LARGE DEVIATIONS FOR MARKOVIAN NONLINEAR HAWKES ...

LARGE DEVIATIONS FOR MARKOVIAN NONLINEAR
HAWKES PROCESSES
LINGJIONG ZHU
Abstract. In the 2007 paper, Bordenave and Torrisi [1] proves the large deviation
principles for Poisson cluster processes and in particular, the linear
Hawkes processes. In this paper, we prove first a large deviation principle for
a special class of nonlinear Hawkes process, i.e. a Markovian Hawkes process
with nonlinear rate and exponential exciting function, and then generalize it
to get the result for sum of exponentials exciting functions. We then provide
an alternative proof for the large deviation principle for linear Hawkes process.
Finally, we use an approximation approach to prove the large deviation principle
for a special class of nonlinear Hawkes processes with general exciting
functions.
Contents
1. Introduction 1
2. An Ergodic Lemma 3
3. LDP for Markovian Nonlinear Hawkes Processes with Exponential
Exciting Function 5
4. LDP for Markovian Nonlinear Hawkes Processes with Sum of
Exponentials Exciting Function 13
5. LDP for Linear Hawkes Processes: An Alternative Proof 16
6. LDP for a Special Class of Nonlinear Hawkes Processes: An
Approximation Approach 20
Acknowledgements 25
References 25
1. Introduction
Let N be a simple point process on R and let Ft := σ(N(C), C ∈ B(R), C ⊂ R + )
be an increasing family of σ-algebras. Any nonnegative Ft-progressively measurable
process λt with
b
(1.1) E [N(a, b]|Fa] = E λsds
Fa Date: 9 August 2011. Revised: 7 June 2012.
2000 Mathematics Subject Classification. 60G55, 60F10.
Key words and phrases. Large deviations, rare events, point processes, Hawkes processes, selfexciting
processes.
This research was supported partially by a grant from the National Science Foundation: DMS-
0904701, DARPA grant and MacCracken Fellowship at NYU.
1
a

2 LINGJIONG ZHU
a.s. for all intervals (a, b] is called an Ft-intensity of N. We use the notation
Nt := N(0, t] to denote the number of points in the interval (0, t].
A general Hawkes process is a simple point process N admitting an Ft-intensity
t
(1.2) λt := λ h(t − s)N(ds) ,
0
where λ(·) : R + → R + , h(·) : R + → R + and we always assume that hL1
=
∞
h(t)dt < ∞. In the literatures, h(·) and λ(·) are usually referred to as exciting
0
function and rate function respectively.
Let Zt =
0

LDP FOR MARKOVIAN NONLINEAR HAWKES PROCESSES 3
and for any open set G ⊂ R,
(1.5) lim inf
t→∞
where
(1.6) I(x) =
1
t log P(Nt/t ∈ G) ≥ − inf
x∈G I(x),
xθx + ν − νx
ν+µx if x ∈ [0, ∞)
+∞ otherwise
where θ = θx is the unique solution in (−∞, µ − 1 − log µ] of E[e θS ] = x
ν+xµ ,
x > 0. It is well known that, for instance, see page 39 of Jagers [12], for all
θ ∈ (−∞, µ − 1 − log µ], E[e θS ] satisfies
(1.7) E[e θS ] = e θ exp µ(E[e θS ] − 1) .
See Dembo and Zeitouni [5] for general background regarding large deviations and
the applications. Also Varadhan [19] has an excellent survey article on this subject.
Once the LDP for Nt
t ∈ · is established, it is easy to study the ruin probability.
Stabile and Torrisi [18] considered risk processes with non-stationary Hawkes claims
arrivals and studied the asymptotic behavior of infinite and finite horizon ruin
probabilities under light-tailed conditions on the claims.
We are interested in general non-linear λ(·). If λ(·) is nonlinear, then the usual
Galton-Watson theory approach no longer works. If the exciting function h is
exponential or a sum of exponentials, the process is Markovian and there exists
a generator of the process. The difficulty arises when h is not exponential or a
sum of exponentials in which case the process is non-Markovian. Another possible
generalization is to consider h to be random. Then, we will get a marked point
process. (For a discussion on marked point processes, see Cox [3].)
When λ(·) is nonlinear, Brémaud and Massoulié [2] proves that under certain
conditions, there exists a unique stationary version of the nonlinear Hawkes process
and Brémaud and Massoulié [2] also proves the convergence to equilibrium of a
nonstationary version, both in distribution and in variation.
In this paper, we will prove the large deviation when h is exponential and λ is
nonlinear first. Then, we will generalize the proof to the case when h is a sum
of exponentials. We will use that to recover the result proved in Bordenave and
Torrisi [1]. Finally, we will prove the result for a special class of nonlinear λ and
general h.
2. An Ergodic Lemma
Let us prove an ergodic theorem first. Assume h(t) = d
i=1 aie −bit . Here bi > 0,
ai = 0 might be negative but we assume that h(t) > 0 for any t ≥ 0. Zt =
τj

4 LINGJIONG ZHU
The lecture notes [9] by Martin Hairer gives the criterion for the existence and
uniqueness of the invariant probability measure for Markov processes.
Suppose we have a jump diffusion process with generator L. If we can find u
such that u ≥ 0, Lu ≤ C1 − C2u for some constants C1, C2 > 0, then, there exists
an invariant probability measure. We thereby have the following lemma.
Lemma 1. Consider h(t) = d i=1 aie−bit > 0. Let ɛi = +1 if ai > 0 and ɛi = −1 if
ai < 0. Assume λ(z1, . . . , zn) ≤ d i=1 αi|zi|+β, where β > 0 and αi > 0, 1 ≤ i ≤ d,
satisfies d |ai|
i=1 bi αi < 1. Then, there exists a unique invariant probability measure
for (Z1(t), . . . , Zd(t)).
Proof. Try u(z1, . . . , zd) = d i=1 ɛicizi ≥ 0, where ci > 0, 1 ≤ i ≤ d. Then,
d
d
(2.2) Au = − biɛicizi + λ(z1, . . . , zd)
Take ci = αi
bi
(2.3)
≤ −
i=1
d
bici|zi| +
i=1
> 0, we get
Au ≤ −
1 −
d
αi|zi|
i=1
i=1
aiɛici
d
|ai|ci + β
i=1
d
d
|ai|αi
αi|zi| + β
i=1
≤ − min
1≤i≤d bi ·
bi
1 −
i=1
d
|ai|αi
u + β
i=1
bi
d
|ai|ci.
i=1
d |ai|αi
i=1
bi
d |ai|αi
.
Next, we will prove the uniqueness of the invariant probability measure. Let
us consider the simplest case h(t) = ae −bt . To get the uniqueness of the invariant
probability measure, it is sufficient to prove that for any x, y > 0, there exists some
T > 0 such that P x (T, ·) and P y (T, ·) are not mutually singular. Here P x (T, ·) =
P(Z x T ∈ ·), where Zx T is ZT starting at Z0 = x, i.e. Z x T = xe−bT +
τj y > 0. Conditional on the event that Z x t and Z y
t have
exactly one jump during the time interval (0, T ) respectively, the law of P x (T, ·)
and P y (T, ·) are absolutely continuous with respect to some probability measures
with positive density on the sets
(2.4)
(a + x)e −bT , xe −bT + a
respectively. Choose T > 1
b
(2.5)
log( x−y+a
a
and
), we have
i=1
bi
(a + y)e −bT , ye −bT + a
(a + x)e −bT , xe −bT + a (a + y)e −bT , ye −bT + a = ∅,
which implies that Px (T, ·) and Py (T, ·) are not mutually singular.
Similarly, we can prove the uniqueness of the invariant probability measure for
the multidimensional case. We need to conditional on the event that we have exactly
d jumps during the time interval (0, T ) for both Zx t and Z y
t , where x, y ∈ Zd. Then,
(2.6) Z x t = (Z x1
t , . . . , Z xd
t ) ∈ Zd,
where Z xi
t = xie −bit +
τj

