω ∈ Ω x , i.e.,1. P τ,ω ∈ P acsem P τ,ω ( B τ ( ω ) = ω τ ( ω ) ) = 1 β P τ,ω u ∈ b ∗ ( B u ) and α P τ,ω u ∈ a ∗ ( B u ) for dP τ,ω ⊗ dt -almost all (˜ ω, u ) ∈ Ω x × ( s, T ] ,for P -a.e. ω ∈ Ω x . Here we denote by β P τ,ω = ( β P τ,ω u ) u ∈ ( s,T ] the absolutely continuous differentialprocess with respect to the probability measure P τ,ω and the filtration F . The same notationis used for the differential process α . The second point follows directly by the definition of theprobability P τ,ω in (2 . , as it is possible to choose the probability measure P τ,ω concentratedon the paths which coincide with ω up to time τ ( ω ) , see Section 2.1 in [38]. The first point is aconsequence of Theorem 3.1 in [33], which contains two main results. First, given a probabilitymeasure P ∈ P acsem it follows that for P -a.e. ω ∈ Ω we have P τ,ω ∈ P acsem . Second, given thedifferential characteristics of the canonical process under P ∈ P acsem with respect to F are ( β P , α P ) ,then the P τ,ω - F -characteristics are given by ( β P τ,ω u , α P τ,ω u ) := (( β Pτ + u ) τ,ω , ( α Pτ + u ) τ,ω ) , (8.2)where the notation introduced in (2 . is used. As in our setting P ∈ A ( s, ω s , Θ) , it holds β Pu ∈ b ∗ ( B u ) for dP ⊗ dt -almost all (˜ ω, u ) ∈ Ω x × ( s, T ] by the definition of the set A ( s, ω s , Θ) . Thisallows to conclude that β P τ,ω u ( · ) = β Pτ + u ( ω ⊗ τ · ) ∈ b ∗ ( B τ + u ( ω ⊗ τ · )) for dP ⊗ dt -almost all (˜ ω, u ) ∈ Ω x × [0 , T ] . As a consequence it holds B τ + u ( ω ⊗ τ · ) = B u ( · ) P τ,ω -a.s. by using (2 . and (2 . , which provesthe affine property. With the same arguments the result follows for the process α P τ,ω s .3) Stability under Pasting:
By using similar arguments as in the proof of the invariance conditionthis property follows by generalizing the results of Proposition 4.1 in [33].
Step 2:
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