Amel Bentata

Rama Cont

Short-time asymptotics for marginal distributions of semimartingales

We study the short-time aymptotics of conditional expectations of smooth and non-smooth functions of a (discontinuous) Ito semimartingale; we compute the leading term in the asymptotics in terms of the local characteristics of the semimartingale. We derive in particular the asymptotic behavior of call options with short maturity in a semimartingale model: whereas the behavior of out-of-the-money options is found to be linear in time, the short time asymptotics of at-the-money options is shown to depend on the fine structure of the semimartingale.

We study the short-time asymptotics of expectations of the form E[f (ξ t )] where ξ t a discontinuous Itô semimartingale. We study two different cases: first, the case where f is a smooth (C 2 ) function, then the case where f (x) = (x -K) + . We compute the leading term in the asymptotics in terms of the local characteristics of the semimartingale.

As an application, we derive the asymptotic behavior of call option prices close to maturity in a semimartingale model: whereas the behavior of out-of-themoney options is found to be linear in time, the short time asymptotics of at-themoney options is shown to depend on the fine structure of the semimartingale.

These results generalize and extend various asymptotic results previously derived for diffusion models [START_REF] Berestycki | Asymptotics and calibration of local volatility models[END_REF][START_REF] Berestycki | Computing the implied volatility in stochastic volatility models[END_REF], Lévy processes [START_REF] Jacod | Asymptotic properties of power variations of Lévy processes[END_REF][START_REF] Figueroa-López | Small-time expansions for the transition distributions of Lévy processes[END_REF][START_REF] Tankov | Pricing and Hedging in Exponential Lévy Models: Review of Recent Results[END_REF], Markov jumpdiffusion models [START_REF] Benhamou | Smart expansion and fast calibration for jump diffusions[END_REF] and one-dimensional martingales [START_REF] Muhle-Karbe | Small-Time Asymptotics of Option Prices and First Absolute Moments[END_REF] to the more general case of a discontinuous semimartingale. In particular, the independence of increments or the Markov property do not play any role in our derivation.

Introduction

In applications such as stochastic control, statistics of processes and mathematical finance, one is often interested in computing or approximating conditional expectations of the type

E [f (ξ t )|F t0 ] , (1)
where ξ is a stochastic process. Whereas for Markov process various wellknown tools -partial differential equations, Monte Carlo simulation, semigroup methods-are available for the computation and approximation of conditional expectations, such tools do not carry over to the more general setting of semimartingales. Even in the Markov case, if the state space is high dimensional exact computations may be computationally prohibitive and there has been a lot of interest in obtaining approximations of (1) as t → t 0 . Knowledge of such short-time asymptotics is very useful not only for computation of conditional expectations but also for the estimation and calibration of such models. Accordingly, short-time asymptotics for (1) (which, in the Markov case, amounts to studying transition densities of the process ξ) has been previously studied for diffusion models [START_REF] Berestycki | Asymptotics and calibration of local volatility models[END_REF][START_REF] Berestycki | Computing the implied volatility in stochastic volatility models[END_REF][START_REF] Feng | Short-maturity asymptotics for a fast mean-reverting Heston stochastic volatility model[END_REF], Lévy processes [START_REF] Jacod | Asymptotic properties of power variations of Lévy processes[END_REF][START_REF] Léandre | Densité en temps petit d'un processus de sauts[END_REF][START_REF] Rüschendorf | Expansion of transition distributions of lévy processes in small time[END_REF][START_REF] Barndorff-Nielsen | Probability measures, Lévy measures and analyticity in time[END_REF][START_REF] Figueroa-López | Small-time expansions for the transition distributions of Lévy processes[END_REF][START_REF] Figueroa-López | The small-maturity smile for exponential Lévy models[END_REF][START_REF] Tankov | Pricing and Hedging in Exponential Lévy Models: Review of Recent Results[END_REF], Markov jump-diffusion models [START_REF] Alòs | On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility[END_REF][START_REF] Benhamou | Smart expansion and fast calibration for jump diffusions[END_REF] and scalar martingales driven by a one-dimensional Poisson random measure [START_REF] Muhle-Karbe | Small-Time Asymptotics of Option Prices and First Absolute Moments[END_REF], using a variety of techniques. The proofs of these results in the case of Lévy processes makes heavy use of the independence of increments; proofs in other case rely on the Markov property, estimates for heat kernels for second-order differential operators or Malliavin calculus. What is striking, however, is the similarity of the results obtained in these different settings.

