An Expansion Formula for Hawkes Processes and Application to Cyber-Insurance Derivatives ∗

An expansion formula for Hawkes processes and application to cyber-insurance derivatives ∗

Caroline Hillairet† Anthony Réveillac‡ Mathieu Rosenbaum§

April 6, 2021

Abstract In this paper we provide an expansion formula for Hawkes processes which involves the addition of jumps at deterministic times to the Hawkes process in the spirit of the well- known integration by parts formula (or more precisely the Mecke formula) for Poisson functional. Our approach allows us to provide an expansion of the premium of a class of cyber insurance derivatives (such as reinsurance contracts including generalized Stop- Loss contracts) or risk management instruments (like Expected Shortfall) in terms of so-called shifted Hawkes processes. From the actuarial point of view, these processes can be seen as "stressed" scenarios. Our expansion formula for Hawkes processes enables us to provide lower and upper bounds on the premium (or the risk evaluation) of such cyber contracts and to quantify the surplus of premium compared to the standard modeling with a homogenous Poisson process.

Keywords: Hawkes process; Malliavin calculus; pricing formulae; cyber insurance derivatives.

1 Introduction

In actuarial science, the classical Cramer-Lundberg model used to describe the surplus process of an insurance portfolio relies on the assumptions of the claims arrival being modeled by a Poisson process, and of independence among claim sizes and between claim sizes and claim inter-occurrence times. However, in practice those assumptions are often too restrictive and there is a need for more general models. A ﬁrst generalization in the modeling of claims arrivals consists in using Cox processes (also known as doubly stochastic Poisson processes), in the context of ruin theory such as in Albrecher and Asmussen (2006) [2], or for pricing stop-loss arXiv:2104.01579v1 [math.PR] 4 Apr 2021 catastrophe insurance contract and catastrophe insurance derivatives such as in Dassios and Jang (2003) [13] and (2013) [14], or Hillairet et al. (2018) [11].

∗This research is supported by a grant of the French National Research Agency (ANR), “Investissements d’Avenir” (LabEx Ecodec/ANR-11-LABX-0047) and the Joint Research Initiative "Cyber Risk Insurance: actuarial modeling" with the partnership of AXA Research Fund. †ENSAE Paris, CREST UMR 9194, 5 avenue Henry Le Chatelier 91120 Palaiseau, France. Email: [email protected] ‡INSA de Toulouse, IMT UMR CNRS 5219, Université de Toulouse, 135 avenue de Rangueil 31077 Toulouse Cedex 4 France. Email: [email protected] §Ecole Polytechnique, CMAP UMR 7641, Route de Saclay, 91120 Palaiseau, France. Email: [email protected]

1 Besides, self-exciting effects have been highlighted in cyber risk, in favor of modeling the claims arrivals by a Hawkes process, that is adapted to model aftershocks of cyber attacks. Such processes have been recently used in the cyber security field, for instance by Peng et al. (2016) [?] who focused on extreme cyber attacks rates. Baldwin et al. (2017) also studied in [?] the threats to 10 important IP services, using industry standard SANS data, and they claim that Hawkes processes provides the adequate modeling of cyber attacks into information systems because they capture both shocks and persistence after shocks that may form attack contagion. In cyber insurance, the statistical analysis of Bessy et al. (2020) [?] on the public Privacy Rights Clearinghouse database highlights the ability of Hawkes models to capture self-excitation and interactions of data-breaches. Although the application in this paper fo- cuses on cyber risk, this methodology can be applied to other risks presenting self-exciting properties, such as credit risk. Hawkes processes, which have wide applications in many fields (such as seismology, finance, neuroscience or social networks), are also beginning to get studied in actuarial sciences. Mag- nusson Thesis (2015) [13] is dedicated to Hawkes processes with exponential decay and their application to insurance. Dassios and Zhao (2012) [5] consider the ruin problem in a model where the claims arrivals follow a Hawkes process with decreasing exponential kernel. Stabile and Torrisi (2010) [18] study the asymptotic behavior of infinite and finite horizon ruin probabilities, assuming non-stationary Hawkes claims arrivals and under light-tailed conditions on the claims. Gao and Zhu (2018) [8] establish large deviations results for Hawkes processes with an exponential kernel and develop approximations for finite-horizon ruin probabilities. Swishchuk (2018) [?] applies limit theorems for risk model based on general compound Hawkes process to compute premium principles and ruin times. Rather than giving explicit computations for probabilities of ruin, our paper proposes to give pricing formulae of insurance contracts, such as Stop-Loss contracts. Stop-Loss is a non- proportional type of reinsurance and works similarly to excess-of-loss reinsurance. While excess-of-loss is related to single loss amounts, either per risk or per event, stop-loss covers are related to the total amount of claims in a year. The reinsurer pays the part of the total loss that exceeds a certain amount K. The reinsurer’s liability is often limited to a given threshold. Stop-loss reinsurance offers protection against an increase in either or both severity and frequency of a company’s loss experience. Various approximations of stop-loss reinsurance premiums are described in literature, some of them assuming certain dependence structure, such as Gerber (1982) [9], Albers (1999) [1], De Lourdes Centeno (2005) [6] or Reijnen et al. (2005) [17]. Stop-loss contracts are the paradigm of reinsurance contracts, but we aim at dealing with more general payoffs (of maturity T ), whose valuation involves the computation of the quantity of the form   KT is the effective loss covered by the reinsurance company, E[KT h(LT )] where  LT is the loss quantity that activates the contract.

For example, for stop loss contracts h(LT ) = 1{LT ≥K}. Similarly, this methodology can be applied to valuation of credit derivatives, such as Credit Default Obligation tranches. It also goes beyond the analysis of pricing and ﬁnds application in the computation of the expected shortfall of contingent claims : the expected shortfall is a useful risk measure, that takes into account the size of the expected loss above the value at risk. We refer to [11] for more details on those analogies.

2 Our paper considers cyber insurance contracts, with underlying a cumulative loss indexed by a Hawkes process. We propose to compute an explicit closed form pricing formula. Although Monte Carlo procedures are certainly the most eﬃcient to compute numerically the premium of such general contracts, the closed form expansion formula we develop allows to compute lower and upper bounds for the premium, and to quantify the surplus of premium compared to the standard modeling with a homogenous Poisson process. A correct estimate of this surplus of premium is a crucial challenge for cyber insurance, to avoid an underestimation of the risk induced by a Poisson process model. Such formula could also be eﬃcient for sensitivity analysis. The formula relies on the so-called shifted Hawkes processes, which, from the actuarial point of view, can be seen as "stressed" scenarios.

From the probabilistic point of view, the quantity E[KT h(LT )] can be expressed (conditioning hR i with respect to the claims) as E (0,T ] ZtdHtF where Z is a predictable process and F := h(LT ) is a functional of the Hawkes process. In the case where the counting process is a Poisson process (or a Cox process), Malliavin calculus enables one to transform this quantity. More precisely, to simplify the discussion, assume H is an homogeneous Poisson process with intensity µ > 0 (in other words the self-exciting kernel Φ is put to 0), the Malliavin integration by parts formula allows us to derive that1 :

"Z # Z T + E ZtdHtF = µ E ZtF ◦ εt dt, (1.1) (0,T ] 0

+ where the notation F ◦ εt denotes the functional on the Poisson space where a deterministic jump is added to the paths of H at time t. This expression turns out to be particularly interesting from an actuarial point of view since adding a jump at some time t corresponds to realising a stress test by adding artiﬁcially a claim at time t. This approach has been followed in [11] for Cox processes (that is doubly stochastic Poisson processes with stochastic but independent intensity). Naturally, in case of a Poisson process, the additional jump at some time t only impacts the payoﬀ of the contract by adding a new claim in the contract but it does not impact the dynamic of the counting process H.

The goal of this paper is two-fold:

1. First we provide in Theorem 3.13 a generalization of Equation (1.1) in case H is a Hawkes process. The main ingredient consists in using a representation of a Hawkes process known as the "Poisson embedding" (related to the "Thinning Algorithm") in terms of a Poisson process N on [0,T ] × R+ to which the Malliavin integration by parts formula can be applied. As the adjunction of a jump at a given time impacts the dynamic of the Hawkes process, we refer to the obtained expression more to an "expansion" rather than an "integration by parts formula" for the Hawkes process, as it involves what we name "shifted Hawkes processes" for which jumps at deterministic times are added to the process accordingly to the self-exciting kernel Φ. We refer to Theorem 3.13 and to Remark 3.14 for a discussion on this expansion and its 1Note that strictly speaking this formula is not the Malliavin integration by parts formula on the Poisson R space as the stochastic integral (0,T ] ZtdHt is not exactly the divergence of Z, which explains why the dual operator on the right-hand side is not exactly the Malliavin diﬀerence operator. However, in case of predictable integrators Z, the classical integration by parts formula can be reduced to this form which is suﬃcient for our purpose.

3 link to the one obtained for homogeneous Poisson process.

