LIMIT THEOREMS FOR MARKED HAWKES PROCESSES WITH ...
LIMIT THEOREMS FOR MARKED HAWKES PROCESSES WITH APPLICATION TO A RISK MODEL DMYTRO KARABASH AND LINGJIONG ZHU
Abstract. This paper focuses on limit theorems for linear Hawkes processes with random marks. We prove a large deviation principle, which answers the question raised by Bordenave and Torrisi. A central limit theorem is also obtained. We conclude with an example of application in finance.
Contents 1. Introduction and Main Results 1.1. Introduction 1.2. Limit Theorems for Unmarked Hawkes Processes 1.3. Main Results 2. Proof of Central Limit Theorem 3. Proof of Large Deviation Principle 3.1. Limit of a Logarithmic Moment Generating Function 3.2. Large Deviation Principle 4. Risk Model with Marked Hawkes Claims Arrivals 5. Examples with Explicit Formulas Acknowledgements References
1 1 2 3 4 5 5 9 10 13 14 14
1. Introduction and Main Results 1.1. Introduction. We consider in this article a linear Hawkes process with random marks. Let Nt be a simple point process. Nt denotes the number of points in the interval [0, t). Let Ft be the natural filtration up to time t. We assume that N (−∞, 0] = 0. At time t, the point process has Ft -predictable intensity X λt := ν + Zt , Zt := h(t − τi , ai ), (1.1) τi 0, the (τi )i≥1 are arrival times of the points, and the (ai )i≥1 are i.i.d. random marks, ai being independent of previous arrival times τj , j ≤ i. Let us assume that ai has a common distribution q(da) on a metric space X. Here, h(·, ·) : Date: 16 November 2012. Revised: 16 November 2012. 2000 Mathematics Subject Classification. 60G55, 60F10, 60F05. Key words and phrases. Central limit theorems, large deviations, rare events, point processes, marked point processes, Hawkes processes, self-exciting processes. This research was supported partially by a grant from the National Science Foundation: DMS0904701, DARPA grant and MacCracken Fellowship at NYU. 1
2
DMYTRO KARABASH AND LINGJIONG ZHU
R∞R R∞ R+ ×X → R+ is integrable, i.e. 0 X h(t, a)q(da)dt < ∞. Let H(a) := 0 h(t, a)dt for any a ∈ X. We also assume that Z H(a)q(da) < 1. (1.2) X
Let Pq denote the probability measure for the ai ’s with the common law q(da). Under assumption (1.2), it is well known that there exists a unique stationary version of the linear marked Hawkes process satisfying the dynamics (1.1) and that by ergodic theorem, a law of large numbers holds, lim
t→∞
Nt ν = . t 1 − Eq [H(a)]
(1.3)
This paper is organized as the following. In Section 1.2, we will review some results about the limit theorems for unmarked Hawkes processes. In Section 1.3, we will introduce the main results of this paper, i.e. the central limit theorem and the large deviation principle for linear marked Hawkes processes. The proof of the central limit theorem will be given in Section 2 and the proof of the large deviation principle will be given in Section 3. Finally, we will discuss an application of our results to a risk model in finance in Section 4. 1.2. Limit Theorems for Unmarked Hawkes Processes. Most of the literature about Hawkes processes considered the unmarked case, i.e. with intensity ! X λt := λ h(t − τ ) , (1.4) τ 0 and khkL1 < 1, it has a very nice immigration-birth representation, see for example Hawkes and Oakes [8]. For linear Hawkes process, limit theorems are very well understood. There is the law of large numbers (see for instance Daley and Vere-Jones [4]), i.e. ν Nt → , t 1 − khkL1
as t → ∞ a.s.
(1.5)
Moreover, Bordenave and Torrisi [2] proved a large deviation principle for ( Ntt ∈ ·) with the rate function ( x x log ν+xkhk − x + xkhkL1 + ν if x ∈ [0, ∞) L1 I(x) = . (1.6) +∞ otherwise Once you have the large deviation principle, you can also study some risk processes in finance. (See Stabile and Torrisi [10].)