LDP FOR MARKOVIAN NONLINEAR HAWKES PROCESSES 5
3. LDP for Markovian Nonlinear Hawkes Processes with
Exponential Exciting Function
We assume first that h(t) = ae −bt , where a, b > 0, i.e. the process Zt jumps
upwards an amount a at each point and decays exponentially between points with
rate b. In this case, Zt is Markovian.
Notice first that Z0 = 0 and
(3.1) dZt = −bZtdt + adNt,
which implies that Nt = 1
aZt + b
t
a 0 Zsds.
We prove first the existence of the limit of the logarithmic moment generating
function of Nt.
Theorem 2. Assume that limz→∞ λ(z)
z
positive constant, then, we have
= 0 and λ(·) is bounded below by some
(3.2)
1
lim
t→∞ t log E[eθNt ] = Γ(θ),
where Γ(θ) is defined as
θb
zˆπ(dz) + (
a ˆ
λ − λ)ˆπ(dz) − log( ˆ
λ/λ) ˆλˆπ(dz) ,
(3.3) Γ(θ) = sup
( ˆ λ,ˆπ)∈Qe
where Qe is defined as
(3.4) Qe = ( ˆ
λ, ˆπ) ∈ Q : Â has unique invariant probability measure ˆπ ,
where
(3.5) Q =
( ˆ λ, ˆπ) : ˆπ ∈ M(R +
),
zˆπ < ∞, ˆ λ ∈ L 1 (ˆπ), ˆ
λ > 0 ,
and for any ˆ λ such that ( ˆ λ, ˆπ) ∈ Q, we define the generator  as
(3.6) Âf(z) = −bz ∂f
∂z + ˆ λ(z)[f(z + a) − f(z)].
for any f : R + → R that is C 1 , i.e. continously differentiable.
Proof. Note that for any real θ, we have E[e θNt ] < ∞ (That is because we can
dominate λ(z) by νɛ + ɛz for arbitrarily small ɛ > 0. When λ(z) = νɛ + ɛz, i.e. in
the linear case, E[e θNt ] < ∞ for any θ ≤ C(ɛ), where C(ɛ) → ∞ as ɛ → 0 and C(ɛ)
does not depend on νɛ (we refer to Bordenave and Torrisi [1]).) and
(3.7) E[e θNt ] = E
e θ
a(Zt+b R t
0 Zsds)
.
By Dynkin’s formula, for any u which satisfies Au + V u ≤ Mu, we have
E u(Zt)e R (3.8)
t
0
V (Zs)ds
= u(Z0) +
t
≤ u(Z0) + M
E
0
t
which implies by Gronwall’s lemma that
(3.9) E u(Zt)e R t
V (Zs)ds
0
0
(Au(Zs) + V (Zs)u(Zs))e R s
0
E
u(Zs)e R s
0
V (Zv)dv
ds,
≤ u(Z0)e Mt = u(0)e Mt .
V (Zv)dv
ds

6 LINGJIONG ZHU
In our case, V (z) = θb
a
z. Now for any u(z) ≥ c1e θ
a z , we have
(3.10) E e θNt ≤ 1
E u(Zt)e
c1
R t θb
0 a Zsds
≤ 1
u(0)e
c1
Mt .
Therefore, we get
(3.11) lim sup
t→∞
where
(3.12) Uθ =
1
t log E e θNt ≤ inf sup
u∈Uθ z≥0
Au(z) + θb
a zu(z)
u(z)
u ∈ C 1 (R + , R + ) : u(z) = e f(z)
, where f ∈ F ,
where
(3.13)
F = f : f(z) = Kz + g(z) + L, K > θ
a , K, L ∈ R, g is C1
with compact support .
Define the tilted probability measure ˆ P as
d
(3.14)
ˆ P
t
dP
= exp (λ(Zs) −
Ft
0
ˆ λ(Zs))ds +
t
0
ˆλ(Zs)
log dNs .
λ(Zs)
Notice that ˆ P defined in (3.14) is indeed a probability measure by Girsanov formula.
(For the theory of absolute continuity for point processes and its Girsanov formula,
we refer to Lipster and Shiryaev [16].)
Now, by Jensen’s inequality, we have
1
lim inf
t→∞ t log E[eθNt (3.15)
]
1
= lim inf log Ê exp θNt − log
t→∞ t dˆ
P
dP
1
≥ lim inf Ê
t→∞ t θNt − 1
t log dˆ
P
dP
1
= lim inf Ê
t→∞ t θNt − 1
t
(λ(Zs) −
t
ˆ
t ˆλ(Zs)
λ(Zs))ds − log dNs .
λ(Zs)
Since Nt − t
0 ˆ λ(Zs)ds is a (local) martingale under ˆ P, we have
t ˆλ(Zs)
(3.16) Ê log (dNs −
λ(Zs)
ˆ
λ(Zs)ds) = 0.
0
0
Therefore, by the usual ergodic theorem, (for a reference, see Chapter 16.4 of Koralov
and Sinai [14]), for any ( ˆ λ, ˆπ) ∈ Qe, we have,
(3.17)
1
lim inf
t→∞ t log E[eθNt ]
1
≥ lim inf Ê
t→∞ t θNt − 1
t
(λ(Zs) −
t 0
ˆ
t ˆλ(Zs)
λ(Zs))ds − log ˆλ(Zs)ds
0 λ(Zs)
θb
= zˆπ(dz) + (
a ˆ
λ − λ)ˆπ(dz) − log( ˆ
λ) − log(λ) ˆλˆπ(dz).
0
,