2. Then, we apply our main result to the specific quantity E[KT h(LT )] which is at the core for determining the premium of a large class of insurance derivatives or risk management instruments. Our main result on that regard is given in Theorem 4.3. As pointed out in the discussion at the beginning of Section 4.3, the shifted processes Hvn,...,v1 (see Definition 3.6 for a precise statement) appearing in the form of the premium are of the same complexity than the original Hawkes process H. However, they exhibit deterministic jumps at some times v1, . . . , vn which are weighted by correlation factors of the form Φ(vi − vi−1). In other words, this formula make appears n-jumps of the Hawkes process at some deterministic times. This provides an additional input compared to classical estimates for Hawkes processes for obtain- ing lower or upper bounds of their CDF in terms of the one of a Poisson process for instance. We benefit from this formulation to derive in Proposition 4.5 and Proposition 4.8 a lower and an upper bound respectively for the quantity E[KT h(LT )].

We proceed as follows. In the next section, we provide general notations and elements of Malliavin calculus on the (classical) Poisson space. In particular, the shift operators (which will play a central role in our analysis) on the Poisson space are introduced. We also explain the representation of a Hawkes process using the Poisson embedding. Section 3 provides the derivation of the expansion formula in Theorem 3.13. Note that it requires the introduction and analysis of what we named the shifted Hawkes processes resulting from the shifts on the Poisson space of the original Hawkes process (this material is presented in Section 3.1). In- surance contracts of interest are presented in Section 4 together with the main result for the representation of the premium of such contracts in Theorem 4.3. Lower and upper bounds for this premium are presented in Propostion 4.5 and Proposition 4.8, and in Corollaries 4.7 and 4.9. Finally, we postponed some technical material in Section 5.

2 Elements of stochastic analysis on the Poisson space, Hawkes process and thinning

This short section provides some generalities and elements of stochastic analysis on the Poisson space, and in particular the integration by parts formula for the Poisson process. Hawkes processes are also deﬁned and their representation through the thinning procedure is presented. Throughout this paper T > 0 denotes a ﬁxed positive real number. For X a topological space, we set B(X) the σ-algebra of Borelian sets.

2.1 Elements of stochastic analysis on the Poisson space Let the space of conﬁgurations

( n ) N N X Ω := ω = δti,θi , i = 1, . . . , n, 0 = t0 < t1 < ··· < tn ≤ T, θi ∈ R+, n ∈ N ∩ {+∞} . i=1

N N Each path of a counting process is represented as an element ω in Ω which is a N-valued N N measure on [0,T ] × R+. Let FT be the σ-ﬁeld associated to the vague topology on Ω , and N N P the Poisson measure on Ω under which the counting process N deﬁned as the canonical

4 process on ΩN as

(N(ω))([0, t] × [0, b])(ω) := ω([0, t] × [0, b]), t ∈ [0,T ], b ∈ R+, is an homogeneous Poisson process with intensity one (so that N([0, t] × [0, b]) is a Poisson N N random variable with intensity bt for any (t, b) ∈ [0,T ] × R+). We set F := (Ft )t∈[0,T ] N the natural ﬁltration of N, that is Ft := σ(N(T × B), T ⊂ B([0, t]),B ∈ B(R+)). The N expectation with respect to P is denoted by E[·].

One of the main ingredient in our approach will be the integration by parts formula for the Poisson process N and the shift operators deﬁned below.

Deﬁnition 2.1 (Shift operator). We deﬁne for (t, θ) in [0,T ] × R+ the measurable map + N N ε(t,θ) :Ω → Ω + ω 7→ ε(t,θ)(ω), + with (ε(t,θ)(ω))(A) := ω(A \ (t, θ)) + 1A(t, θ),A ∈ B([0,T ] × R+) and where 1, if (t, θ) ∈ A, 1A(t, θ) := 0, else.

Remark 2.2. Let (t0, θ0) in (0,T ) × R+, t0 < s < t, T ∈ {(s, t), (s, t], [s, t), [s, t]} and B in B(R+). We have that N ◦ ε+ (T × B) = ε+ (T × B) = N(T × B), − a.s.. (t0,θ0) (t0,θ0) P N Lemma 2.3. Let t in [0,T ] and F be an Ft -measurable random variable. Let v > t and θ0 ≥ 0. It holds that F ◦ ε+ = F, − a.s.. (v,θ0) P N Proof. The proof consists in noticing that a Ft -measurable random variable is a functional of N·∧t. N − Similarly, for any ω in Ω and (t, θ) in [0,T ] × R+, we set the measure ε(t,θ)(ω) deﬁned as − (ε(t,θ)(ω))(A) := ω(A \{(t, θ)}),A ∈ B([0,T ] × R+). We conclude this section with the integration by parts formula (ore more speciﬁcally the Mecke formula) on the Poisson space (see [16, Corollaire 5] or [14]). 1 N Proposition 2.4 (Mecke’s Formula). Let F be in L (Ω, FT , P) and Z = (Z(t, θ))t∈[0,T ],θ∈R+ N 2 hR T i be a F -adapted process with E 0 |Z(t, θ)|dt < +∞ and such that − Z(t, θ) ◦ ε(t,θ) = Z(t, θ), P ⊗ dt ⊗ dθ, a.e.. (2.1) We have that " Z # Z T Z + E F Z(t, θ)N(dt, dθ) = E Z(t, θ)(F ◦ ε(t,θ))dtdθ . [0,T ]×R+ 0 R+ For the expansion formula in Theorem 3.13, the Mecke’s formula will be applied for the process

Z(t, θ) = Zt1{θ≤Λt} where Λ will denote the intensity of the Hawkes process (we refer to the proof of Theorem 3.13 for a more precise relation between this formula and our result).

2 N Note that only measurability with respect to FT is necessary here.

5 2.2 Representation of Hawkes processes We ﬁrst recall the deﬁnition of a Hawkes process.

Deﬁnition 2.5 (Standard Hawkes process, [10]). Let (Ω, FT , P, F := (Ft)t∈[0,T ]) be a ﬁltered probability space, µ > 0 and Φ : [0,T ] → R+ be a bounded non-negative map with kΦk1 < 1. A standard Hawkes process H := (Ht)t∈[0,T ] with parameters µ and Φ is a counting process such that

(i) H0 = 0, P − a.s.,

(ii) its (F-predictable) intensity process is given by Z Λt := µ + Φ(t − s)dHs, t ∈ [0,T ], (0,t)

that is for any 0 ≤ s ≤ t ≤ T and A ∈ Fs, "Z # E [1A(Ht − Hs)] = E 1AΛrdr . (s,t]

This deﬁnition can be generalized as follows, by considering a starting date v > 0 and allowing starting points (for the Hawkes process itself and its intensity) that are Fv-measurable (that is that are known at time v).

Deﬁnition 2.6 (Generalized Hawkes process). Let (Ω, FT , P, F := (Ft)t∈[0,T ]) be a ﬁltered v probability space. Let v in [0,T ], h be a Fv-measurable random variable with valued in v v v N, µ := (µ (t))t∈[v,T ] a positive map such that µ (t) is Fv-measurable for any t ≥ v, and Φ : [0,T ] → R+ be a bounded non-negative map with kΦk1 < 1. A Hawkes process on [v, T ] v v with parameters µ , h and Φ : [0,T ] → R+ is a (F-adapted) counting process H := (Ht)t∈[v,T ] such that

v (i) Hv = h , P − a.s.,

(ii) its (F-predictable) intensity process is given by Z v Λt := µ (t) + Φ(t − s)dHs, t ∈ [v, T ], (v,t)

that is for any v ≤ s ≤ t ≤ T and A ∈ Fs, " # Z

E [1A(Ht − Hs)|Fv] = E 1AΛrdr Fv . (s,t]

Our main result relies on the following representation of a Hawkes process known as the "Poisson embedding" and related to the "Thinning Algorithm" (see e.g. [2, 3, 4, 15] and references therein). To this end we consider the ﬁltered probability space (Ω, FT , P, F) as follows

N N N N Ω := Ω , FT := FT , F := (Ft)t∈[0,T ], Ft := Ft , t ∈ [0,T ], P := P .

6 v Theorem 2.7. Let v in [0,T ] and (µ (t))t∈[v,T ] be a non-negative stochastic process such v N v N that for any t ≥ v, µ (t) is a Fv -measurable random variable. Let in addition h be a Fv - measurable random with values in N. On the probability space (Ω, FT , P), the SDE (2.3) with

 ˆ v v R R Ht = h + 1 ˆ v N(ds, dθ), t ∈ [v, T ]  (v,t] R+ {θ≤Λs } (2.2)  ˆ v v R ˆ v  Λt = µ (t) + (v,t) Φ(t − u)dHu

N admits a unique F -adapted solution Hˆ . Uniqueness is understood in the strong sense, that is, if Hˆ 1, Hˆ 2 denote two solutions then " # ˆ 1 ˆ 2 P sup |Ht − Ht |= 6 0 = 0. t∈[0,T ]

This result can be deduced from the general construction of point process of Jacod et al [12]. Nevertheless for seek of completeness, a direct proof for Hawkes process is postponed to Section 5.1.