LIMIT THEOREMS FOR MARKED HAWKES PROCESSES
3
Recently, Bacry et al. [1] proved a functional central limit theorem for the linear multivariate Hawkes process under certain assumptions which includes the linear Hawkes process as a special case and they proved that N·t − ·µt √ → σB(·), t
as t → ∞,
weakly on D[0, 1] equipped with Skorokhod topology, where ν ν µ= . and σ 2 = 1 − khkL1 (1 − khkL1 )3
(1.7)
(1.8)
Moderate deviation principle for linear Hawkes processes is obtained in Zhu [16], which fills in the gap between central limit theorem and large deviation principle. For nonlinear Hawkes processes, a central limit theorem is obtained in Zhu [15]. In Bordenave and Torrisi [2], they raised two questions about large deviations for Hawkes processes. One question is about large deviations for nonlinear Hawkes process and the other is about large deviations for linear marked Hawkes processes. Recently, Zhu [13] considered a special case for nonlinear Hawkes processes when h(·) is exponential or sums of exponentials and proved the large deviations. In another paper, Zhu [14] proved a process-level, i.e. level-3 large deviation principle for nonlinear Hawkes processes for general h(·) and hence by contraction principle, the level-1 large deviation principle for (Nt /t ∈ ·). In this paper, we will prove the large deviations for linear marked Hawkes processes and thus both questions raised in Bordenave and Torrisi [2] have been answered. 1.3. Main Results. Before we proceed, recall that a sequence (Pn )n∈N of probability measures on a topological space X satisfies the large deviation principle (LDP) with rate function I : X → R if I is non-negative, lower semicontinuous and for any measurable set A, we have 1 1 (1.9) − inf o I(x) ≤ lim inf log Pn (A) ≤ lim sup log Pn (A) ≤ − inf I(x). n→∞ n x∈A n→∞ n x∈A Here, Ao is the interior of A and A is its closure. See Dembo and Zeitouni [5] or Varadhan [12] for general background regarding large deviations and their applications. Also Varadhan [11] has an excellent survey article on this subject. For a linear marked Hawkes process satisfying the dynamics (1.1), we prove the following large deviation principle in this article. Theorem 1 (Large Deviation Principle). Assume the conditions (1.2) and Z H(a)x lim e q(da) − x = ∞. (1.10) x→∞
X
Then, (Nt /t ∈ ·) satisfies a large deviation principle with rate function, n h io ( q x inf qˆ xEqˆ[H(a)] + ν − x + x log xEqˆ[H(a)]+ν + xEqˆ log dˆ dq Λ(x) := +∞ ( θ∗ x − ν(x∗ − 1) x ≥ 0 = , +∞ x 0, we have Z t FN (t) := E[exp(θNt )] = exp ν (FS (t − s) − 1)ds . (3.25) 0
LIMIT THEOREMS FOR MARKED HAWKES PROCESSES
9
But since FS (s) ↑ x∗ as s → ∞ we obtain the main result 1 1 log FN (t) = ν t t
Z
t
(FS (s) − 1) ds −→ ν(x∗ − 1), t→∞
0
(3.26)
which proves the desired formula. Note that x∗ = ∞ when there is no solution to (3.24). The proof is complete. 3.2. Large Deviation Principle. In this section, we prove the main result, i.e. Theorem 1 by using the G¨ artner-Ellis theorem for the upper bound and tilting method for the lower bound. Proof of Theorem 1. For the upper bound, since we have Theorem 4, we can simply apply G¨ artner-Ellis theorem. To prove the lower bound, it suffices to show that for any x > 0, > 0, we have Nt 1 ∈ B (x) ≥ − sup{θx − Γ(θ)}, (3.27) lim inf log P t→∞ t t θ∈R ˆ denote the where B (x) denotes the open ball centered at x with radius . Let P ˆ tilted probability measure with rate λ and marks distributed by qˆ(da) as defined in Lemma 6. By Jensen’s inequality, 1 Nt log P ∈ B (x) (3.28) t t Z 1 dP ˆ dP ≥ log N ˆ t t dP t ∈B (x) " # Z ˆ 1 N 1 1 d P t ˆ ˆ = log P ∈ B (x) − log dP ˆ Nt ∈ B (x) Nt ∈B (x) dP t t t P t t " # ˆ 1 N 1 1 d P t ˆ ˆ 1 Nt ≥ log P ∈ B (x) − log . · ·E ˆ Nt ∈ B (x) t t ∈B (x) t t dP P t By the ergodic theorem, 1 Nt ˆ qˆ, π lim inf log P ∈ B (x) ≥ − inf H(λ, ˆ ). t→∞ t t 0 0 is the constant premium and the Ci ’s are i.i.d. positive random variables with the common distribution µ(dC). Ci represents
LIMIT THEOREMS FOR MARKED HAWKES PROCESSES
11
the claim size at the ith arrival time, these being independent of Nt , a marked Hawkes process. For u > 0, let τu = inf{t > 0 : Rt ≤ 0}, (4.2) and denote the infinite and finite horizon ruin probabilities by ψ(u) = P(τu < ∞),
ψ(u, uz) = P(τu ≤ uz),
u, z > 0.
(4.3)
By the law of large numbers, N
t Eµ [C]ν 1X Ci = . lim t→∞ t 1 − Eq [H(a)] i=1
(4.4)
Therefore, to exclude the trivial case, we need to assume that ν(xc − 1) Eµ [C]ν u} and ψ(u) = P(τu < ∞). Assume that there exist γ, > 0 such that (i) κn (θ) = log E[eθSn ] is well defined and finite for γ − < θ < γ + . (ii) lim supn→∞ E[eθ(Sn −Sn−1 ) ] < ∞ for − < θ < . (iii) κ(θ) = limn→∞ n1 κn (θ) exists and is finite for γ − < θ < γ + . (iv) κ(γ) = 0 and κ is differentiable at γ with 0 < κ0 (γ) < ∞. Then, limu→∞ u1 log ψ(u) = −γ. Remark 9. We claim that ΓC (θ) = ρθ has a unique positive solution θ† < θc . Let G(θ) = ΓC (θ) − ρθ. Notice that G(0) = 0, G(∞) = ∞, and that G is convex. We Eµ [C]ν also have G0 (0) = 1−E q [H(a)] − ρ < 0 and ΓC (θc ) − ρθc > 0 since we assume that ρ< ρθ† .