Hence, we have
(3.18)
LDP FOR MARKOVIAN NONLINEAR HAWKES PROCESSES 7
lim inf
t→∞
≥ sup
( ˆ λ,ˆπ)∈Qe
1
t log E[eθNt ]
θb
zˆπ +
a
( ˆ
λ − λ)ˆπ − log( ˆ
λ) − log(λ) ˆλˆπ .
Recall that
(3.19)
F = f : f(z) = Kz + g(z) + L, K > θ
a , K, L ∈ R, g is C1
with compact support .
We claim that
(3.20) inf
f∈F
0 if (
Âf(z)ˆπ(dz) =
ˆ λ, ˆπ) ∈ Qe,
−∞ if ( ˆ λ, ˆπ) ∈ Q\Qe.
It is easy to see that for ( ˆ λ, ˆπ) ∈ Qe, g C1 with compact support, Agˆπ = 0.
Next, we can find a sequence fn(z) → z pointwisely and |fn(z)| ≤ αz + β, for some
α, β > 0, where fn(z) is C1
with compact support.
But by our definition for Q,
zˆπ < ∞. So by dominated convergence theorem, Âzˆπ = 0. The nontrivial part
is to prove that if for any g ∈ G = {g(z) + L, g is C1 with compact support} such
that Âgˆπ = 0, then ( ˆ λ, ˆπ) ∈ Qe. We can easily check the conditions in Echevrría
[6]. (For instance, G is dense in C(R + ), the set of continuous and bounded functions
on R + with limit that exists at infinity and  satisfies the minimum principle, i.e.
Âf(z0) ≥ 0 for any f(z0) = infz∈R + f(z). This is because at minimum, the first
derivative of f vanishes and ˆ λ(z0)(f(z0 + a) − f(z0)) ≥ 0. The other conditions in
Echeverría [6] can also be easily verified.) Thus, Echevrría [6] implies that ˆπ is an
invariant measure. Now, our proof in Lemma 1 shows that ˆπ has to be unique as
well. Therefore, ( ˆ λ, ˆπ) ∈ Qe. This implies that if ( ˆ λ, ˆπ) ∈ Q\Qe, there exists some
g ∈ G, such that Âgˆπ = 0. Now, any constant multiplier of g still belongs to G
and thus infg∈G Âgˆπ = −∞ and hence inff∈F Âf ˆπ = −∞ if ( λ, ˆ ˆπ) ∈ Q\Qe.
Therefore, we have
1
lim inf
t→∞ t log E[eθNt ] ≥ sup
( ˆ
θb
inf
f∈F
λ,ˆπ)∈Q a zˆπ − ˆ H( ˆ
(3.21)
λ, ˆπ) + Âf ˆπ
θb
≥ sup inf
f∈F a zˆπ − ˆ H( ˆ
(3.22)
λ, ˆπ) + Âf ˆπ ,
( ˆ λˆπ,ˆπ)∈R
where R = {( ˆ λˆπ, ˆπ) : ( ˆ λ, ˆπ) ∈ Q} and
(3.23)
H( ˆ λ, ˆ ˆπ) = (λ − ˆ
λ) + log ˆλ/λ ˆλ ˆπ.
Define
(3.24)
F ( ˆ
θb
λˆπ, ˆπ, f) =
a zˆπ − ˆ H( ˆ
λ, ˆπ) + Âf ˆπ
θb
=
a zˆπ − ˆ H( ˆ
λ, ˆπ) − bz ∂f
ˆπ + (f(z + a) − f(z))
∂z ˆ λˆπ.

8 LINGJIONG ZHU
Notice that F is linear in f and hence convex in f and also
(3.25) H( ˆ λ, ˆ ˆπ) = sup
f∈Cb(R + )
ˆλf
f
+ λ(1 − e ) ˆπ ,
where Cb(R + ) denotes the set of bounded functions on R + . Inside the bracket
above, it is linear in both ˆπ and ˆ λˆπ. Hence ˆ H is weakly lower semicontinuous
and convex in ( ˆ λˆπ, ˆπ). Therefore, F is concave in ( ˆ λˆπ, ˆπ). Furthermore, for any
f = Kz + g + L ∈ F,
(3.26)
F ( ˆ
θ
λˆπ, ˆπ, f) = − K bzˆπ −
a ˆ H( ˆ
λ, ˆπ) − bz ∂g
∂z ˆπ
+ (g(z + a) − g(z)) ˆ
λˆπ + Ka ˆλˆπ.
If λnπn → γ∞ and πn → π∞ weakly, then, since g is C1 with compact support, we
have
(3.27)
− bz ∂g
∂z πn
+ (g(z + a) − g(z))λnπn + Ka
→ − bz ∂g
∂z π∞
+ (g(z + a) − g(z))γ∞ + Ka
λnπn
as n → ∞. Moreover, in general, if Pn → P weakly, then, for any f which is upper
semicontinuous and bounded from above, we have lim supn fdPn ≤
fdP . Since
θ
a − K bz is continuous and nonpositive on R + , we have
θ
θ
(3.28) lim sup − K bzπn ≤ − K bzπ∞.
n→∞ a a
Hence, we conclude that F is upper semicontinuous in the weak topology.
In order to switch the supremum and infimum in (3.22), since we have already
proved that F is concave, upper semicontinuous in ( ˆ λˆπ, ˆπ) and convex in f, it is
sufficient to prove the compactness of R to apply Ky Fan’s minmax theorem (see
Fan [7]). Indeed, Joó developed some level set method and proved that it is sufficient
to show the compactness of the level set (see Joó [13] and Frenk and Kassay [8]).
In other words, it suffices to prove that, for any C ∈ R and f ∈ F, the level set
(3.29)
( ˆ λˆπ, ˆπ) ∈ R : ˆ
H +
γ∞,
bz ∂f θb
ˆπ −
∂z a zˆπ − ˆ
λ[f(z + a) − f(z)]ˆπ ≤ C
is compact.
Fix any f = Kz + g + L ∈ F, where K > θ
a and g is C1 with compact support
and L is some constant, uniformly for any pair ( ˆ λˆπ, ˆπ) that is in the level set of

LDP FOR MARKOVIAN NONLINEAR HAWKES PROCESSES 9
(3.29), there exists some C1, C2 > 0 such that
C1 ≥ ˆ
H + K − θ
(3.30)
b zˆπ − C2
ˆλˆπ
a
≥ λ −
ˆλ≥cz+ℓ
ˆ λ + ˆ λ log( ˆ
λ/λ) ˆπ + K − θ
b zˆπ
a
− C2
ˆλˆπ − C2
ˆλˆπ
ˆλ≥cz+ℓ
ˆλ 0, where we used the fact that limz→∞ λ(z)
z
and minz λ(z) > 0. Hence, we proved that
(3.31)
zˆπ ≤ C3,
where
ˆλ≥cz+ℓ
C1 + ℓC2
(3.32) C3 =
−c · C2 + K − θ
, C4 =
a b
Therefore, we have
(3.33)
ˆλˆπ =
and hence
(3.34)
(3.35) lim
ℓ→∞ sup
n
ˆλ≥cz+ℓ
ˆλˆπ ≤ C4,
C1 + ℓC2
minz≥0 log cz+ℓ
λ(z) − 1 − C2
ˆλˆπ + ˆλˆπ ≤ C4 + c · C3 + ℓ,
ˆλ

10 LINGJIONG ZHU
To prove (ii), notice that (λ − λn) + λn log(λn/λ) ≥ 0. That is because x − 1 −
log x ≥ 0 for any x > 0 and hence
(3.37) λ − ˆ λ + ˆ λ log( ˆ λ/λ) = ˆ
λ (λ/ ˆ λ) − 1 − log(λ/ ˆ
λ) ≥ 0.
Notice that
(3.38)
lim
ℓ→∞ sup
λnπn ≤ lim sup
n z≥ℓ
ℓ→∞ n
+ lim
ℓ→∞ sup
n
λn< √ λnπn
λz,z≥ℓ
λn≥ √ λz,z≥ℓ
For the first term, since supn zπn < ∞ and limz→∞ λ(z)
z
(3.39) lim
ℓ→∞ sup
n
λn< √ λnπn ≤ lim sup
λz,z≥ℓ
ℓ→∞ n
For the second term, since lim supz→∞
(3.40)
lim
ℓ→∞ sup
n
λn≥ √ λnπn
λz,z≥ℓ
λ(z)
z
= 0,
≤ lim sup ˆH(λn, πn) sup
ℓ→∞ n
λn≥ √ λz,z≥ℓ
z≥ℓ
λnπn.
= 0,
√ λzπn = 0.
λn
= 0.
λ − λn + λn log(λn/λ)
Therefore, passing to some subsequence if necessary, we have λnπn → γ∞ and
πn → π∞ weakly. Since we proved that F is upper semicontinuous in the weak
topology, hence the level set is compact in the weak topology. Therefore, we can
switch the supremum and infimum in (3.22) and get
(3.41)
(3.42)
(3.43)
(3.44)
(3.45)
(3.46)
lim inf
t→∞
≥ inf
f∈F
= inf
f∈F
= inf
f∈F sup
1
t log E e θNt
sup
sup
ˆπ: R zˆπ