We reconsider here the short-time asymptotics of conditional expectations in a more general framework which contains existing models but allows to go beyond the Markovian setting and to incorporate path-dependent features. Such a framework is provided by the class of Itô semimartingales, which contains all the examples cited above but allows to use the tools of stochastic analysis.

An Itô semimartingale on a filtered probability space (Ω, F, (F t ) t≥0 , P) is a stochastic process ξ with the representation

ξ t = ξ 0 + t 0 β s ds+ t 0 δ s dW s + t 0 R d κ(y) M (dsdy)+ t 0 R d
(y -κ(y)) M (dsdy),

(2) where ξ 0 is in R d , W is a standard R n -valued Wiener process, M is an integervalued random measure on [0, ∞]×R d with compensator µ(ω, dt, dy) = m(ω, t, dy)dt and M = M -µ its compensated random measure, β (resp. δ) is an adapted process with values in R d (resp. M d×n (R)) and κ(y) = y 1 + y 2 is a truncation function.

We study the short-time asymptotics of conditional expectations of the form (1) where ξ is an Ito semimartingale of the form (2), for various classes of functions f : R d → R. First, we prove a general result for the case of f ∈ C 2 b (R d , R). Then we will treat, when d = 1, the case of

E (ξ t -K) + |F t0 , (3)
which corresponds to the value at t 0 of a call option with strike K and maturity t in a model described by equation [START_REF] Barndorff-Nielsen | Probability measures, Lévy measures and analyticity in time[END_REF]. We show that whereas the behavior of (3) in the case K > ξ t0 ( out-of-the-money options) is linear in t -t 0 , the asymptotics in the case K = ξ t0 (which corresponds to at-the-money options) depends on the fine structure of the semimartingale ξ at t 0 . In particular, we show that for continuous semimartingales the short-maturity asymptotics of atthe-money options is determined by the local time of ξ at t 0 . In each case we identify the leading term in the asymptotics and express this term in terms of the local characteristics of the semimartingale at t 0 .

Our results unify various asymptotic results previously derived for particular examples of stochastic models and extend them to the more general case of a discontinuous semimartingale. In particular, we show that the independence of increments or the Markov property do not play any role in the derivation of such results.

Short-time asymptotics for expectations of the form (1) have been studied in the context of statistics of processes [START_REF] Jacod | Asymptotic properties of power variations of Lévy processes[END_REF] and option pricing [START_REF] Alòs | On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility[END_REF][START_REF] Berestycki | Asymptotics and calibration of local volatility models[END_REF][START_REF] Berestycki | Computing the implied volatility in stochastic volatility models[END_REF][START_REF] Benhamou | Smart expansion and fast calibration for jump diffusions[END_REF][START_REF] Figueroa-López | Small-time expansions for the transition distributions of Lévy processes[END_REF][START_REF] Tankov | Pricing and Hedging in Exponential Lévy Models: Review of Recent Results[END_REF][START_REF] Muhle-Karbe | Small-Time Asymptotics of Option Prices and First Absolute Moments[END_REF]. Berestycki, Busca and Florent [START_REF] Berestycki | Asymptotics and calibration of local volatility models[END_REF][START_REF] Berestycki | Computing the implied volatility in stochastic volatility models[END_REF] derive short maturity asymptotics for call options when ξ t is a diffusion, using analytical methods. Durrleman [START_REF]From implied to spot volatilities[END_REF] studied the asymptotics of implied volatility in a general, non-Markovian stochastic volatility model. Jacod [START_REF] Jacod | Asymptotic properties of power variations of Lévy processes[END_REF] derived asymptotics for (1) for various classes of functions f , when ξ t is a Lévy process. Lopez [START_REF] Figueroa-López | Small-time expansions for the transition distributions of Lévy processes[END_REF] and Tankov [START_REF] Tankov | Pricing and Hedging in Exponential Lévy Models: Review of Recent Results[END_REF] study the asymptotics of (3) when ξ t is the exponential of a Lévy process. Lopez [START_REF] Figueroa-López | Small-time expansions for the transition distributions of Lévy processes[END_REF] also studies short-time asymptotic expansions for [START_REF] Alòs | On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility[END_REF], by iterating the infinitesimal generator of the Lévy process ξ t . Alos et al [START_REF] Alòs | On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility[END_REF] derive short-maturity expansions for call options and implied volatility in a Heston model using Malliavin calculus. Benhamou et al. [START_REF] Benhamou | Smart expansion and fast calibration for jump diffusions[END_REF] derive short-maturity expansions for call options in a model where ξ is the solution of a Markovian SDE whose jumps are described by a compound Poisson process. These results apply to processes with independence of increments or solutions of a "Markovian" stochastic differential equation.