ν(f (θc )−1) . θc
Therefore, there exists only one solution θ† ∈ (0, θc ) of ΓC (θ† ) =
Theorem 10 (Infinite Horizon). Assume all the assumptions in Theorem 1 and in addition (4.5), we have limu→∞ u1 log ψ(u) = −θ† , where θ† ∈ (0, θc ) is the unique positive solution of ΓC (θ) = ρθ.
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DMYTRO KARABASH AND LINGJIONG ZHU
PNt Ci − ρt and κt (θ) = log E[eθSt ]. Then limt→∞ 1t κt (θ) = Proof. Take St = i=1 ΓC (θ) − ρθ. Consider {Snh }n∈N . We have limn→∞ n1 κnh (θ) = hΓC (θ) − hρθ. Checking the conditions in Theorem 8 and applying it, we get 1 lim log P sup Snh > u = −θ† . (4.9) u→∞ u n∈N Finally, notice that sup St ≥ sup Snh ≥ sup St − ρh. t∈R+
Hence, limu→∞
1 u
n∈N
(4.10)
t∈R+
log ψ(u) = −θ† .
Theorem 11 (Finite Horizon). Under the same assumptions as in Theorem 10, we have 1 (4.11) lim log ψ(u, uz) = −w(z), for any z > 0. u→∞ u Here ( zΛC z1 + ρ if 0 < z < Γ0 (θ1† )−ρ , (4.12) w(z) = θ† if z ≥ Γ0 (θ1† )−ρ ΛC (x) = supθ∈R {θx − ΓC (θ)} and θ† ∈ (0, θc ) is the unique positive solution of ΓC (θ) = ρθ, as before. Proof. The proof is similar to that in Stabile and Torrisi [10] and we omit it here.
Next, we are interested to study the case when the claim sizes have heavy tails, R i.e. R+ eθC µ(dC) = +∞ for any θ > 0. A distribution function B is subexponential, i.e. B ∈ S if lim
x→∞
P(C1 + C2 > x) = 2, P(C1 > x)
(4.13)
where C1 , C2 are i.i.d. random variables with distribution function B. Let us denote R x B(x) := P(C1 ≥ x) and let us assume that E[C1 ] < ∞ and define B0 (x) := 1 E[C] 0 B(y)dy, where F (x) = 1 − F (x) is the complement of any distribution function F (x). Goldie and Resnick [7] showed that if B ∈ S and satisfies some smoothness conditions, then B belongs to the maximum domain of attraction of either the Frechet distribution or the Gumbel distribution. In the former case, B is regularly varying, i.e. B(x) = L(x)/xα+1 , for some α > 0 and we write it as B ∈ R(−α − 1), α > 0. We assume that B0 ∈ S and either B ∈ R(−α − 1) or B ∈ G, i.e. the maximum domain of attraction of Gumbel distribution. G includes Weibull and lognormal distributions. When the arrival process Nt satisfies a large deviation result, the probability that it deviates away from its mean is exponentially small, which is dominated by subexonential distributions. By using the techniques for the asymptotics of ruin probabilities for risk processes with non-stationary, non-renewal arrivals and subexponential claims from Zhu [17], we have the following infinite-horizon and finite-horizon ruin probability estimates when the claim sizes are subexponential.
LIMIT THEOREMS FOR MARKED HAWKES PROCESSES
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Theorem 12. Assume the net profit condition ρ > E[C1 ] 1−Eqν[H(a)] . (i) (Infinite-Horizon) lim
u→∞
νE[C1 ] ψ(u) = . q ρ(1 − E [H(a)]) − νE[C1 ] B 0 (u)
(4.14)
(ii) (Finite-Horizon) For any T > 0, ψ(u, uz) lim B 0 (u) −α ρ(1−Eq [H(a)])−νE[C1 ] T νE[C1 ] ρ(1−Eq [H(a)])−νE[C1 ] 1 − 1 + ρ(1−Eq [H(a)]) α = ρ(1−Eq [H(a)])−νE[C1 ] νE[C1 ] 1 − e− ρ(1−Eq [H(a)]) T ρ(1−Eq [H(a)])−νE[C 1]
(4.15)
u→∞
if B ∈ R(−α − 1) . if B ∈ G
5. Examples with Explicit Formulas In this section, we discuss two examples where an explicit formula exists. Example 13 is about the exponential asymptotics of the infinite-horizon ruin probability when H(a) and the claim size C are exponentially distributed. Example 14 gives an explicit expression for the rate function of the large deviation principle when H(a) is exponentially distributed. Example 13. Recall that x is the minimal solution of Z Z eθC+(x−1)H(a) q(da)µ(dC). x= R+
(5.1)
X
Now, assume that H(a) is exponentially distributed with parameter λ > 0, then, we have λ x = Eµ [eθC ] , (5.2) λ − (x − 1) which implies that q 1 (5.3) x= λ + 1 − (λ + 1)2 − 4λEµ [eθC ] . 2 Now, assume that C is exponentially distributed with parameter γ > 0. Then, r 1 γ x= λ + 1 − (λ + 1)2 − 4λ . (5.4) 2 γ−θ The infinite horizon probability satisfies limu→∞ u1 log ψ(u) = −θ† , where θ† satisfies r 1 γ ρθ† = ν λ + 1 − (λ + 1)2 − 4λ − 1 , (5.5) 2 γ − θ† which implies s 2ρθ† +1−λ=− ν
(λ + 1)2 −
4λγ , γ − θ†
(5.6)
and thus
ρ2 † 2 ρθ† λγ −λθ† (θ ) + (1 − λ) = λ − = . ν2 ν γ − θ† γ − θ† Since we are looking for positive θ† , we get the quadratic equation, ρ2 (θ† )2 − (ρ2 γ − ρν(1 − λ))θ† − (ρνγ(1 − λ) + λν 2 ) = 0.