Corollary 2.8. Let µ ∈ R+. We consider (H, Λ) the unique solution to SDE  R R Ht = 1{θ≤Λ }N(ds, dθ), t ∈ [0,T ]  (0,t] R+ s (2.3)  R  Λt = µ + (0,t) Φ(t − u)dHu.

H H H N We set F := (Ft )t∈[0,T ] the natural ﬁltration of H (obviously Ft ⊂ Ft ). Then H is a standard Hawkes process.

H Proof. Let 0 ≤ s ≤ t ≤ T and A in Fs . Using Lemma 2.3, Z(u, θ) = 1{θ≤Λu} satisﬁes Relation (2.1) (as Λ is predictable), one can thus apply Proposition 2.4 (Mecke’s Formula) for H Z(u, θ) = 1{θ≤Λu}. Then, we have (by conditioning with respect to Fs )

E [1As (Ht − Hs)] " Z # = E 1As dHu (s,t] " Z Z #

= E 1As 1{θ≤Λu}N(du, dθ) (s,t] R+ " Z Z #

= E 1As 1{θ≤Λu}dudθ (s,t] R+ " Z # = E 1As Λudu , (s,t]

+ where we have used again Lemma 2.3 to prove that 1As ◦ ε(u,θ) = 1As for u > s.

7 3 An expansion formula for Hawkes processes

Thanks to the previous representation of a Hawkes process using the Poisson embedding, we derive in this section an expansion formula. The main result is stated in Theorem 3.13. It requires an accurate deﬁnition and analysis of what we named the shifted Hawkes processes resulting from the shifts on the Poisson space of the original Hawkes process.

3.1 The shifted Hawkes processes We now introduce shifted Hawkes processes that is the eﬀect of the shift operators ε+ on the Hawkes process. As we will see the resulting Hawkes process can also be described in terms of a Poisson SDE. Deﬁnition 3.1 (One shift Hawkes process). Let H be a standard Hawkes process with initial intensity µ > 0 and bounded excitation function Φ : [0,T ] → R+ such that kΦk1 < 1. Let v in (0,T ). We set  Z Z !  v v  H = 1[0,v)(t)Ht + 1[v,T ](t) H + 1 + 1{θ≤Λv}N(ds, dθ) ,  t v− s  (v,t] R+ (3.1) !  Z  v v,1 v  Λt = 1(0,v](t)Λt + 1(v,T ](t) µ (t) + Φ(t − u)dHu ,  (v,t)

v,1 R v R with µ (t) := µ + (0,v] Φ(t − u)dHu = µ + (0,v) Φ(t − u)dHu + Φ(t − v). Remark 3.2. (Hv, Λv) is called a shifted (at time v) Hawkes process but it is not a Hawkes process as it possesses a jump at the deterministic time v. However, it is a Hawkes process on (0, v) (and coincides with the original Hawkes process (H, Λ)) and is a Hawkes process in the sense of Deﬁnition 2.6 (generalized Hawkes process) on [v, T ]. Deﬁnition 3.3. Let v in [0,T ], we set ε+ :ΩN → ΩN (v,Λv) + ω 7→ (ε(v,θ)(ω))θ=Λv(ω).

Remark 3.4. Let v in [0,T ] and θ0 ≥ 0. As P[N({v} × R+) > 0] = 0, a direct computation gives that Z Z ! + + Hv ◦ ε = 1 N(ds, dθ) ◦ ε (v,θ0) {θ≤Λs} (v,θ0) [0,v] R+

= Hv− + 1{θ0≤Λv}, P − a.s.. Hence + + 1 Hv ◦ ε = 1 Hv ◦ ε , − . {θ0≤Λv} (v,θ0) {θ0≤Λv} (v,Λv) P a.s. (3.2) v v As Equation (2.2) is completely determined by the values h and (µ (t))t∈[0,T ] (recalling that Φ does not depend on v), we deduce that for any θ0 ≥ 0, on the set {θ0 ≤ Λv}, + + + + (Ht ◦ ε , Λt ◦ ε ) = (Ht ◦ ε , Λt ◦ ε ) . (v,θ0) (v,θ0) t∈[v,T ] (v,Λv) (v,Λv) t∈[v,T ] This remark leads us to the following lemma.

8 Lemma 3.5. Let v in [0,T ]. We have that

(H ◦ ε+ , Λ ◦ ε+ ) = (Hv, Λv), (v,Λv) (v,Λv) where (Hv, Λv) is deﬁned in (3.1).

Proof. Recall that  R R Ht = Hv + 1{θ≤Λ(s)}N(ds, dθ), t ∈ [0,T ]  (v,t] R+  R R  Λ(t) = µ + (0,v] Φ(t − u)dHu + (v,t) Φ(t − u)dHu.

H ◦ ε+ = H t < v Λ ◦ ε+ = Λ t ≤ v t ≥ v Thus, by Lemma 2.3, t (v,Λv) t for and t (v,Λv) t for . Let , we have that (recall that the jump times of N shifted by Λv after time v coincide with those of N)

 + R R Ht ◦ ε(v,Λ ) = Hv− + 1 + (v,t] 1{θ≤Λ◦ε+ (s)}N(ds, dθ), t ∈ [0,T ]  v R+ (v,Λv)

 Λ ◦ ε+ = µ + R Φ(t − u)dH + Φ(t − v) + R Φ(t − u)dH ◦ ε+ . t (v,Λv) (0,v) u (v,t) u (v,Λv)

The result follows by uniqueness of the solution to this SDE.

We now proceed iteratively to construct a Multi-shifted Hawkes process: the shifts (v1, ··· , vn−1) th being chosen, the n shift is taken in the interval ]0, vn[.

Deﬁnition 3.6 (Multi-shifted Hawkes process). ∗ Let n ∈ N , and 0 < vn < vn−1 < ··· < v1 < T . We set  n Z Z !  vn,...,v1 X vn,...,v1  vn,...,v  Ht = 1[0,vn)(t)Ht + 1[v ,v )(t) Hv − + 1 + 1 1 N(ds, dθ) ,  i i−1 i {θ≤Λs }  i=1 (vi,t] R+ !  n Z  vn,...,v1 X vi,n vn,...,v1  Λ = 1 (t)Λt + 1 (t) µ (t) + Φ(t − u)dH ,  t (0,vn] (vi,vi−1] u i=1 (vi,t) (3.3) v ,n R vn,...,v1 R vn,...,v1 µ i (t) := µ + Φ(t − u)dHu = µ + Φ(t − u)dHu + Φ(t − vi) with (0,vi] (0,vi) . Remark 3.7. Note that the process Hvn,...,v1 is not a Hawkes process as it has deterministic jumps at times vn, . . . , v1 but it is a generalized Hawkes process on each interval (vi−1, vi).

∗ Proposition 3.8. Let n ∈ N , and 0 < vn < vn−1 < ··· < v1 < T . We have that

(H, Λ) ◦ ε+ ◦ · · · ◦ ε+ = (Hvn,...,v1 , Λvn,...,v1 ). (v1,Λv1 ) (vn,Λvn ) The proof is postponed to Section 5.2. We conclude this section on shifted Hawkes processes with the following remark.

Remark 3.9. Let v < vn and θ0 ≥ 0. Following the lines of Remark 3.4, we get that

1 (Hvn,...,v1 ◦ε+ , Λvn,...,v1 ◦ε+ ) = 1 (Hvn,...,v1 ◦ε+ , Λvn,...,v1 ◦ε+ ) . {θ0≤Λv} t (v,θ0) t (v,θ0) t∈[v,T ] {θ0≤Λv} t (v,Λv) t (v,Λv) t∈[v,T ]

9 3.2 Expansion formula for the Hawkes process

Let H = (Ht)t∈[0,T ] be a standard Hawkes process with parameters µ > 0 and bounded non- negative kernel Φ with kΦk1 < 1, solution to the SDE (2.3) and build on the Poisson space N N N (Ω , FT , P ), according to Section 2. In line of the results obtained in the previous section, we can derive the lemma below.

N Lemma 3.10. Let F be a FT -measurable random variable and v > 0. We have that Z (F ◦ ε+ )1 dθ = Λ (F ◦ ε+ ), − a.s.. (v,θ) {θ≤Λv} v (v,Λv) P R+ Proof. Let θ ≥ 0. Remarks 3.5 and 3.9 entail

1 (F ◦ ε+ ) = 1 (F ◦ ε+ ), {θ≤Λv} (v,θ) {θ≤Λv} (v,Λv)

which concludes the proof.