LDP FOR MARKOVIAN NONLINEAR HAWKES PROCESSES 11
whose supremum is achieved at some finite z∗ > 0 since limz→∞ λ(z)
z
= 0, K > θ
a
and g ∈ C 1 with compact support. Hence zˆπ < ∞ is satisified for the optimal ˆπ.
This gives us (3.44). Finally, for any f ∈ F, u = e f ∈ Uθ, which implies (3.46).
Now, we are ready to prove the large deviations result.
Theorem 3. Assume limz→∞ λ(z)
z = 0 and λ(·) is bounded below by some positive
constant. Then, ( Nt
t ∈ ·) satisfies the large deviation principle with the rate function
I(·) as the Fenchel-Legendre transform of Γ(·),
(3.48) I(x) = sup {θx − Γ(θ)} .
θ∈R
λ(z)
Proof. If lim supz→∞ z = 0, then the forthcoming Lemma 5 implies that Γ(θ) <
∞ for any θ. Thus, by Gärtner-Ellis Theorem, we have the upper bound. For
Gärtner-Ellis Theorem and a general theory of large deviations, see for example [5].
To prove the lower bound, it suffices to show that for any x > 0, ɛ > 0, we have
1 Nt
(3.49) lim inf log P ∈ Bɛ(x) ≥ − sup{θx
− Γ(θ)},
t→∞ t t θ
where Bɛ(x) denotes the open ball centered at x with radius ɛ. Let ˆ P denote
the tilted probability measure with rate ˆ λ as defined in Theorem 2. By Jensen’s
inequality, we have
(3.50)
1
log P
t
Nt
t
≥ 1
t log
∈ Bɛ(x)
N t
t ∈Bɛ(x)
Nt
= 1
t log ˆ P
t
≥ 1
t log ˆ
Nt
P
t
By ergodic theorem, we get
(3.51) lim inf
t→∞
where Λ(x) is defined as
dP
d ˆ P dˆ P
∈ Bɛ(x)
∈ Bɛ(x) −
+ 1
t log
ˆP Nt
t
1
ˆP
Nt
t ∈ Bɛ(x)
1 1
·
∈ Bɛ(x) t Ê
1 Nt
log P ∈ Bɛ(x) ≥ −Λ(x),
t t
(3.52) Λ(x) = inf
( ˆ λ,ˆπ)∈Qx
e
(λ − ˆ
λ)ˆπ +
where Q x e is defined as
(3.53) Q x e =
( ˆ
λ, ˆπ) ∈ Qe :
log( ˆ λ/λ) ˆ
λˆπ ,
ˆλ(z)ˆπ(dz) = x .
N t
t ∈Bɛ(x)
dP
dˆ P dˆ
P
1 N t
t ∈Bɛ(x) log dˆ P
dP
.

12 LINGJIONG ZHU
Notice that
(3.54)
Γ(θ) = sup
( ˆ
λ,ˆπ)∈Qe
= sup
x
sup
( ˆ λ,ˆπ)∈Q x e
θˆ
λˆπ + ( ˆ
λ − λ)ˆπ −
= sup{θx
− Λ(x)}.
x
θˆ
λˆπ + ( ˆ
λ − λ)ˆπ −
log( ˆ λ/λ) ˆ
λˆπ
log( ˆ λ/λ) ˆ
λˆπ
We prove in Lemma 4 that Λ(x) is convex in x, identifying it as the convex conjugate
of Γ(θ) thus concluding the proof.
Lemma 4. Λ(x) in (3.52) is convex in x.
Proof. Define
(3.55)
Then, we have
H( ˆ λ, ˆ ˆπ) =
(λ − ˆ
λ)ˆπ +
(3.56) Λ(x) = inf
( ˆ λ,ˆπ)∈Q x e
log( ˆ λ/λ) ˆ λˆπ.
ˆH( ˆ λ, ˆπ).
We want to prove that Λ(αx1 + βx2) ≤ αΛ(x1) + βΛ(x2) for any α, β ≥ 0 and
α + β = 1. For any ɛ > 0, we can choose ( ˆ λk, ˆπk) ∈ Qxk e such that ˆ H( ˆ λk, ˆπk) ≤
Λ(xk) + ɛ/2, for k = 1, 2. Set
(3.57) ˆπ3 = αˆπ1 + βˆπ2, ˆ λ3 =
d(αˆπ1)
d(αˆπ1 + βˆπ2) ˆ d(βˆπ2)
λ1 +
d(αˆπ1 + βˆπ2) ˆ λ2.
Then, for any test function f, we have
(3.58)
Â3f ˆπ3 = α Â1f ˆπ1 + β Â2f ˆπ2 = 0,
which implies that ( ˆ λ3, ˆπ3) ∈ Qe. Furthermore,
(3.59)
ˆλ3ˆπ3 = α ˆλ1ˆπ1 + β ˆλ2ˆπ2 = αx1 + βx2.
Therefore, we have ( ˆ λ3, ˆπ3) ∈ Qαx1+βx2 e . Finally, since x log x is a convex function
and if we apply Jensen’s inequality, we get
ˆH( ˆ
λ3, ˆπ3) = (λ − ˆ λ3 − ˆ λ3 log λ) + ˆ λ3 log ˆ
(3.60)
λ3 ˆπ3
≤ (λ − ˆ λ3 − ˆ λ3 log λ) + α dˆπ1 ˆλ1 log
dˆπ3
ˆ λ1 + β dˆπ2 ˆλ2 log
dˆπ3
ˆ
λ2 ˆπ3
Therefore, we conclude that
(3.61)
= α ˆ H( ˆ λ1, ˆπ1) + β ˆ H( ˆ λ2, ˆπ2).
Λ(αx1 + βx2) ≤ ˆ H( ˆ λ3, ˆπ3) ≤ α ˆ H( ˆ λ1, ˆπ1) + β ˆ H( ˆ λ2, ˆπ2) ≤ αΛ(x1) + βΛ(x2) + ɛ.

Lemma 5. If lim supz→∞
(3.62) θ < log
LDP FOR MARKOVIAN NONLINEAR HAWKES PROCESSES 13
λ(z)
bz
< 1
a
b
a lim sup z→∞
we have Γ(θ) < ∞. If lim sup z→∞
, then, for any
− 1 + a λ(z)
· lim sup
b z→∞ z ,
λ(z)
z
λ(z)
z
Proof. For K ≥ θ
a , we have eKz ∈ Uθ and
(3.63)
Define the function
Γ(θ) ≤ inf sup
g∈Uθ
= sup
z≥0
z≥0
−
Ag(z) + θb
a zg(z)
g(z)
bK − θb
a
(3.64) F (K) = −K + lim sup
z→∞
= 0, then Γ(θ) < ∞ for any θ ∈ R.
Kz Ae
≤ sup
z≥0
z + λ(z)(e Ka − 1)
θb
+
eKz
.
λ(z)
bz · (eKa − 1).
a z
Then F (0) = 0, F is convex and F (K) → ∞ as K → ∞ and its minimum is
attained at
(3.65) K ∗ = 1
a log
b
a lim supz→∞ > 0,
and F (K∗ ) < 0. Therefore, Γ(θ) < ∞ for any
λ(z)
θ < −a min −K + lim sup
K>0
z→∞ bz · (eKa (3.66)
− 1)
= log
b
a lim supz→∞ − 1 + a λ(z)
· lim sup
b z→∞ z < K∗a. λ(z)
z
If lim supz→∞ a , we have Γ(θ) < ∞ for any
θ.
λ(z)
z
λ(z)
z = 0, trying eKz ∈ Uθ for any K > θ
4. LDP for Markovian Nonlinear Hawkes Processes with Sum of
Exponentials Exciting Function
Now, let h be a sum of exponentials, i.e. h(t) = d i=1 aie−bit and let
(4.1) Zi(t) =
aie −bi(t−τj) , 1 ≤ i ≤ d,
τj 0.
In particular, h(0) = d i=1 ai > 0. If ai > 0, then Zi(t) ≥ 0 almost surely; if ai < 0,
then Zi(t) ≤ 0 almost surely.