(5.7)
(5.8)
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DMYTRO KARABASH AND LINGJIONG ZHU
Since ρ > †
θ =
Eµ [C]ν 1−Eq [H(a)]
=
νλ γ(λ−1) ,
(ρ2 γ − ρν(1 − λ)) +
p
we have ρνγ(1 − λ) + λν 2 > 0. Therefore, (ρ2 γ − ρν(1 − λ))2 + 4ρ2 (ρνγ(1 − λ) + λν 2 ) . 2ρ2
(5.9)
Example 14. Now, let H(a) be exponentially distributed with parameter λ > 0. We want an explicit expression for the rate function of the large deviation principle for (Nt /t ∈ ·). Notice that, o ( n p 2 ν 21 λ + 1 − (λ + 1)2 − 4λeθ − 1 for θ ≤ log (λ+1) 4λ Γ(θ) = . (5.10) +∞ otherwise To get I(x) = supθ∈R {θx − Γ(θ)}, we optimize over θ and consider x = Γ0 (θ). Evidently, 1 1 = 0, (5.11) x + ν(−4λ)eθ p 2 2 (λ + 1)2 − 4λeθ which gives us θ = log
! p −2x2 + x 4x2 + ν 2 (λ + 1)2 , λν 2
(5.12)
whence,
I(x) =
√ −2x2 +x 4x2 +ν 2 (λ+1)2 x log λν 2
−ν
1 2
λ+1−
√
−2x+
4x2 +ν 2 (λ+1)2 ν
+∞
−1
if x ≥ 0 . otherwise (5.13)
Acknowledgements The authors are both supported by NSF grant DMS-0904701, DARPA grant and MacCracken Fellowship at New York University. References [1] Bacry, E., Delattre, S., Hoffmann, M., and J. F. Muzy, Scaling Limits for Hawkes Processes and Application to Financial Statistics, available on arXiv, 2012 [2] Bordenave, C. and G. L. Torrisi, Large Deviations of Poisson Cluster Processes, Stochastic Models, 23:593-625, 2007 [3] Br´ emaud, P., and L. Massouli´ e, Stability of Nonlinear Hawkes Processes, The Annals of Probability, Vol.24, No.3, 1563-1588, 1996 [4] Daley, D. J. and D. Vere-Jones, An Introduction to the Theory of Point Processes, Volume I and II, Springer, Second Edition, 2003 [5] Dembo, A. and O. Zeitouni, Large Deviations Techniques and Applications, 2nd Edition, Springer, 1998 [6] Glynn, P. W. and W. Whitt, Logarithmic Asymptotics for Steady-State Tail Probabilities in a Single-Server Queue, J. Appl. Probab. 31A, 131-156, 1994 [7] Goldie, C. M. and S. Resnick. (1988). Distributions that are both subexponential and in the domain of attraction of an extreme value distribution. Adv. Appl. Probab. 20, 706-718. [8] Hawkes, A. G. and D. Oakes, A Cluster Process Representation of a Self-Exciting Process, J. Appl. Prob.1 1,4 93-503, 1974 [9] Karabash, D., On Stability of Hawkes Process, preprint, 2011. [10] Stabile, G. and G. L. Torrisi, Risk Processes with Non-Stationary Hawkes Arrivals, Methodol. Comput. Appl. Prob. 12:415-429, 2010
LIMIT THEOREMS FOR MARKED HAWKES PROCESSES
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[11] Varadhan, S. R. S., Special Invited Paper: Large Deviations, The Annals of Probability, Vol. 36, No. 2, 397-419, 2008 [12] Varadhan, S. R. S., Large Deviations and Applications, SIAM, 1984 [13] Zhu, L., Large Deviations for Markovian Nonlinear Hawkes Processes, preprint, 2011 [14] Zhu, L., Process-Level Large Deviations for Nonlinear Hawkes Point Processes, to appear in Annales de l’Institut Henri Poincar´ e, 2011 [15] Zhu, L., Central Limit Theorem for Nonlinear Hawkes Processes, preprint, 2012 [16] Zhu, L., Moderate Deviations for Hawkes Processes, preprint, 2012 [17] Zhu, L. Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims. Preprint. arXiv:1304.1940. Courant Institute of Mathematical Sciences New York University 251 Mercer Street New York, NY-10012 United States of America E-mail address: [email protected] Courant Institute of Mathematical Sciences New York University 251 Mercer Street New York, NY-10012 United States of America E-mail address: [email protected]
Abstract. This paper focuses on limit theorems for linear Hawkes processes with random marks. We prove a large deviation principle, which answers the question raised by Bordenave and Torrisi. A central limit theorem is also obtained. We conclude with an example of application in finance.