N N Deﬁnition 3.11. Let F be a FT -measurable random variable and Z a F -predictable process. We set

(i) for v in (0,T ), F v := F ◦ ε+ ,Zv := Z ◦ ε+ ; (v,Λv) (v,Λv)

(ii) for n ≥ 2 and 0 < vn < vn−1 < ··· < v1 < T ,

F vn,...,v1 := F vn−1,...,v1 ◦ ε+ ,Zvn,...,v2 := Zvn−1,...,v2 ◦ ε+ . (3.4) (vn,Λvn ) (vn,Λvn )

1 Notation 3.12. For n = 1, we set mΦ(∆ ) := 1 and for n ≥ 2,

n Z T Z v1 Z vn−1 n Y mΦ(∆ ) := ··· Φ(vi−1 − vi)dvn ··· dv1, 0 0 0 i=2

To ensure the convergence of the forthcoming expansion formula, one will need the following assumption n lim mΦ(∆ ) = 0. (3.5) n→+∞ n T n Remark that if kΦk∞ < 1, mΦ(∆ ) ≤ n! and Relation (3.5) is satisﬁed. We have now introduced all the ingredients to state the expansion formula for the Hawkes process.

Theorem 3.13 (Expansion formula for the Hawkes process). N Let F be a bounded FT -measurable random variable and Z = (Zt)t∈[0,T ] be a bounded H ∗ F -predictable process. Then for any M ∈ N , " Z # E F ZtdHt [0,T ] Z T v = µ E [ZvF ] dv 0

10 M n Z T Z v1 Z vn−1 X Y vn,...,v2 vn,...,v1 + µ ··· Φ(vi−1 − vi)E Zv1 F dvn ··· dv1 n=2 0 0 0 i=2 M+1 Z T Z v1 Z vM Y vM+1,...,v2 vM+1,...,v1 + ··· Φ(vi−1 − vi)E Zv1 F ΛvM+1 dvM+1 ··· dv1. (3.6) 0 0 0 i=2 n In addition if Relation (3.5) is satisfied (limn→+∞ mΦ(∆ ) = 0), we have that " Z # E F ZtdHt [0,T ] Z T v = µ E [ZvF ] dv 0 +∞ n Z T Z v1 Z vn−1 X Y vn,...,v2 vn,...,v1 + µ ··· Φ(vi−1 − vi)E Zv1 F dvn ··· dv1. (3.7) n=2 0 0 0 i=2 R T v Remark 3.14. Remark that the first term µ 0 E [ZvF ] dv corresponds to the formula for a Poisson process (setting the self-exciting kernel Φ at zero). Therefore the sum in the second term can be interpreted as a correcting term due to the self-exciting property of the counting process H. Besides, this formula can be used to provide lower and upper bounds on the premium of insurance contracts. H N Proof. Set Z(t, θ) = Zt1{θ≤Λt}. As Z is F -predictable, it is F -predictable and thus Relation (2.1) is satisfied. Thus Mecke’s formula for Poisson functionals (see Proposition 2.4) gives that " Z # E F ZtdHt [0,T ] " Z Z #

= E F Zt1{θ≤Λt}N(dt, dθ) [0,T ] R+ "Z Z # + = E Zt(F ◦ ε(t,θ))1{θ≤Λt}dtdθ [0,T ] R+ Z Z + = E Zt (F ◦ ε(t,θ))1{θ≤Λt}dθ dt [0,T ] R+ Z h i = Z (F ◦ ε+ )Λ dt E t (t,Λt) t [0,T ]

= µ m1 + I1, with Z h i m := Z (F ◦ ε+ ) dt, 1 E t (t,Λt) [0,T ] Z " Z # I := Z (F ◦ ε+ ) Φ(v − u)dH dv. 1 E v (v,Λv) u [0,T ] (0,v) F v := F ◦ ε+ Setting (v,Λv), once again by Proposition 2.4 we have that Z " Z # v1 I1 = E Zv1 F Φ(v1 − v2)dHv2 dv1 [0,T ] (0,v1)

11 Z " Z Z # = Z F v1 Φ(v − v ) 1 N(dv , dθ) dv E v1 1 2 {θ≤Λv2 } 2 1 [0,T ] (0,v1) R+ Z Z v1 Z + v1 + = Φ(v1 − v2) Zv ◦ ε F ◦ ε 1 dθ dv2dv1 E 1 (v2,θ) (v2,θ) {θ≤Λv2 } [0,T ] 0 R+ Z Z v1 h i + v1 + = Φ(v1 − v2) Zv ◦ ε F ◦ ε Λv dv2dv1 E 1 (v2,Λv ) (v2,Λv ) 2 [0,T ] 0 2 2

= µ m2 + I2, with Z Z v1 h i + v1 + m2 := Φ(v1 − v2) Zv ◦ ε F ◦ ε dv2dv1, E 1 (v2,Λv ) (v2,Λv ) [0,T ] 0 2 2 and " # Z Z v1 Z + v1 + I2 := Φ(v1 − v2)E Zv1 ◦ ε F ◦ ε Φ(v3 − v2)dHv3 dv2dv1. (v2,Λv2 ) (v2,Λv2 ) [0,T ] 0 (0,v2] For n ≥ 2 we set n Z T Z v1 Z vn−1 Y vn,...,v2 vn,...,v1 mn := ··· Φ(vi−1 − vi)E Zv1 F dvn ··· dv1, 0 0 0 i=2 and n " # Z T Z v1 Z vn−1 Z Y vn,...,v2 vn,...,v1 In := ··· Φ(vi−1−vi)E Zv1 F Φ(vn+1 − vn)dHvn+1 dvn ··· dv1. 0 0 0 i=2 (0,vn] We have that

In n " # Z T Z v1 Z vn−1 Y Z = ··· Φ(v − v ) Zvn,...,v2 F vn,...,v1 Φ(v − v )1 N(dv , dθ) dv ··· dv i−1 i E v1 n+1 n {θ≤Λvn+1 } n+1 n 1 0 0 0 i=2 (0,vn] n+1 Z T Z v1 Z vn Z Y vn,...,v2 vn,...,v1 + = ··· Φ(vi−1 − vi) (Z F ) ◦ ε 1 dθ dvn+1 ··· dv1 E v1 (vn+1,θ) {θ≤Λvn+1 } 0 0 0 i=2 R+ n+1 Z T Z v1 Z vn h i Y vn,...,v2 vn,...,v1 + = ··· Φ(vi−1 − vi)E (Zv F ) ◦ ε Λvn+1 dvn+1 ··· dv1 1 (vn+1,Λvn+1 ) 0 0 0 i=2 n+1 Z T Z v1 Z vn Y vn+1,...,v2 vn+1,...,v1 = ··· Φ(vi−1 − vi)E Zv1 F Λvn+1 dvn+1 ··· dv1 0 0 0 i=2

= µ mn+1 + In+1. Hence, by induction, we obtain the ﬁrst part of Theorem 3.13 (Equation (3.6)). Classical estimates (using the expression of Λ in Deﬁnition 2.5 and [?, Lemma 3]) yield that kΦk1 [Λt] ≤ µ 1 + t Z F E 1−kΦk1 , for any . Therefore, as and are assumed to be bounded, there exists C > 0 (depending on kZk∞, kF k∞, µ and kΦk1) such that M+1 Z T Z v1 Z vM Y vM+1,...,v2 vM+1,...,v1 ··· Φ(vi−1 − vi)E Zv1 F ΛvM+1 dvM+1 ··· dv1 0 0 0 i=2

12 M+1 Z T Z v1 Z vM Y M+1 ≤ C ··· Φ(vi−1 − vi)dvM+1 ··· dv1 = CmΦ(∆ ) 0 0 0 i=2 which converges to zero as M → +∞ due to Relation (3.5).

An alternative form can be given for the representation (3.7). Let n ≥ 1 and ∆n the n- dimensional simplex of [0,T ] :

n ∆ := {0 < vn < ··· < v1 < T } .

Let U n be a ﬂat Dirichlet distribution on ∆n that is a continuous random variable uniformly n n distributed on ∆ , then for any Borelian integrable map ϕ : R → R,

Z T Z vn−1 n n! E [ϕ(U )] = n ··· ϕ(vn, . . . , v1)dvn ··· dv1. T 0 0 n Remark 3.15. Let (U )n≥2 be a sequence of independent random variables such that i i i i U := (U1,..., Ui ) is a Dirichlet distribution on ∆ . Then " Z # Z T v E F ZtdHt = µ E [ZvF ] dv [0,T ] 0 +∞ n " n # T U n,...,U n n n X Y n n n 2 Un ,...,U1 + µ E Φ(Ui−1 − Ui )ZU n F . (3.8) n! 1 n=2 i=2

4 Insurance derivatives for cyber risk

The expansion formula can be applied to compute closed formula for the premium of a class of insurance and ﬁnancial derivatives (such as reinsurance contracts including generalized Stop- Loss contracts, or CDO tranches) or risk management instruments (like Expected Shortfall), to provide generalizations of results that have been proved in [11] in a Cox model setting. In this section, we choose to focus on cyber reinsurance contracts. Indeed, one important feature of cyber risk is the presence of accumulation phenomena and contagion, than can not be accurately modeled by Poisson processes (see the statistical analysis of Bessy et al. [?], based on the public Privacy Rights Clearinghouse database). This means that assuming a Poisson process to model the frequency of cyber risk may induce an underestimation of the risk, due to a misspeciﬁcation of the dependency and self-exciting components of the risk. Developing a valuation formula for cyber contracts, taking into account this self-exciting feature is thus a crucial challenge for cyber insurance.