14 LINGJIONG ZHU
Theorem 6. Assume limz→∞ λ(z)
z
1
(4.3) lim
t→∞ t log E[eθNt ] = inf
= 0. Then,
sup
u∈Uθ (z1,...,zd)∈Z
where Z = {(z1, . . . , zd) : aizi ≥ 0, 1 ≤ i ≤ d} and
Au
u +
θ
d
i=1 ai
(4.4) Uθ = u ∈ C1(R d , R + ), u = e f , f ∈ F ,
where
(4.5) F = f = g + θ d i=1 zi
d i=1 ai
+ L, L ∈ R, g ∈ G ,
where
(4.6) G =
d
Proof. Notice that
i=1
d
i=1
Kɛizi + g, K > 0, g is C1 with compact support
(4.7) dZi(t) = −biZi(t)dt + aidNt, 1 ≤ i ≤ d.
Hence, aiNt = Zi(t) − Zi(0) + t
0 biZi(s)ds and
(4.8) E[e θNt
θ
] = E exp
d i=1 Zi(t) − Zi(0)
d i=1 ai
θ
+ d i=1 ai
t
0
Therefore, by Feynman-Kac formula, we obtain the upper bound
(4.9) lim sup
t→∞
1
t log E[eθNt ] ≤ inf
sup
u∈Uθ (z1,...,zd)∈Z
As before, we can obtain the lower bound
(4.10)
lim inf
t→∞
≥ sup
( ˆ λ,ˆπ)∈Qe
≥ sup
Au
u +
bizi
.
d
biZi(s)ds .
i=1
θ
d
i=1 ai
d
i=1
1
t log E[eθNt ]
θˆ λ − λ + ˆ λ − ˆ
λ log ˆλ/λ ˆπ(dz1, . . . , dzd)
inf
( ˆ g∈G
λ,ˆπ)∈Q
= sup
inf
( ˆ f∈F
λ,ˆπ)∈Q
θˆ λ − λ + ˆ λ − ˆ
λ log ˆλ/λ + Âg
ˆπ
,
bizi
θ d i=1 bizi
d i=1 ai
− λ + ˆ λ − ˆ
λ log ˆλ/λ + Âf
ˆπ.
The last line above is by taking f = g + L + θ Pd i=1 zi
Pd i=1 ai
∈ F for g ∈ G, where
(4.11) G =
d
i=1
Kɛizi + g, K > 0, g is C1 with compact support
Here, ɛi = ai/|ai|, 1 ≤ i ≤ d. Define
(4.12) F ( ˆ
λˆπ, ˆπ, f) =
θ d i=1 bizi
d i=1 ai
+ Âf
ˆπ − ˆ H( ˆ λ, ˆπ).
.
.

LDP FOR MARKOVIAN NONLINEAR HAWKES PROCESSES 15
F is linear in f and hence convex in f. Also ˆ H is weakly lower semicontinuous
and convex in ( ˆ λˆπ, ˆπ). Therefore, F is concave in ( ˆ λˆπ, ˆπ). Furthermore, for any
f = θ Pd i=1 zi
Pd i=1 ai
+ d i=1 Kɛizi + g + L ∈ F,
(4.13) F ( ˆ
λˆπ, ˆπ, f) =
θ +
d
d
Kɛiai
ˆλˆπ − Kɛibiziˆπ − ˆ H( ˆ
λ, ˆπ) +
i=1
i=1
Âgˆπ.
If λnπn → γ∞ and πn → π∞ weakly, then, since g is C1 with compact support, we
have
(4.14)
d
θ + Kɛiai λnπn + Âgπn →
d
θ + Kɛiai γ∞ + Âgπ∞.
i=1
Since − d i=1 Kɛibizi is continuous and nonpositive on Z, we have
(4.15) lim sup
n→∞
d
− Kɛibizi πn ≤
d
− Kɛibizi π∞.
i=1
Hence, we conclude that F is upper semicontinuous in the weak topology.
In order to apply the minmax theorem, we want to prove the compactness of the
level set in the weak topology
(4.16) ( ˆ
λˆπ, ˆπ) :
− θ d i=1 bizi
d i=1 ai
− Âf
ˆπ + ˆ H( ˆ
λ, ˆπ) ≤ C .
For any f = θ P d
i=1 zi
P d
i=1 ai
i=1
+ d
i=1 Kɛizi + g + L ∈ F, where g is C1 with compact
support etc., there exist some C1, C2 > 0 such that
(4.17)
C1 ≥ ˆ d
H + Kbiɛi ziˆπ − C2
ˆλˆπ
≥
− C2
i=1
ˆλ≥ Pd i=1 cizi+ℓ
ˆλ≥ P d
i=1 cizi+ℓ
λ − ˆ λ + ˆ λ log( ˆ
λ/λ) ˆπ +
ˆλˆπ − C2
ˆλ< P d
i=1 cizi+ℓ
i=1
d
Kbiɛi ziˆπ
i=1
≥ min
(z1,...,zd)∈Z log c1z1
+ · · · + cdzd + ℓ
− 1 − C2
λ(z1 + · · · + zd)
d
+ [−ci · C2 + Kbiɛi] ziˆπ − ℓC2.
i=1
ˆλˆπ
ˆλ≥ P d
i=1 cizi+ℓ
If ai > 0, then ɛi > 0, pick up ci > 0 such that −ci · C2 + Kbiɛi > 0. If ai < 0, then
ɛi < 0, pick up ci such that −ci · C2 + Kbiɛi < 0. Finally, choose ℓ big enough such
that the big bracket above is positive. Therefore, we have
(4.18)
|zi|ˆπ ≤ C3,
ˆλ≥ P d
i=1 cizi+ℓ
ˆλˆπ ≤ C4.
ˆλˆπ