Contents 1. Introduction and Main Results 1.1. Introduction 1.2. Limit Theorems for Unmarked Hawkes Processes 1.3. Main Results 2. Proof of Central Limit Theorem 3. Proof of Large Deviation Principle 3.1. Limit of a Logarithmic Moment Generating Function 3.2. Large Deviation Principle 4. Risk Model with Marked Hawkes Claims Arrivals 5. Examples with Explicit Formulas Acknowledgements References
1 1 2 3 4 5 5 9 10 13 14 14
1. Introduction and Main Results 1.1. Introduction. We consider in this article a linear Hawkes process with random marks. Let Nt be a simple point process. Nt denotes the number of points in the interval [0, t). Let Ft be the natural filtration up to time t. We assume that N (−∞, 0] = 0. At time t, the point process has Ft -predictable intensity X λt := ν + Zt , Zt := h(t − τi , ai ), (1.1) τi 0, the (τi )i≥1 are arrival times of the points, and the (ai )i≥1 are i.i.d. random marks, ai being independent of previous arrival times τj , j ≤ i. Let us assume that ai has a common distribution q(da) on a metric space X. Here, h(·, ·) : Date: 16 November 2012. Revised: 16 November 2012. 2000 Mathematics Subject Classification. 60G55, 60F10, 60F05. Key words and phrases. Central limit theorems, large deviations, rare events, point processes, marked point processes, Hawkes processes, self-exciting processes. This research was supported partially by a grant from the National Science Foundation: DMS0904701, DARPA grant and MacCracken Fellowship at NYU. 1
2
DMYTRO KARABASH AND LINGJIONG ZHU
R∞R R∞ R+ ×X → R+ is integrable, i.e. 0 X h(t, a)q(da)dt < ∞. Let H(a) := 0 h(t, a)dt for any a ∈ X. We also assume that Z H(a)q(da) < 1. (1.2) X
Let Pq denote the probability measure for the ai ’s with the common law q(da). Under assumption (1.2), it is well known that there exists a unique stationary version of the linear marked Hawkes process satisfying the dynamics (1.1) and that by ergodic theorem, a law of large numbers holds, lim
t→∞
Nt ν = . t 1 − Eq [H(a)]
(1.3)
This paper is organized as the following. In Section 1.2, we will review some results about the limit theorems for unmarked Hawkes processes. In Section 1.3, we will introduce the main results of this paper, i.e. the central limit theorem and the large deviation principle for linear marked Hawkes processes. The proof of the central limit theorem will be given in Section 2 and the proof of the large deviation principle will be given in Section 3. Finally, we will discuss an application of our results to a risk model in finance in Section 4. 1.2. Limit Theorems for Unmarked Hawkes Processes. Most of the literature about Hawkes processes considered the unmarked case, i.e. with intensity ! X λt := λ h(t − τ ) , (1.4) τ 0 and khkL1 < 1, it has a very nice immigration-birth representation, see for example Hawkes and Oakes [8]. For linear Hawkes process, limit theorems are very well understood. There is the law of large numbers (see for instance Daley and Vere-Jones [4]), i.e. ν Nt → , t 1 − khkL1
as t → ∞ a.s.
(1.5)
Moreover, Bordenave and Torrisi [2] proved a large deviation principle for ( Ntt ∈ ·) with the rate function ( x x log ν+xkhk − x + xkhkL1 + ν if x ∈ [0, ∞) L1 I(x) = . (1.6) +∞ otherwise Once you have the large deviation principle, you can also study some risk processes in finance. (See Stabile and Torrisi [10].)