4.1 The cumulative loss processes and derivatives payoﬀs The so-called cumulative loss process is a key process for risk analysis in insurance and reinsurance. It corresponds to the cumulative sum of claims amounts, the sum being indexed by a counting process modeling the arrival times of the claims. In standard models, the counting process is assumed to be a Poisson process, that is the claims inter-arrivals are assumed iid with exponential distribution. Nevertheless, for cyber risk, a Hawkes process modeling is more appropriate, due to the auto-excitation feature of cyber risk (see [?], [?] or [?]). In the following, we propose closed-form formula for insurance derivatives, such as stop-loss contracts, for

13 PHt which the cumulative loss process is indexed by a Hawkes process Lt := i=1 Xi where (Ht) th H is a Hawkes process and Xi models the i -claim amount, that arrives at the random time τi .

N N N More precisely, the mathematical framework is the following. Let (Ω , F , P ) the proba- C C C bility space as deﬁned previously. We consider (Ω , F , P ) a probability space on which we 2 deﬁne (η1, ϑ1) a R+-valued random variable (we denote by µ its distribution) and (ηi, ϑi)i≥2 ¯ and (¯ηi, ϑi)i≥1 that are independent copies of (η1, ϑ1). We set

N C H C N C Ω := Ω × Ω , F := (Ft)t∈[0,T ], Ft := Ft ⊗ F , P := P ⊗ P .

Note that variables (or processes) defined only on ΩN (respectively ΩC ) naturally extend to Ω. In addition, H and the underlying Poisson process N are independent of the variables ¯ ηi, η¯i, ϑi, ϑi (i ≥ 1). We now define the cumulative loss processes and the insurance derivatives we consider. One typical example of insurance contract is Stop-Loss contract. A Stop-Loss contract provides to its buyer protection against losses which are larger than a given threshold K. If the ith- H H P t −κ(t−τi ) claim size is f(ηi), then the cumulative loss process is given by Lt := i=1 e f(ηi), −κ(t−τ H ) where e i is a discount factor. The process (Lt) is the loss that activates the contract. Sometimes the compensation amount are not exactly the ones that are computed to activate the reinsurance contract and may depend on other losses ϑi. The "generalized" loss process H H P t −κ(t−τi ) is then given by Kt := i=1 e g(ηi, ϑi) . For example, for a stop loss contract, the reinsurance company pays the loss amount above a threshold K, up to a given amount (K¯ −K) fixed by the contract. The payoff of a generalized stop loss contract is then given by  0, if LT ≤ K  ¯ KT − K, if K ≤ LT ≤ K , (4.1)  ¯ ¯ K − K, if LT ≥ K.

Then, the premium3 of such a contract is ¯ ¯ ¯ E KT 1{LT >K} − KP LT ∈ [K, K] + (K − K)P LT ≥ K . (4.2)

Therefore one has to compute quantities of the form E [KT h (LT )]. The deﬁnitions are gathered below.

Deﬁnition 4.1 (Insurance derivative contract). 4 H We denote by (τi )i∈N the jump times of the counting process H.

(i) Given f : R+ → R+ a bounded deterministic function, the loss process is

Ht X −κ(t−τ H ) Lt := e i f(ηi), t ∈ [0,T ]. (4.3) i=1 3Remark that when the risk of the contract is neither hedgeable nor even related to a ﬁnancial market, the premium of the contract relies on the computation of the expectation of the payoﬀ, under the physical probability measure P. 4We put strong a condition of boundedness on the functions f, g and h since this condition is in force for actuarial derivatives, but it can be relaxed to a weaker integrability condition

14 2 (ii) Given g : R+ → R+ a bounded deterministic function, the generalized loss process is

Ht X −κ(t−τ H ) Kt := e i g(ηi, ϑi), t ∈ [0,T ]. (4.4) i=1

(iii) Let h : R+ → R+ be a bounded deterministic function. We aim at computing the expectation of the derivatives payoﬀ, that is the quantity

E [KT h (LT )] . (4.5)

4.2 General pricing formula

For the computation of E [KT h (LT )], we will rely on the formula in Theorem 3.13, by writing the cumulative generalized loss process Kt as an integral with respect to the Hawkes process. Namely, Z KT = ZsdHs, t ∈ [0,T ] (0,T ] with +∞ X −κ(T −s) Zs := g(ηi, ϑi)e 1 H H (s), s ∈ [0,T ]. (τi−1,τi ] i=1 We ﬁrst introduce the following deﬁnition.

∗ Deﬁnition 4.2. Let n ∈ N and 0 < vn < ··· < v1 < T . vn,...,v1 th vn,...,v1 We set τi the i -jump time of the shifted Hawkes process HT and

vn,...,v1 HT v ,...,v X vn,...,v1 n 1 −κ(T −τi ) LT := e f(ηi). i=1

Theorem 4.3. Under the previous assumptions and Relation (3.5), it holds that

E[KT h(LT )]      (Hv1 )−1 Z T T    X v1   −κ(T −v1)  ¯   −κ(T −v1) −κ(T −τi )   = µ e E g(¯η1, ϑ1)E h e f(¯η1) + e f(ηi) η¯1 dv1 0    i=1   v τ 1 6=v i 1 +∞ n Z T Z v1 Z vn−1 X −κ(T −v1) Y + µ ··· e Φ(vi−1 − vi) (4.6) n=2 0 0 0 i=2      vn,...,v1 n (HT )−n   X X vn,...,v1    ¯   −κ(T −vk) −κ(T −τi )   E g(¯η1, ϑ1)E h  e f(¯ηk) + e f(ηi) η¯1 dvn ··· dv1.   k=1 i=1   v ,...,v τ n 1 6=v ,...,v i n 1

where (¯ηi, ϑ¯i)i≥1 are independent copies of (η1, ϑ1), independent of all other variables.

15 R P+∞ −κ(T −s) Proof. For all t ∈ [0,T ],Kt = ZsdHs with Zs := e g(ηi, ϑi)1 H H (s). Note (0,t] i=1 (τi−1,τi ] −κ(T −s) PHT −κ(T −τi) that Zs = e g(η1+Hs− , ϑ1+Hs− ) for any s in (0,T ]. Recall that LT = i=1 e f(ηi). By Theorem 3.13, we have that

E[KT h(LT )] " Z # = E h(LT ) ZtdHt (0,T ] " Z # −κ(T −t) = E h(LT ) e g(η1+Ht− , ϑ1+Ht− )dHt (0,T ] " " Z ## C = E E h(LT ) ZtdHt|F (0,T ] +∞ Z T h i X = µ e−κ(T −v1)g(η , ϑ )h(Lv1 ) dv + µ I E 1+Hv1− 1+Hv1− T 1 n 0 n=2 with for n ≥ 2

n Z T Z v1 Z vn−1 Y h vn,...,v1 i −κ(T −v1) v ,...,v v ,...,v In := ··· Φ(vi−1−vi)E e g(η1+H n 2 , ϑ1+H n 2 )h(LT ) dvn ··· dv1 v1− v1− 0 0 0 i=2 and for n = 1 Z T h i I := e−κ(T −v1)g(η , ϑ )h(Lv1 ) dv . 1 E 1+Hv1− 1+Hv1− T 1 0 vn,...,v2 vn,...,v1 By deﬁnition, 1 + Hv − = Hv1 , P-a.s.. Using the fact that the (ηi) are iid, we can 1 ¯ separate the claim at time v1 (that we will represent using the random variables (η¯1, ϑ1)) from the other claims. Thus

In n Z T Z v1 Z vn−1 Y = ··· Φ(vi−1 − vi) 0 0 0 i=2    vn,...,v1 HT −1   X vn,...,v1   −κ(T −v1) ¯  −κ(T −v1) −κ(T −τi )  E e g(¯η1, ϑ1)h e f(¯η1) + e f(ηi) dvn ··· dv1   i=1  v ,...,v τ n 1 6=v i 1 n Z T Z v1 Z vn−1 Y = ··· Φ(vi−1 − vi) 0 0 0 i=2      vn,...,v1 HT −1    X vn,...,v1    −κ(T −v1) ¯   −κ(T −v1) −κ(T −τi )  ¯  E e g(¯η1, ϑ1)E h e f(¯η1) + e f(ηi) (¯η1, ϑ1) dvn ··· dv1    i=1   v ,...,v τ n 1 6=v i 1 n Z T Z v1 Z vn−1 Y = ··· Φ(vi−1 − vi) 0 0 0 i=2

16      vn,...,v1 n (HT )−n   X X vn,...,v1    −κ(T −v1) ¯   −κ(T −vk) −κ(T −τi )   E e g(¯η1, ϑ1)E h  e f(¯ηk) + e f(ηi) η¯1 dvn ··· dv1.   k=1 i=1   v ,...,v τ n 1 6=v ,...,v i n 1

The treatment for I1 is similar, which concludes the proof.