16 LINGJIONG ZHU
Hence, λˆπ ˆ ≤ C5 and ˆ H ≤ C6. We can use the similar method as in the proof of
Theorem 2 to show that
(4.19) lim sup
ℓ→∞ n
λnπn = 0,
|zi|>ℓ
1 ≤ i ≤ d.
For any (λnπn, πn) ∈ R, we can find a subsequence that converges in the weak
topology by Prokhorov’s Theorem. Therefore,
(4.20)
1
lim inf
t→∞ t log E[eθNt ]
≥ sup
( ˆ
inf
f∈F
λ,ˆπ)∈Q
θ d i=1 bizi
d i=1 ai
− λ + ˆ λ − ˆ
λ log ˆλ/λ + Âf
ˆπ
= inf
f∈F sup
sup
ˆπ ˆλ
θ d i=1 bizi
d i=1 ai
− λ + ˆ λ − ˆ
λ log ˆλ/λ + Âf
ˆπ
= inf
sup
f∈F (z1,...,zd)∈Z
≥ inf
sup
u∈Uθ (z1,...,zd)∈Z
θ d
i=1 bizi
d
i=1 ai
Au
u +
+ λ(e f(z1+a1,...,zd+ad)−f(z1,...,zd) − 1) −
θ
d
i=1 ai
d
i=1
bizi
That is because optimizing over ˆ λ, we get ˆ λ = λe f(z1+a1,...,zd+ad)−f(z1,...,zd) and
finally for each f ∈ F, u = e f ∈ Uθ.
Theorem 7. Assume limz→∞ λ(z)
z = 0 and λ(·) is bounded below by some positive
constant. Then, ( Nt
t ∈ ·) satisfies the large deviation principle with the rate function
I(·) as the Fenchel-Legendre transform of Γ(·),
(4.21) I(x) = sup {θx − Γ(θ)} ,
θ∈R
where
(4.22) Γ(θ) = sup
( ˆ λ,ˆπ)∈Qe
.
θˆ λ − λ + ˆ λ − ˆ
λ log ˆλ/λ ˆπ.
Proof. The proof is the same as in the case of exponential h(·).
5. LDP for Linear Hawkes Processes: An Alternative Proof
In this section, we use our method to recover the result proved in Bordenave and
Torrisi [1]. We prove the existence of the limit of logarithmic moment generating
function first. The strategy is to use the tilting method to prove the lower bound.
That requires an ergodic lemma, which we state as Lemma 8. For the upper bound,
we can opitimize over a special class of testing functions for the linear rate with
sum of exponential exciting function hn. Any continuous and integrable h can
be approximated by a sequence hn. By a coupling argument, we can use that
to approximate the upper bound for the logarithmic moment generating function
when the exciting function is h. Finally, by tilting argument for the lower bound
and Gärtner-Ellis theorem for the upper bound, we can prove the large deviations
for the linear Hawkes processes.
d
i=1
bizi
∂f
∂zi

LDP FOR MARKOVIAN NONLINEAR HAWKES PROCESSES 17
Lemma 8. Assume λ(z) = α + βz and µ = ∞
h(t)dt < ∞. If βµ < 1, then there
0
exists a stationary and ergodic probability measure π for Zt and zπ = αµ
1−βµ .
Proof. The ergodicity is a well known result for linear Hawkes process. (See Hawkes
and Oakes [11].) Let π be the invariant probability measure for Zt, then
Nt
(5.1) lim
t→∞ t =
λ(z)π(dz) = α + β zπ(dz).
If Zt is invariant in t, taking expectations to Zt = t
h(t − s)dNs,
−∞
t
(5.2) E[Zt] = zπ(dz) = λ(z)π(dz) h(t − s)ds = µ λ(z)π(dz),
which implies that
−∞
zπ = αµ
1−βµ .
Remark 9. In Lemma 8, we assumed that λ(z) = α + βz and βh L 1 < 1. However,
when do the LDP for linear Hawkes process and when we prove Theorem 11,
we assume that λ(z) = ν+z since λ(z) = ν+βz is equivalent to the case λ(z) = ν+z
if we change h(·) to βh(·). The reason we used λ(z) = α+βz in Lemma 8 is because
we need to use that when we tilt λ(z) = ν + z to Kλ(z) = Kν + Kz in the proof of
lower bound in Theorem 11.
Lemma 10. If h(t) > 0, ∞
h(t)dt < ∞, h(∞) = 0 and h is continuous, then h
0
can be approximated by a sum of exponentials both in L1 and L∞ norms.
Proof. The Stone-Weierstrass Theorem says that if X is a compact Hausdorff space
and suppose A is a subspace of C(X) with the following properties. (i) If f, g ∈ A,
then f × g ∈ A. (ii) 1 ∈ A. (iii) If x, y ∈ X then we can find an f ∈ A such that
f(x) = f(y). Then A is dense in C(X) in L∞ norm. Consider X = R + ∪ {∞} =
[0, ∞] and C[0, ∞] consists of continuous functions vanishing at ∞ and the constant
function 1.
By Stone-Weierstrass Theorem, the linear combination of 1, e−t , e−2t etc. is
dense in C[0, ∞]. In other words, for any continuous function h on C[0, ∞], we
have
n
(5.3) sup
h(t)
− aje
t≥0 −jt
≤ ɛ.
j=0
In fact, since h(∞) = 0, we get |a0| ≤ ɛ. Thus
n
(5.4) sup
h(t)
− aje
t≥0 −jt
≤ 2ɛ.
j=1
However, n
j=1 aje −jt may not be positive. We can approximate h(t) first by
a sum of exponentials and then approximate h(t) by the square of that sum of
exponentials, which is again a sum of exponentials but positive this time.
Indeed, we can approximate h(t) by the sum of exponentials in L 1 norm as well.
Suppose h − hnL ∞ → 0, where hn is a sum of exponentials. Then, by Dominated
Convergence Theorem, for any δ > 0, |h − hn|e −δt dt → 0 as n → ∞. Thus, we
can find a sequence δn > 0 such that δn → 0 as n → ∞ and |h − hn|e −δnt dt → 0.
By Dominated Convergence Theorem again, h(1 − e −δnt )dt → 0. Hence, we have
|h − hne −δnt |dt → 0 as n → ∞, where hne −δnt is a sum of exponentials.

18 LINGJIONG ZHU
We will show that hne −δnt converges to h in L ∞ as well.
(5.5) h − hne −δnt L ∞ ≤ h − hnL ∞ + hn − hne −δnt L ∞.
Notice that (1−e −δnt )hn ≤ (1−e −δnt )(h(t)+ɛ). Since h(∞) = 0, there exists some
M > 0, such that for t > M, h(t) ≤ ɛ so that (1 − e −δnt )hn ≤ 2ɛ for t > M. For
t ≤ M, (1 − e −δnt )hn ≤ (1 − e −δnM )(hL ∞ + ɛ) which is small if δn is small.
Theorem 11. Assume λ(z) = ν + z, ν > 0. h(·) satisfies the assumptions in
Lemma 10 and ∞
h(t)dt < 1. We have
0
1
(5.6) lim
t→∞ t log E[eθNt ] = ν(x − 1),
where x is the minimal solution to x = e θ+µ(x−1) , where µ = ∞
0 h(t)dt.
Proof. By Lemma 8, we have
(5.7)
lim inf
t→∞
1
t log E[eθNt ] ≥ sup
( ˆ λ,ˆπ)∈Qe
≥ sup
(Kλ,ˆπ)∈Qe,K∈R +
≥ sup
θˆ λ + ˆ λ − λ − ˆ
λ log ˆλ/λ ˆπ
0

Now, for any N ∈ N,
E e θ PN j=1 Dj(t)
(5.9)
= E
LDP FOR MARKOVIAN NONLINEAR HAWKES PROCESSES 19
e θ P N−1
j=1 Dj(t) e (eθ −1) R t
0 λ
≤ E e θ PN−2 j=1 Dj(t) e ((eθ−1)hn+hɛL1 +θ)DN−1(t)
≤ · · · · · ·
≤ E e
≤ E
θD1(t)+fN−1(θ)D2(t)
e θD1(t)+(ef N−1 (θ) −1)hɛ L 1 D1(t)
,
„
P
τ∈ SN−1
i=1 Di ,τ