LIMIT THEOREMS FOR MARKED HAWKES PROCESSES
3
Recently, Bacry et al. [1] proved a functional central limit theorem for the linear multivariate Hawkes process under certain assumptions which includes the linear Hawkes process as a special case and they proved that N·t − ·µt √ → σB(·), t
as t → ∞,
weakly on D[0, 1] equipped with Skorokhod topology, where ν ν µ= . and σ 2 = 1 − khkL1 (1 − khkL1 )3
(1.7)
(1.8)
Moderate deviation principle for linear Hawkes processes is obtained in Zhu [16], which fills in the gap between central limit theorem and large deviation principle. For nonlinear Hawkes processes, a central limit theorem is obtained in Zhu [15]. In Bordenave and Torrisi [2], they raised two questions about large deviations for Hawkes processes. One question is about large deviations for nonlinear Hawkes process and the other is about large deviations for linear marked Hawkes processes. Recently, Zhu [13] considered a special case for nonlinear Hawkes processes when h(·) is exponential or sums of exponentials and proved the large deviations. In another paper, Zhu [14] proved a process-level, i.e. level-3 large deviation principle for nonlinear Hawkes processes for general h(·) and hence by contraction principle, the level-1 large deviation principle for (Nt /t ∈ ·). In this paper, we will prove the large deviations for linear marked Hawkes processes and thus both questions raised in Bordenave and Torrisi [2] have been answered. 1.3. Main Results. Before we proceed, recall that a sequence (Pn )n∈N of probability measures on a topological space X satisfies the large deviation principle (LDP) with rate function I : X → R if I is non-negative, lower semicontinuous and for any measurable set A, we have 1 1 (1.9) − inf o I(x) ≤ lim inf log Pn (A) ≤ lim sup log Pn (A) ≤ − inf I(x). n→∞ n x∈A n→∞ n x∈A Here, Ao is the interior of A and A is its closure. See Dembo and Zeitouni [5] or Varadhan [12] for general background regarding large deviations and their applications. Also Varadhan [11] has an excellent survey article on this subject. For a linear marked Hawkes process satisfying the dynamics (1.1), we prove the following large deviation principle in this article. Theorem 1 (Large Deviation Principle). Assume the conditions (1.2) and Z H(a)x lim e q(da) − x = ∞. (1.10) x→∞
X
Then, (Nt /t ∈ ·) satisfies a large deviation principle with rate function, n h io ( q x inf qˆ xEqˆ[H(a)] + ν − x + x log xEqˆ[H(a)]+ν + xEqˆ log dˆ dq Λ(x) := +∞ ( θ∗ x − ν(x∗ − 1) x ≥ 0 = , +∞ x 0, we have Z t FN (t) := E[exp(θNt )] = exp ν (FS (t − s) − 1)ds . (3.25) 0
LIMIT THEOREMS FOR MARKED HAWKES PROCESSES
9
But since FS (s) ↑ x∗ as s → ∞ we obtain the main result 1 1 log FN (t) = ν t t
Z
t
(FS (s) − 1) ds −→ ν(x∗ − 1), t→∞
0
(3.26)
which proves the desired formula. Note that x∗ = ∞ when there is no solution to (3.24). The proof is complete. 3.2. Large Deviation Principle. In this section, we prove the main result, i.e. Theorem 1 by using the G¨ artner-Ellis theorem for the upper bound and tilting method for the lower bound. Proof of Theorem 1. For the upper bound, since we have Theorem 4, we can simply apply G¨ artner-Ellis theorem. To prove the lower bound, it suffices to show that for any x > 0, > 0, we have Nt 1 ∈ B (x) ≥ − sup{θx − Γ(θ)}, (3.27) lim inf log P t→∞ t t θ∈R ˆ denote the where B (x) denotes the open ball centered at x with radius . Let P ˆ tilted probability measure with rate λ and marks distributed by qˆ(da) as defined in Lemma 6. By Jensen’s inequality, 1 Nt log P ∈ B (x) (3.28) t t Z 1 dP ˆ dP ≥ log N ˆ t t dP t ∈B (x) " # Z ˆ 1 N 1 1 d P t ˆ ˆ = log P ∈ B (x) − log dP ˆ Nt ∈ B (x) Nt ∈B (x) dP t t t P t t " # ˆ 1 N 1 1 d P t ˆ ˆ 1 Nt ≥ log P ∈ B (x) − log . · ·E ˆ Nt ∈ B (x) t t ∈B (x) t t dP P t By the ergodic theorem, 1 Nt ˆ qˆ, π lim inf log P ∈ B (x) ≥ − inf H(λ, ˆ ). t→∞ t t 0 0 is the constant premium and the Ci ’s are i.i.d. positive random variables with the common distribution µ(dC). Ci represents
LIMIT THEOREMS FOR MARKED HAWKES PROCESSES
11
the claim size at the ith arrival time, these being independent of Nt , a marked Hawkes process. For u > 0, let τu = inf{t > 0 : Rt ≤ 0}, (4.2) and denote the infinite and finite horizon ruin probabilities by ψ(u) = P(τu < ∞),
ψ(u, uz) = P(τu ≤ uz),
u, z > 0.
(4.3)
By the law of large numbers, N
t Eµ [C]ν 1X Ci = . lim t→∞ t 1 − Eq [H(a)] i=1
(4.4)
Therefore, to exclude the trivial case, we need to assume that ν(xc − 1) Eµ [C]ν u} and ψ(u) = P(τu < ∞). Assume that there exist γ, > 0 such that (i) κn (θ) = log E[eθSn ] is well defined and finite for γ − < θ < γ + . (ii) lim supn→∞ E[eθ(Sn −Sn−1 ) ] < ∞ for − < θ < . (iii) κ(θ) = limn→∞ n1 κn (θ) exists and is finite for γ − < θ < γ + . (iv) κ(γ) = 0 and κ is differentiable at γ with 0 < κ0 (γ) < ∞. Then, limu→∞ u1 log ψ(u) = −γ. Remark 9. We claim that ΓC (θ) = ρθ has a unique positive solution θ† < θc . Let G(θ) = ΓC (θ) − ρθ. Notice that G(0) = 0, G(∞) = ∞, and that G is convex. We Eµ [C]ν also have G0 (0) = 1−E q [H(a)] − ρ < 0 and ΓC (θc ) − ρθc > 0 since we assume that ρ< ρθ† .