The expansion formula (4.6) is written in terms of the shifted Hawkes process Hvn,...,v1 . The shifted Hawkes process has three "types" of jumps

• the spontaneous jumps induced by an homogeneous Poisson with intensity µ

• the deterministic enforced jumps at time vn < ··· < v1

• the auto-excited jumps that are induced by previous jumps of the process

Controlling this diﬀerent types of jumps allows us to perform bounds on the premium.

4.3 Lower and upper bounds We now use Relation (4.6) to perform lower and upper bounds on the premium. H H P T −κ(T −τi ) Recall that the premium takes the form E[KT h(LT )] with KT = i=1 e Xi. In the more simple case without discounting (κ = 0), computing lower or upper bounds on the premium relies on two types of estimates : Pn 1) estimates on the CDF of a given sum of claims i=1 Xi (where n is prescribed) 2) estimates on the CDF of the counting process H. Usually, for insurance derivatives a speciﬁc model of claims is considered so that the former Pn quantity i=1 Xi is accessible. The main issue of course lies in the estimates for the value of HT . For instance, one can make the following type of estimates on the Hawkes process :

(i) Obtain an upper bounds on P[HT ≥ C] for some constant C (by Markov’s inequality for instance), but this is only an upper bound.

(ii) Get a lower bound on the premium by noting that H ≥ N˜ where N˜ is an homogeneous Poisson process with intensity µ (as Λt ≥ µ for any t ≥ 0).

In both approaches one makes rough estimates on the Hawkes process. Our approach allows for an intermediary situation as the processes Hvn,...,v1 have deterministic jumps at times vn, . . . , v1 which are weighted by the kernels Φ(vi−1 −vi) over the simplex. So we can make less stringent estimates by at least knowing n jumps of the (shifted) Hawkes process. Obviously, on each intervals [vi−1, vi] we have to consider a Hawkes process for which estimates of the form (i)-(ii) above are the best tools available.

4.3.1 Lower bound

Using Theorem 4.3, one can already obtain a ﬁrst lower bound by just considering the n enforced jumps of the shifted Hawkes process and ignoring the other jumps (which is clearly a very rough estimate).

17 Proposition 4.4. Assume h is non-decreasing. We have that Z T h i −κ(T −v1) ¯ −κ(T −v1) E[KT h(LT )] ≥ µ e E g(¯η1, ϑ1)h e f(¯η1) dv1 0 +∞ X Z T Z v1 Z vn−1 + µ ··· e−κ(T −v1) n=2 0 0 0 n " " n ! ## Y X ¯ −κ(T −vk) Φ(vi−1 − vi)E g(¯η1, ϑ1)E h e f(¯ηk) η¯1 dvn ··· dv1. i=2 k=1 If κ = 0, the lower bound simpliﬁes as " " ! ## +∞ n X ¯ X n E[KT h(LT )] ≥ µ E g(¯η1, ϑ1)E h f(¯ηk) η¯1 mΦ(∆ ). (4.7) n=1 k=1 Proof. We apply Relation (4.6) and note that for any n ≥ 1, we have simply used the fact vn,...,v1 that HT ≥ n. The formula above takes only into account the deterministic jumps added to the process. The proposition below is more accurate by including the jumps of a homogeneous Poisson process with constant intensity µ. p Proposition 4.5. Assume h is non-decreasing. We consider a family (U )p≥1 of independent random variables (which are constructed from N only), where for each p, U p is a ﬂat Dirichlet p distributions on ∆ . Set for n ≥ 1, 0 < vn < ··· < v1 < T :

αn(vn, . . . , v1) +∞ p " " n p ! ## X −(T µ) (T µ) X −κ(T −v ) X −κ(T −U p) := e g(¯η , ϑ¯ ) h e k f(¯η ) + e i f(η ) η¯ . p! E 1 1 E k i 1 p=0 k=1 i=1 It holds that Z T −κ(T −v1) E[KT h(LT )] ≥µ e α1(v1)dv1 0 +∞ n Z T Z vn−1 X −κ(T −v1) Y + µ ··· e Φ(vi−1 − vi)αn(vn, . . . , v1)dvn ··· dv1. (4.8) n=2 0 0 i=2

vn,...,v1 Proof. Let n ≥ 1 and 0 < vn < ··· < v1 < T . By Proposition 3.8, HT ≥ n + HT , P − ˜ ˜ a.s.. In addition by Lemma 5.1 (see Section 5.3), we have that HT ≥ NT , with Nt := N([0, t]× [0, µ]) which is an homogeneous Poisson process with intensity µ and by construction, any jump N˜ ˜ vn,...,v1 τi of N diﬀerent of vn, . . . , v1 is a jump of HT (diﬀerent of vn, . . . , v1). Hence, using Relation (4.6), we have that

E[KT h(LT )]      ˜ Z T NT    X N˜   −κ(T −v1)  ¯   −κ(T −v1) −κ(T −τi )   ≥ µ e E g(¯η1, ϑ1)E h e f(¯η1) + e f(ηi) η¯1 dv1 0    i=1   ˜ τN 6=v i 1

18 +∞ n Z T Z vn−1 X −κ(T −v1) Y + µ ··· e Φ(vi−1 − vi) n=2 0 0 i=2     

n N˜T   X X N˜    ¯   −κ(T −vk) −κ(T −τi )   E g(¯η1, ϑ1)E h  e f(¯ηk) + e f(ηi) η¯1 dvn ··· dv1.   k=1 i=1   ˜ τN 6=v ,...,v i n 1

For n ≥ 1 and 0 < vn < ··· < v1 < T , it holds that :    

n N˜T  X X N˜     −κ(T −vk) −κ(T −τi )   E h  e f(¯ηk) + e f(ηi) η¯1  k=1 i=1   ˜ τN 6=v ,...,v i n 1     +∞ p n p X −(T µ) (T µ)  X −κ(T −v ) X −κ(T −τ N˜ )   = e h  e k f(¯η ) + e i f(η ) η¯  p! E   k i  1 p=0  k=1 i=1   ˜ τN 6=v ,...,v i n 1 +∞ p " n p ! # X −(T µ) (T µ) X −κ(T −v ) X −κ(T −U p) = e h e k f(¯η ) + e i f(η ) η¯ , p! E k i 1 p=1 k=1 i=1 which concludes the proof.

Remark 4.6. A direct approach for exhibiting p jumps of the Hawkes process would require conditioning on HT ≥ p and then to obtain a lower bound for P[HT ≥ p] by for example ˜ ˜ P[NT ≥ p] (where N is the homogeneous Poisson process obtained from N with intensity µ) which is what we do. But the advantage of our approach lies in the fact that on top of these jumps of N˜ we can go further and consider n jumps of the Hawkes process which somehow are really produced by the self-excitation phenomenon as these jumps are weighted by the kernel Φ along the jump times vn, . . . , v1.

As a corollary to Proposition 4.5, we can deduce the following lower bound in case the discounting factor κ is equal to 0. Corollary 4.7. Assume h is non-decreasing and κ = 0. It holds that

+∞ +∞ " " n p ! ## X X (T µ)p X X [K h(L )] ≥ µ m (∆n) e−(T µ) g(¯η , ϑ¯ ) h f(¯η ) + f(η ) η¯ , E T T Φ p! E 1 1 E k i 1 n=1 p=0 k=1 i=1 (4.9) n where we recall that mΦ(∆ ) is deﬁned in Notation 3.12.

As pointed out in Remark 3.14, the ﬁrst term in the sum indexed by n (that is for n = 1) corresponds to the formula for a Poisson process with intensity µ. Therefore the sum of the remaining terms (for n ≥ 2) corresponds to a lower bound for the correcting term due to the self-exciting property of the counting process H. This quantity should be at least added to a computation of the premium based on a standard Poisson process model.

19 Application for Stop Loss contract with deductible. We consider the following setting of Stop Loss contract (h(x) = 1x≥K ) with no discounting (κ = 0), and with a deductible on the reporting of the claims such that only claims whose amount exceeds a threshold f are reported. A lower bound for the surplus of premium due to the self-exciting property of the counting process H is

+∞ +∞ X X (T µ)p µ g(¯η , ϑ¯ ) m (∆n) e−(T µ) E 1 1 Φ p! n=2 p+n≥bK/fc+1 where µ is the intensity of the "spontaneous" jumps (induced an homogeneous Poisson with ¯ intensity µ) and E g(¯η1, ϑ1) is the mean cost of one claim. If we assume furthermore a n n−1 T n decreasing excitation kernel Φ, then mΦ(∆ ) ≥ Φ(T ) n! and the lower bound becomes

+∞ +∞ X T n X (T µ)p µ g(¯η , ϑ¯ ) Φ(T )n−1 e−(T µ) . E 1 1 n! p! n=2 p+n≥bK/fc+1

4.3.2 Upper bound

We now turn to the upper bound. We introduce the following quantities, as in [7].