20 LINGJIONG ZHU
function I(x) given by
(5.12) I(x) =
x x log ν+xµ − x + µx + ν if x ∈ [0, ∞)
.
+∞ otherwise
Proof. For the upper bound, apply Gärtner-Ellis theorem. For the lower bound, use
the tilting method and identify I(x) as the Fenchel-Legendre transform of Γ(θ).
Remark 13. In Bordenave and Torrisi [1], their I(x) has the form
xθx + ν −
(5.13) I(x) =
νx
ν+µx if x ∈ [0, ∞)
,
+∞ otherwise
where θ = θx is the unique solution in (−∞, µ − 1 − log µ] of E[eθS ] = x
ν+xµ , x > 0.
Here, E[eθS ] satisfies the equation
(5.14) E[e θS ] = e θ exp µ(E[e θS ] − 1) ,
x
which implies that θx = log ν+xµ
x − µ ν+xµ − 1 . Substituting into the formula,
their rate function is the same as what we got.
6. LDP for a Special Class of Nonlinear Hawkes Processes: An
Approximation Approach
In this section, we prove the large deviation results for (Nt/t ∈ ·) for a very
special class of nonlinear λ(·) and h(·) that satisfies the assumptions in Lemma 10.
Let Pn denotes the probability measure under which Nt follows the Hawkes
process with exciting function hn = n
i=1 aie −bit such that hn → h as n → ∞ in
both L 1 and L ∞ norms. Let us define
1
(6.1) Γn(θ) = lim
t→∞ t log EPn e θNt .
We have the following results.
Lemma 14. For any K > 0 and θ1, θ2 ∈ [−K, K], there exists some constant
C(K) only depending on K such that for any n,
(6.2) |Γn(θ1) − Γn(θ2)| ≤ C(K)|θ1 − θ2|.
Proof. Without loss of generality, assume that θ2 > θ1 such that
(6.3)
where
(6.4) Q ∗ e =
Γn(θ1) ≤ Γn(θ2)
= sup
( ˆ λ,ˆπ)∈Q ∗ e
≤ sup
( ˆ λ,ˆπ)∈Q ∗ e
( ˆ
λ, ˆπ) ∈ Qe :
(θ2 − θ1) ˆ λˆπ + θ1 ˆ λˆπ − ˆ H( ˆ λ, ˆπ)
(θ2 − θ1) ˆ λˆπ + Γn(θ1),
θ1 ˆ λˆπ − ˆ H( ˆ
λ, ˆπ) ≥ Γn(θ1) − 1 .

LDP FOR MARKOVIAN NONLINEAR HAWKES PROCESSES 21
The key is to prove that sup ( ˆ λ,ˆπ)∈Q ∗ e
ˆλˆπ ≤ C(K) for some constant C(K) > 0
only depending on K. Define u(z1, . . . , zn) = e P n
i=1 cizi , where
(6.5) ci = 3K
n ai
i=1 bi
Define V = − Au
u
such that
(6.6) V (z1, . . . , zn) = 3K
n ai
i=1 bi
· 1
, 1 ≤ i ≤ n.
bi
n
zi − λ(z1 + · · · + zn)(e 3K − 1).
i=1
Notice that Âf ˆπ = 0 for any test function f with certain regularities. If we try
f = zi
, 1 ≤ i ≤ n, we get
bi
(6.7) − ziˆπ + ai
ˆλˆπ = 0, 1 ≤ i ≤ n.
Summing over 1 ≤ i ≤ n, we get
(6.8)
ˆλˆπ
1
= n ai
i=1 bi
Notice that n
bi
n
ziˆπ.
ai
i=1 bi = hnL1 which is approximately hL1 when n is large. Since
λ(z)
i=1
i=1
lim supz→∞ z = 0 and n i=1 zi ≥ 0, we have
(6.9) θ1
ˆλˆπ ≤ K ˆλˆπ
K n
= n ziˆπ ≤ ai
i=1 bi
1
2
V ˆπ + C 1/2(K),
where C 1/2(K) is some positive constant only depending on K.
We claim that V (z)ˆπ ≤ ˆ H(ˆπ) for any ˆπ ∈ Q ∗ e. Let us prove it. By ergodic
theorem and Jensen inequality, we have
(6.10)
V (z)ˆπ = lim E
t→∞ ˆπ
1
t
t
0
1
V (Zs)ds ≤ lim sup log Eπ
t→∞ t
e R t
0
V (Zs)ds
+ ˆ H(ˆπ).
Next, we will show that u ≥ 1. That is equivalent to proving that n zi
i=1 bi
Consider the process
n Zi(t)
(6.11) Yt = = n ai
e −bi(t−τj) =
g(t − τj),
i=1
bi
bi
τj

22 LINGJIONG ZHU
Notice that
(6.14) −∞ < Γn(θ1) − 1 ≤ θ1
Hence, we have
(6.15) Γn(θ1) − 1 + 1
2 ˆ H ≤ θ1
ˆλˆπ − ˆ H ≤ Γn(θ1) < ∞.
ˆλˆπ − 1
2 ˆ H ≤ C 1/2(K),
which implies that ˆ H ≤ 2(C1/2(K) − Γn(θ1) + 1). Hence,
(6.16) ˆλˆπ ≤ 1
2K
V ˆπ + 1
K C1/2(K) ≤ 1
K (C1/2(K)−Γn(θ1)+1)+ 1
K C1/2(K). Finally, notice that since hn → h in both L 1 and L ∞ norms, we can find a function
g such that sup n hn ≤ g and g L 1 < ∞. and thus
(6.17) Γn(θ1) ≥ Γn(−K) ≥ Γg(−K),
where Γg denotes the case when the rate function is still λ(·) but the exciting
function is g(·) (instead of hn(·)). Notice that here gL1 < ∞ but it may not
be less than 1. It is still well defined because of the assumption limz→∞ λ(z)
z = 0.
Indeed, we can find λ(z) = νɛ + ɛz that dominates the original λ(·) for νɛ > 0
big enough and ɛ > 0 is small enough such that ɛgL1 < 1. Now, we have
Γg(−K) ≥ Γνɛ ɛg(−K) which is finite (see Theorem 11), where Γνɛ ɛg(−K) corresponds
to the case when λ(z) = νɛ + ɛz. Hence, we conclude that
ˆλˆπ ≤ C(K),
(6.18) sup
( ˆ λ,ˆπ)∈Q ∗ e
for some C(K) > 0 only depending on K.
Lemma 15. Assume that λ(·) ≥ c for some c > 0, limz→∞ λ(z)
z = 0 and λ(·)α is
Lipschitz with constant Lα for any α ≥ 1. For any K > 0, Γn(θ) is Cauchy with θ
uniformly in [−K, K].
Proof. Let us write Hn(t) =
τj 1, 1
p
get
(6.20)
E Pm [e θNt ] = E Pn
= E Pn
≤ E Pn
dPm θNt
e
dPn
e θNt−R t
0 (λ(Hm(s))−λ(Hn(s)))ds−R t
0
λ(Hn(s))
log( λ(Hm(s)))dNs
e pθNt−p R t
0 (λ(Hm(s))−λ(Hn(s)))ds 1/p
E Pn
e q R t
0
+ 1
q
ds
= 1, we
log( λ(Hm(s))
λ(Hn(s)) )dNs
1/q
.