ν(f (θc )−1) . θc
Therefore, there exists only one solution θ† ∈ (0, θc ) of ΓC (θ† ) =
Theorem 10 (Infinite Horizon). Assume all the assumptions in Theorem 1 and in addition (4.5), we have limu→∞ u1 log ψ(u) = −θ† , where θ† ∈ (0, θc ) is the unique positive solution of ΓC (θ) = ρθ.
12
DMYTRO KARABASH AND LINGJIONG ZHU
PNt Ci − ρt and κt (θ) = log E[eθSt ]. Then limt→∞ 1t κt (θ) = Proof. Take St = i=1 ΓC (θ) − ρθ. Consider {Snh }n∈N . We have limn→∞ n1 κnh (θ) = hΓC (θ) − hρθ. Checking the conditions in Theorem 8 and applying it, we get 1 lim log P sup Snh > u = −θ† . (4.9) u→∞ u n∈N Finally, notice that sup St ≥ sup Snh ≥ sup St − ρh. t∈R+
Hence, limu→∞
1 u
n∈N
(4.10)
t∈R+
log ψ(u) = −θ† .
Theorem 11 (Finite Horizon). Under the same assumptions as in Theorem 10, we have 1 (4.11) lim log ψ(u, uz) = −w(z), for any z > 0. u→∞ u Here ( zΛC z1 + ρ if 0 < z < Γ0 (θ1† )−ρ , (4.12) w(z) = θ† if z ≥ Γ0 (θ1† )−ρ ΛC (x) = supθ∈R {θx − ΓC (θ)} and θ† ∈ (0, θc ) is the unique positive solution of ΓC (θ) = ρθ, as before. Proof. The proof is similar to that in Stabile and Torrisi [10] and we omit it here.
Next, we are interested to study the case when the claim sizes have heavy tails, R i.e. R+ eθC µ(dC) = +∞ for any θ > 0. A distribution function B is subexponential, i.e. B ∈ S if lim
x→∞
P(C1 + C2 > x) = 2, P(C1 > x)
(4.13)
where C1 , C2 are i.i.d. random variables with distribution function B. Let us denote R x B(x) := P(C1 ≥ x) and let us assume that E[C1 ] < ∞ and define B0 (x) := 1 E[C] 0 B(y)dy, where F (x) = 1 − F (x) is the complement of any distribution function F (x). Goldie and Resnick [7] showed that if B ∈ S and satisfies some smoothness conditions, then B belongs to the maximum domain of attraction of either the Frechet distribution or the Gumbel distribution. In the former case, B is regularly varying, i.e. B(x) = L(x)/xα+1 , for some α > 0 and we write it as B ∈ R(−α − 1), α > 0. We assume that B0 ∈ S and either B ∈ R(−α − 1) or B ∈ G, i.e. the maximum domain of attraction of Gumbel distribution. G includes Weibull and lognormal distributions. When the arrival process Nt satisfies a large deviation result, the probability that it deviates away from its mean is exponentially small, which is dominated by subexonential distributions. By using the techniques for the asymptotics of ruin probabilities for risk processes with non-stationary, non-renewal arrivals and subexponential claims from Zhu [17], we have the following infinite-horizon and finite-horizon ruin probability estimates when the claim sizes are subexponential.
LIMIT THEOREMS FOR MARKED HAWKES PROCESSES
13
Theorem 12. Assume the net profit condition ρ > E[C1 ] 1−Eqν[H(a)] . (i) (Infinite-Horizon) lim
u→∞
νE[C1 ] ψ(u) = . q ρ(1 − E [H(a)]) − νE[C1 ] B 0 (u)
(4.14)
(ii) (Finite-Horizon) For any T > 0, ψ(u, uz) lim B 0 (u) −α ρ(1−Eq [H(a)])−νE[C1 ] T νE[C1 ] ρ(1−Eq [H(a)])−νE[C1 ] 1 − 1 + ρ(1−Eq [H(a)]) α = ρ(1−Eq [H(a)])−νE[C1 ] νE[C1 ] 1 − e− ρ(1−Eq [H(a)]) T ρ(1−Eq [H(a)])−νE[C 1]
(4.15)
u→∞
if B ∈ R(−α − 1) . if B ∈ G
5. Examples with Explicit Formulas In this section, we discuss two examples where an explicit formula exists. Example 13 is about the exponential asymptotics of the infinite-horizon ruin probability when H(a) and the claim size C are exponentially distributed. Example 14 gives an explicit expression for the rate function of the large deviation principle when H(a) is exponentially distributed. Example 13. Recall that x is the minimal solution of Z Z eθC+(x−1)H(a) q(da)µ(dC). x= R+
(5.1)
X
Now, assume that H(a) is exponentially distributed with parameter λ > 0, then, we have λ x = Eµ [eθC ] , (5.2) λ − (x − 1) which implies that q 1 (5.3) x= λ + 1 − (λ + 1)2 − 4λEµ [eθC ] . 2 Now, assume that C is exponentially distributed with parameter γ > 0. Then, r 1 γ x= λ + 1 − (λ + 1)2 − 4λ . (5.4) 2 γ−θ The infinite horizon probability satisfies limu→∞ u1 log ψ(u) = −θ† , where θ† satisfies r 1 γ ρθ† = ν λ + 1 − (λ + 1)2 − 4λ − 1 , (5.5) 2 γ − θ† which implies s 2ρθ† +1−λ=− ν
(λ + 1)2 −
4λγ , γ − θ†
(5.6)
and thus
ρ2 † 2 ρθ† λγ −λθ† (θ ) + (1 − λ) = λ − = . ν2 ν γ − θ† γ − θ† Since we are looking for positive θ† , we get the quadratic equation, ρ2 (θ† )2 − (ρ2 γ − ρν(1 − λ))θ† − (ρνγ(1 − λ) + λν 2 ) = 0.