R T 2 Proposition 4.8. Assume h is non-decreasing and Φ is non-increasing. Let C2 := 0 Ψ1(t)dt with Ψ1 solution to Z t Ψ1(t) = 1 + Φ(t − s)Ψ1(s)ds, t ∈ [0,T ] 0 R T and C1 := 0 Ψ2(t)dt, with Ψ2 solution to Z t 2 Ψ2(t) = (Ψ1(t)) + Φ(s)Ψ2(t − s)ds, t ∈ [0,T ]. 0 For n ≥ 1, let also 2 cn := (µ + nΦ(0))C1 + (µ + nΦ(0)) C2.

Finally set for n ≥ 1 and 0 < vn < ··· < v1 < T

βn(vn, . . . , v1) " " n ! ## X −T (µ+nΦ(0)) ¯ −κ(T −vk) := e E g(¯η1, ϑ1)E h e f(¯ηk) η¯1 k=1 +∞ " " n p ! ## X cn X X + ∧ 1 g(¯η , ϑ¯ ) h e−κ(T −vk)f(¯η ) + f(η ) η¯ p2 E 1 1 E k i 1 p=1 k=1 i=1 We have that Z T −κ(T −v1) E[KT h(LT )] ≤µ e β1(v1)dv1 0 +∞ n Z T Z vn−1 X Y −κ(T −v1) + µ ··· Φ(vi−1 − vi)e βn(vn, . . . , v1)dvn ··· dv1. (4.10) n=2 0 0 i=2

20 vn,...,v1 µ+nΦ(0) µ+nΦ(0) Proof. Fix n ≥ 1. By construction (see Lemma 5.2), HT ≤ HT +n, where H denotes the Hawkes process (constructed from N) with initial intensity µ replaced by µ+nΦ(0) (in other words, Hµ+nΦ(0) is solution to Equation (2.3) with initial intensity µ+nΦ(0)). Denote n µ+nΦ(0) H by (τi )i the jump times of HT ; by the thinning procedure, the jump times (τi )i of H n µ+nΦ(0) are necessarily included in the jump times (τi )i of H . By Lemma 5.2, we can make the estimate below. We have that

E[KT h(LT )]      Hµ+nΦ(0) Z T T X n  −κ(T −v1) ¯   −κ(T −v1) −κ(T −τi )   ≤ µ E e g(¯η1, ϑ1)E h e f(¯η1) + e f(ηi) η¯1 dv1 0 i=1

+∞ n X Z T Z vn−1 Y + µ ··· Φ(vi−1 − vi) n=2 0 0 i=2    µ+nΦ(0)   n HT X X n  −κ(T −v1) ¯   −κ(T −vk) −κ(T −τi )   E e g(¯η1, ϑ1)E h  e f(¯ηk) + e f(ηi) η¯1 dvn ··· dv1 k=1 i=1      Hµ+nΦ(0) Z T T  −κ(T −v1) ¯   −κ(T −v1) X   ≤ µ E e g(¯η1, ϑ1)E h e f(¯η1) + f(ηi) η¯1 dv1 0 i=1

+∞ n X Z T Z vn−1 Y + µ ··· Φ(vi−1 − vi) n=2 0 0 i=2    µ+nΦ(0)   n H X TX  −κ(T −v1) ¯   −κ(T −vk)   E e g(¯η1, ϑ1)E h  e f(¯ηk) + f(ηi) η¯1 dvn ··· dv1 k=1 i=1

−κ(T −τ n) n where we used the upper bound e i ≤ 1 to get rid oﬀ the unknown jump times τi . Let n ≥ 1, 0 < vn < ··· < v1 < T . We have that

  µ+nΦ(0)   n H X TX   −κ(T −vk)   E h  e f(¯ηk) + f(ηi) η¯1 k=1 i=1

+∞ " n p ! # X X X µ+nΦ(0) −κ(T −vk) = E h e f(¯ηk) + f(ηi) η¯1 P[HT = p] p=0 k=1 i=1 " n ! # X µ+nΦ(0) −κ(T −vk) ≤ E h e f(¯ηk) η¯1 P[HT = 0] k=1 +∞ " n p ! # X X X µ+nΦ(0) −κ(T −vk) + E h e f(¯ηk) + f(ηi) η¯1 P[HT ≥ p] p=1 k=1 i=1 " n ! # X µ+nΦ(0) −κ(T −vk) ≤ E h e f(¯ηk) η¯1 P[HT = 0] k=1

21 +∞ " n p ! # X 1 X X µ+nΦ(0)2 + h e−κ(T −vk)f(¯η ) + f(η ) η¯ H . p2 E k i 1 E T p=1 k=1 i=1

2 µ+nΦ(0) 2 Then by [7, Proposition 5], E HT = (µ + nΦ(0))C1 + (µ + nΦ(0)) C2 and by µ+nΦ(0) −(µ+nΦ(0))T [7, Proposition 7], P[HT = 0] = e . The result follows by injecting these estimates in the previous one.

Once again, we consider as a corollary the case where κ = 0.

Corollary 4.9. Assume h is non-decreasing, κ = 0 and Φ non-increasing. We consider C1, C2 and cn as deﬁned in Proposition 4.8. For n ≥ 1 set " " ! ## n −T (µ+nΦ(0)) ¯ X βn :=e E g(¯η1, ϑ1)E h f(¯ηk) η¯1 k=1 +∞ " " n p ! ## X cn X X + g(¯η , ϑ¯ ) h f(¯η ) + f(η ) η¯ . p2 E 1 1 E k i 1 p=1 k=1 i=1 We have that +∞ X n E[KT h(LT )] ≤ µ mΦ(∆ )βn, (4.11) n=1 n where we recall mΦ(∆ ) is deﬁned in Notation 3.12. Remark 4.10. Note that in case of a Poisson process, the sum in the right-hand-side of (4.11) resumes to the term n = 1 (since Φ ≡ 0) which is exactly equal to E[KT h(LT )]. Remark 4.11. One can relax the monotony assumption on Φ above by allowing a general bounded Φ map. In that case, the shifts by nΦ(0) are replaced with nΦ∗ with Φ∗ := sup Φ(x) x∈R+ .

5 Technical material

5.1 Proof of Theorem 2.7 We prove the existence and uniqueness of (2.2)  ˆ v v R R Ht = h + 1 ˆ v N(ds, dθ), t ∈ [v, T ]  (v,t] R+ {θ≤Λs }

 ˆ v v R ˆ v  Λt = µ (t) + (v,t) Φ(t − u)dHu v for v in [0,T ], where (µ (t))t∈[v,T ] a non-negative stochastic process such that for any t ≥ v, v N v N µ (t) is a Fv -measurable random variable and h a Fv -measurable random with values in N. The proof is composed of two parts : existence and uniqueness.

Existence We start with the existence part. We set

(1) v Λt := µ (t), t ∈ [v, T ].

22 N We make use the following notation : if τi denotes the ith jump time of N, then there exists N a unique element θi ≥ 0 such that N({(τi } × {θi}) = 1 (in other words we denote by θi the mark associated to the ith-jump time of N). Let

Hˆ n N (1) o τ1 := inf τi ≥ v, θ1 ≤ Λ N , i ≥ 1 ∧ T. τi

Hˆ N In addition, τ1 is a F -stopping time, indeed for any t ∈ [0,T ],

Hˆ [ h N n (1) oi N {τ1 ≤ t} = {v ≤ τi ≤ t} ∩ θi ≤ Λ N ∈ Ft . τi i≥1

For i ≥ 1, we set

(i+1) (i) Hˆ Λt := Λt 1 Hˆ + Φ(t − τi )1 Hˆ , t ∈ [0,T ], v≤t≤τi τi

Hˆ n N N Hˆ (i+1) o τi+1 := inf τk , τk > τi , θk ≤ Λ N , k ≥ 1 ∧ T. (5.1) τk Hˆ N (i) N By induction, one proves that for any i ≥ 1, τi is a F -stopping time and Λ is a F - N predictable stochastic process as a càglàd, F -adapted process. In addition, by construction, (i+1) (i) Hˆ Hˆ Hˆ N Λ and Λ coincide on [0, τi ]. Furthermore, limi→+∞ τi = T, P − p.s. as τi ≥ τi , P−a.s. for any i ≥ 1. We set : (i) Λt := lim Λt , t ∈ [0,T ], (5.2) i→+∞ which is a F-predictable process. We then set : +∞ ˆ v X Ht := h + 1 Hˆ , t ∈ [v, T ]. {t≥τi } i=1

Uniqueness We now turn to the uniqueness of the solution. To do so we give some immediate properties of any solution to (2.3). Consider, Hˆ be a solution to (2.3). Then, by deﬁnition, we have that

Hˆ N {τk , k ∈ N} ⊂ {τk , k ∈ N}.