LDP FOR MARKOVIAN NONLINEAR HAWKES PROCESSES 23
By Cauchy-Schwarz inequality, we get
(6.21)
E Pn
e q R t
0
1/q
λ(Hm(s))
log( λ(Hn(s)) )dNs ≤ E Pn
e
≤ E Pn
≤ E Pn
„ «
R t λ(Hm(s))
2q
0 λ(Hn(s)) 2q−1 −λ(Hn(s)) ds
e 1
c2q−1 L2q
1
2q
R t P
1
0 τ 1. Letting p ↓ 1, we get the desired result.
Remark 16. If λ(·) ≥ c > 0 and limz→∞ λ(z)
zα = 0 for any α > 0, then, λ(·) σ is
Lipschitz for any σ ≥ 1. For instance, λ(z) = [log(z + c)] β satisfies the conditions,
where β > 0 and c > 1.
Theorem 17. Assume that λ(·) ≥ c for some c > 0, limz→∞ λ(z)
z = 0 and λ(·)α is
Lipschitz with constant Lα for any α ≥ 1. We have limt→∞ 1
t log E[eθNt ] exists for
any θ ∈ R and
1
(6.25) lim
t→∞ t log E[eθNt ] = Γ(θ),
where Γ(θ) = limn→∞ Γn(θ).
Proof. By Lemma 15, we have that Γ(θ) = limn→∞ Γn(θ) exists and Γn(θ) converges
to the limit uniformly on any compact set [−K, K]. Since Γn(θ) is Lipschitz
by Lemma 14, it is continuous and the limit Γ is also continuous. Let
ɛn = hn − h L 1 ≤ ɛ. As in the proof of Lemma 15, for any θ ∈ [−K, K], p, q > 1,

24 LINGJIONG ZHU
1 1
p + q = 1, we get
(6.26)
lim sup
t→∞
1
t log E[eθNt ]
≤ Γn(θ) + C(K)L1ɛn + C(K)
2q
· L2qɛn
+ 2 1 −
c2q−1 1
C(K)K.
p
Letting n → ∞ first and then p ↓ 1, we get lim sup t→∞ 1
t log E[eθNt ] ≤ Γ(θ).
Similarly, for any p ′ , q ′ > 1, 1
p ′ + 1
q ′ = 1,
(6.27)
Γn(θ) ≤ lim inf
t→∞
≤ lim inf
t→∞
+ lim inf
t→∞
1
pt log E[e(pθ+pL1ɛn)Nt ] + lim inf
t→∞
1
pp ′ t log E[epp′ θNt ] + lim inf
t→∞
1
log E
2qt
e L2q ɛn
c2q−1 Nt
.
1
log E
2qt
e L2q ɛn
c2q−1 Nt
1
pq ′ t log E[eq′ pL1ɛnNt ]
Since we can dominate λ(·) by the linear case λ(z) = ν +z and under linear case the
limit of logarithmic moment generating function Γν(θ) is continuous in θ, letting
n → ∞, we have
(6.28) Γ(θ) ≤ lim inf
t→∞
This holds for any θ and thus
(6.29) lim inf
t→∞
1
pp ′ t log E[epp′ θNt ].
1
t log E[eθNt ] ≥ pp ′
θ
Γ
pp ′
.
Letting p, p ′ ↓ 1 and using the continuity of Γ(·), we get the desired result.
Theorem 18. Assume that λ(·) ≥ c for some c > 0, limz→∞ λ(z)
z = 0 and λ(·)α
is Lipschitz with constant Lα for any α ≥ 1. We have (Nt/t ∈ ·) satisfies the large
deviation principle with the rate function
(6.30) I(x) = sup{θx
− Γ(θ)}.
θ∈R
Proof. For the upper bound, apply Gärtner-Ellis Theorem. Let us prove the lower
bound. Let Bɛ(x) denote the open ball centered at x with radius ɛ > 0. By Hölder’s
inequality, for any p, q > 1, 1 1
p + q = 1, we have
1/q Nt
dPn
Nt
(6.31) Pn ∈ Bɛ(x) ≤
t P ∈ Bɛ(x) .
dP
t
Therefore, letting t → ∞, we have
(6.32)
1
sup{θx
− Γn(θ)} = lim
θ∈R
t→∞ t
≤ 1
pp ′ Γ(pp′ L1ɛn) + 1
Γ
2pq ′
log Pn
L p (P)
Nt
L2pq ′ɛn
c 2pq′ −1
∈ Bɛ(x)
t
+ 1
1 Nt
lim inf log P
q t→∞ t t
where ɛn = hn − hL1. Hence, letting n → ∞, we have
1 1 Nt
(6.33) lim inf log P ∈ Bɛ(x) ≥ lim sup sup{θx
− Γn(θ)}.
q t→∞ t t n→∞
θ∈R
∈ Bɛ(x) ,

LDP FOR MARKOVIAN NONLINEAR HAWKES PROCESSES 25
We know that Γn(θ) → Γ(θ) uniformly on any compact set K, thus
(6.34) sup{θx
− Γn(θ)} → sup{θx
− Γ(θ)},
θ∈K
θ∈K
as n → ∞ for any compact set K. Notice that λ(·) ≥ c > 0 and recall that the
limit for the logarithmic moment generating function with parameter θ for Poisson
process with constant rate c is (e θ − 1)c. Hence
Γn(θ) (e
(6.35) lim inf ≥ lim inf
θ→+∞ θ θ→+∞
θ − 1)c
= +∞,
θ
which implies that supθ∈R{θx − Γn(θ)} → supθ∈R{θx − Γ(θ)}. Therefore,
1 1 Nt
(6.36)
lim inf log P ∈ Bɛ(x) ≥ sup{θx
− Γ(θ)}.
q t→∞ t t θ∈R
Letting q ↓ 1, we get the desired result.
Remark 19. The class of nonlinear Hawkes process with general exciting function
h for which we proved the large deviation principle here is unfortunately a big too
special. It works for the rate function like λ(z) = [log(c + z)] β for example but does
not work for λ(·) that has sublinear polynomial growth. In fact, by the coupling
argument we used in the proof of the case of linear λ(·) in Theorem 11, we can prove
that in the case when limz→∞ λ(z)
z = 0 and λ(·) is α-Lipshcitz and λ(·) ≥ c > 0,
Γ(θ) = limn→∞ Γn(θ) for θ ≤ µ − 1 − log µ, where µ = ∞
0 h(t)dt and Γ and Γn are
the limit of logarithmic moment generating functions when the exciting functions
are h and hn respectively and hn → h in L1 . For the linear case, since Γ(θ) = ∞
for θ ≥ µ − 1 − log µ, the coupling argument is good enough. However, for the
sublinear λ(·), Γ(θ) < ∞ for any θ and the coupling argument is not enough. In
fact, it will appear in Zhu [20] that under the condition that limz→∞ λ(z)
z = 0, λ(·)
is positive, increasing, α-Lipshcitz and λ(·) ≥ c > 0 and h(·) is positive, decreasing
and ∞
h(t)dt < ∞, there is a level-3 large deviation principle from which we can
0
use the contraction principle to get the level-1 large deviation principle for (Nt/t ∈
·). Therefore, we conjecture that in the sublinear case, Γ(θ) = limn→∞ Γn(θ) for
any θ and (Nt/t ∈ ·) satisfies the large deviation principle with rate function I(x) =
supθ∈R{θx − Γ(θ)}. The advantage of approximating the general case by the case
when h is a sum of exponentials is that Γn(θ) can be evaluated as an optimization
problem, which should be computable by some numerical schemes.
Acknowledgements
The author is enormously grateful to his advisor Professor S. R. S. Varadhan
for suggesting this topic and for his superb guidance, understanding, patience, and
generosity. He would also like to thank his colleague Dmytro Karabash for valuable
discussions on this project. The author would also thank an anonymous referee who
provided very helpful suggestions for the improvement of this paper. to whom the
author is much indebted to. The author is supported by NSF grant DMS-0904701,
DARPA grant and MacCracken Fellowship at New York University.
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Courant Institute of Mathematical Sciences
New York University
251 Mercer Street
New York, NY-10012
United States of America
E-mail address: ling@cims.nyu.edu