(5.7)
(5.8)
14
DMYTRO KARABASH AND LINGJIONG ZHU
Since ρ > †
θ =
Eµ [C]ν 1−Eq [H(a)]
=
νλ γ(λ−1) ,
(ρ2 γ − ρν(1 − λ)) +
p
we have ρνγ(1 − λ) + λν 2 > 0. Therefore, (ρ2 γ − ρν(1 − λ))2 + 4ρ2 (ρνγ(1 − λ) + λν 2 ) . 2ρ2
(5.9)
Example 14. Now, let H(a) be exponentially distributed with parameter λ > 0. We want an explicit expression for the rate function of the large deviation principle for (Nt /t ∈ ·). Notice that, o ( n p 2 ν 21 λ + 1 − (λ + 1)2 − 4λeθ − 1 for θ ≤ log (λ+1) 4λ Γ(θ) = . (5.10) +∞ otherwise To get I(x) = supθ∈R {θx − Γ(θ)}, we optimize over θ and consider x = Γ0 (θ). Evidently, 1 1 = 0, (5.11) x + ν(−4λ)eθ p 2 2 (λ + 1)2 − 4λeθ which gives us θ = log
! p −2x2 + x 4x2 + ν 2 (λ + 1)2 , λν 2
(5.12)
whence,
I(x) =
√ −2x2 +x 4x2 +ν 2 (λ+1)2 x log λν 2
−ν
1 2
λ+1−
√
−2x+
4x2 +ν 2 (λ+1)2 ν
+∞
−1
if x ≥ 0 . otherwise (5.13)
Acknowledgements The authors are both supported by NSF grant DMS-0904701, DARPA grant and MacCracken Fellowship at New York University. References [1] Bacry, E., Delattre, S., Hoffmann, M., and J. F. Muzy, Scaling Limits for Hawkes Processes and Application to Financial Statistics, available on arXiv, 2012 [2] Bordenave, C. and G. L. Torrisi, Large Deviations of Poisson Cluster Processes, Stochastic Models, 23:593-625, 2007 [3] Br´ emaud, P., and L. Massouli´ e, Stability of Nonlinear Hawkes Processes, The Annals of Probability, Vol.24, No.3, 1563-1588, 1996 [4] Daley, D. J. and D. Vere-Jones, An Introduction to the Theory of Point Processes, Volume I and II, Springer, Second Edition, 2003 [5] Dembo, A. and O. Zeitouni, Large Deviations Techniques and Applications, 2nd Edition, Springer, 1998 [6] Glynn, P. W. and W. Whitt, Logarithmic Asymptotics for Steady-State Tail Probabilities in a Single-Server Queue, J. Appl. Probab. 31A, 131-156, 1994 [7] Goldie, C. M. and S. Resnick. (1988). Distributions that are both subexponential and in the domain of attraction of an extreme value distribution. Adv. Appl. Probab. 20, 706-718. [8] Hawkes, A. G. and D. Oakes, A Cluster Process Representation of a Self-Exciting Process, J. Appl. Prob.1 1,4 93-503, 1974 [9] Karabash, D., On Stability of Hawkes Process, preprint, 2011. [10] Stabile, G. and G. L. Torrisi, Risk Processes with Non-Stationary Hawkes Arrivals, Methodol. Comput. Appl. Prob. 12:415-429, 2010
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[11] Varadhan, S. R. S., Special Invited Paper: Large Deviations, The Annals of Probability, Vol. 36, No. 2, 397-419, 2008 [12] Varadhan, S. R. S., Large Deviations and Applications, SIAM, 1984 [13] Zhu, L., Large Deviations for Markovian Nonlinear Hawkes Processes, preprint, 2011 [14] Zhu, L., Process-Level Large Deviations for Nonlinear Hawkes Point Processes, to appear in Annales de l’Institut Henri Poincar´ e, 2011 [15] Zhu, L., Central Limit Theorem for Nonlinear Hawkes Processes, preprint, 2012 [16] Zhu, L., Moderate Deviations for Hawkes Processes, preprint, 2012 [17] Zhu, L. Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims. Preprint. arXiv:1304.1940. Courant Institute of Mathematical Sciences New York University 251 Mercer Street New York, NY-10012 United States of America E-mail address: [email protected] Courant Institute of Mathematical Sciences New York University 251 Mercer Street New York, NY-10012 United States of America E-mail address: [email protected]