In addition, by deﬁnition, the jump times of Hˆ are given by

Hˆ n N N Hˆ o τ := inf τ , τ > τ , θk ≤ Λ N , k ≥ 1 ∧ T, i k k i−1 τk where Z v Λt = µ (t) + Φ(t − s)dHˆs, t ∈ [v, T ]. (5.3) (v,t) ˆ 1 ˆ 2 Hˆ 1 Hˆ 2 Consider, H , H , two solution processes to (2.3). Denote by τk (respectively τk ) the jump ˆ 1 ˆ 2 times of Hˆ 1, ΛH the intensity function of Hˆ 1 (respectively ΛH the one of Hˆ 2), with Z Hˆ i v ˆ i Λt = µ (t) + Φ(t − s)dHs, t ∈ [v, T ], i ∈ {1, 2}. (v,t)

23 Hˆ 1 Hˆ 2 Hˆ 1 Hˆ 2 Hˆ 1 Hˆ 2 By the previous remark, Λ = Λ on [0, τ1 ∧ τ1 ) and so τ1 = τ1 , P-a.s.. Let Hˆ 1 Hˆ 2 Hˆ 1 Hˆ 2 τ1 := τ1 (= τ1 ). We have thus that Λ (τ1) = Λ (τ1). Let i ≥ 1. Assume that

Hˆ 1 Hˆ 2 ˆ 1 ˆ 2 Hˆ 1 Hˆ 2 τi = τi (=: τi), H = H on [0, τi], and Λ = Λ on [0, τi], P − a.s..

Then, as the intensity functions agree up to τi and as the jump times of any solution is Hˆ 1 Hˆ 2 characterized by (5.2), we deduce that τi+1 = τi+1(=: τi+1), P-a.s.. Hence,

ˆ 1 ˆ 2 Hˆ 1 Hˆ 2 H = H on [0, τi + 1], and Λ = Λ on [0, τi+1], P − a.s.. ˆ 1 ˆ 2 Thus, Ht = Ht for any t in [0,T ], P-a.s. (as limi→+∞ τi = T , P − a.s.).

5.2 Proof of Proposition 3.8 ∗ Let n ∈ N , and 0 < vn < vn−1 < ··· < v1 < T . We prove that

(H, Λ) ◦ ε+ ◦ · · · ◦ ε+ = (Hvn,...,v1 , Λvn,...,v1 ). (v1,Λv1 ) (vn,Λvn )

Proof. We set (Hv0 , Λv0 ) := (H, Λ). By Lemma 3.5, (Hv1 , Λv1 ) = (H0 ◦ε+ , Λ0 ◦ε+ ). (v1,Λv1 ) (v1,Λv1 ) Let n ≥ 2 and assume that

(Hvn−2,...,v1 ◦ ε+ , Λvn−2,...,v1 ◦ ε+ ) = (Hvn−1,...,v1 , Λvn−1,...,v1 ). (vn−1,Λvn−1 ) (vn−1,Λvn−1 )

We prove that

(Hvn−1,...,v1 ◦ ε+ , Λvn−1,...,v1 ◦ ε+ ) = (Hvn,...,v1 , Λvn,...,v1 ). (vn,Λvn ) (vn,Λvn )

For simplicity of notations, we set : (Hn−1, Λn−1) := (Hvn−1,...,v1 , Λvn−1,...,v1 ) and (Hn−1,+, Λn−1,+) := (Hvn−1,...,v1 ◦ ε+ , Λvn−1,...,v1 ◦ ε+ ). Since (vn,Λvn ) (vn,Λvn )

 Hn−1 = 1 (t)H + 1 (t)H  t [0,vn) t [vn,vn−1) t  n−1 Z Z !  X n−1  + 1[v ,v )(t) Hv − + 1 + 1 n−1 N(ds, dθ) ,  i i−1 i {θ≤Λs }  i=1 (vi,t] R+

 Λn−1 = 1 (t)Λ + 1 (t)Λ  t (0,vn] t (vn,vn−1] t  n−1 !  X Z Z  + 1 (t) µ + Φ(t − u)dHn−1 + Φ(t − u)dHn−1 ,  (vi,vi−1] u u i=1 (0,vi] (vi,t)

Z Z ! n−1,+ H = 1 (t)Ht + 1 (t) Hv − + 1 + 1 n−1,+ N(ds, dθ) t [0,vn) [vn,vn−1) n {θ≤Λs } (vn,t] R+ 1 Z Z ! X n−1,+ + 1 (t) H + 1 + 1 n−1,+ N(ds, dθ) [vi,vi−1) vi− {θ≤Λs } i=n−1 (vi,t] R+ 1 Z Z ! X n−1,+ = 1 (t)Ht + 1 (t) H + 1 + 1 n−1,+ N(ds, dθ) . [0,vn) [vi,vi−1) vi− {θ≤Λs } i=n (vi,t] R+

24 In addition,

n−1,+ Λt = 1(0,vn](t)Λt Z Z ! + 1 (t) µ + Φ(t − u)dH + Φ(t − v ) + Φ(t − u)dH ◦ ε+ (vn,vn−1] u n u ((vn,Λ(vn)) (0,vn) (vn,t) n−1 Z Z ! X n−1,+ n−1,+ + 1(vi,vi−1](t) µ + Φ(t − u)dHu + Φ(t − u)dHu i=1 (0,vi] (vi,t) Z Z ! n−1,+ n−1,+ = 1(0,vn](t)Λt + 1(vn,vn−1](t) µ + Φ(t − u)dHu + Φ(t − u)dHu (0,vn] (vn,t) n−1 Z Z ! X n−1,+ n−1,+ + 1(vi,vi−1](t) µ + Φ(t − u)dHu + Φ(t − u)dHu i=1 (0,vi] (vi,t) n Z Z ! X n−1,+ n−1,+ = 1(0,vn](t)Λt + 1(vi,vi−1](t) µ + Φ(t − u)dHu + Φ(t − u)dHu . i=1 (0,vi] (vi,t)

Hence, (Hn−1,+, Λn−1,+) solves the same equation than (Hvn,...,v1 , Λvn,...,v1 ), which concludes the proof.

5.3 Two comparison lemma We provide two comparison lemma based on the thinning algorithm. Lemma 5.1. Let µ > 0 and (H,ˆ Λ)ˆ the unique solution to  R R Hˆt = 1 ˆ N(ds, dθ), t ∈ [0,T ]  (0,t] R+ {θ≤Λs}  ˆ R ˆ  Λt = µ + (0,t) Φ(t − u)dHu.

Consider the homogeneous Poisson process H˜ deﬁned as Z Z ˜ Ht = 1{θ≤µ}N(ds, dθ), t ∈ [0,T ]. (0,t] R+ It holds that ˆ ˜ Ht ≥ Ht, ∀t ∈ [0,T ], P − a.s..

Proof. Let t in [0,T ]. As Λˆ t ≥ µ we have that Z Z Hˆ − H˜ = 1 − 1 N(ds, dθ) t t {θ≤Λˆ s} {θ≤µ} (0,t] R+ Z Z = 1 N(ds, dθ) {µ<θ≤Λˆ s} (0,t] R+ ≥ 0.

25 Lemma 5.2. Assume Φ is non-increasing. Let n ≥ 1, 0 < vn < ··· < v1 < T . Let (Hµ+nΦ(0), Λµ+nΦ(0)) the solution to SDE (2.3) with initial intensity µ + nΦ(0) instead of µ. Consider as well (Hvn,...,v1 , Λvn,...,v1 ) the process deﬁned in Deﬁnition 3.6. It holds that :

vn,...,v1 µ+nΦ(0) Ht ≤ n + Ht , ∀t ∈ [0,T ], P − a.s..

µ+nΦ(0) µ+nΦ(0) In addition, if τi denotes a jump time of H , then

µ+nΦ(0) P[∃k ∈ {1, . . . , n}, τi = vk] = 0

Hvn,...,v1 Hµ+nΦ(0) and {τi , i ≥ 1}\{v1, ··· , vn} ⊂ {τi , i ≥ 1} almost surely.

Proof. On [0, vn), we have that

vn,...,v1 µ+nΦ(0) Ht = Ht ≤ Ht as vn,...,v1 µ+nΦ(0) Λt = Λt ≤ Λt , t ∈ [0, vn].

vn,...,v1 µ+nΦ(0) At t = vn, we have that Ht = Ht + 1 ≤ Ht + n. In addition,

vn,...,v1 µ+nΦ(0) Λvn+ = Λvn + Φ(0) ≤ Λvn + nΦ(0)≤Λvn+ .

vn,...,v It is important to note that the jump at time vn for H 1 will impact the self-exciting part of the intensity while for Hµ+nΦ(0) we have only shifted the baseline intensity. Hence, as Φ is decreasing, the previous inequality propagates beyond vn. The previous inequalities transfer vn,...,v1 µ+nΦ(0) to any interval [vi, vi−1) until time T . As Λt ≤ Λt , using the thinning algorithm, vn,...,v1 µ+nΦ(0) any non-deterministic jump of Ht is a jump H (the deterministic jumps being vn, . . . , v1).

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