Durrleman studied the convergence of implied volatility to spot volatility in a stochastic volatility model with finite-variation jumps [START_REF] Durrleman | Convergence of at-the-money implied volatilities to the spot volatility[END_REF]. More recently, Nutz and Muhle-Karbe [START_REF] Muhle-Karbe | Small-Time Asymptotics of Option Prices and First Absolute Moments[END_REF] study short-maturity asymptotics for call options in the case where ξ t is a one-dimensional Itô semimartingale driven by a (onedimensional) Poisson random measure whose Lévy measure is absolutely continuous. Their approach consists in "freezing" the characteristic triplet of ξ at t 0 , approximating ξ t by the corresponding Lévy process and using the results cited above [START_REF] Jacod | Asymptotic properties of power variations of Lévy processes[END_REF][START_REF] Figueroa-López | Small-time expansions for the transition distributions of Lévy processes[END_REF] to derive asymptotics for call option prices.

Our contribution is to extend these results to the more general case when ξ is a d-dimensional semimartingale with jumps. In contrast to previous derivations, our approach is purely based on Itô calculus and makes no use of the Markov property or independence of increments. Also, our multidimensional setting allows to treat examples which are not accessible using previous results. For instance, when studying index options in jump-diffusion model, one considers an index I t = w i S i t where (S 1 , ..., S d ) are Itô semimartingales. In this framework, I is indeed an Itô semimartingale whose stochastic integral representation is implied by those of S i but it is naturally represented in terms of a d-dimensional integer-valued random measure, not a one-dimensional Poisson random measure, so the setting of [START_REF] Muhle-Karbe | Small-Time Asymptotics of Option Prices and First Absolute Moments[END_REF] does not apply. Our setting provides a natural framework for treating such examples.

Note that these 'short-time' asymptotics are different from the 'extremestrike' asymptotics studied by Lee [START_REF] Lee | The moment formula for implied volatility at extreme strikes[END_REF] and extended by Friz, Gulisashvili and others [START_REF] Benaim | On black-scholes implied volatility at extreme strikes[END_REF][START_REF] Friz | On refined volatility smile expansion in the Heston model[END_REF][START_REF] Gulisashvili | Asymptotic formulas with error estimates for call pricing functions and the implied volatility at extreme strikes[END_REF][START_REF] Gulisashvili | Asymptotic behavior of distribution densities in models with stochastic volatility[END_REF]. But, in specific models, the two asymptotic regimes may be related using scaling arguments.

2 Short time asymptotics for conditional expectations

Main result

We make the following assumptions on the characteristics of the semimartingale ξ:

Assumption 2.1 (Right-continuity of characteristics at t 0 ). lim t→t0, t>t0 E [ β t -β t0 |F t0 ] = 0, lim t→t0, t>t0 E δ t -δ t0 2 |F t0 = 0,
where . denotes the Euclidean norm on R d and for

ϕ ∈ C b 0 (R d × R d , R), lim t→t0, t>t0 E R d y 2 ϕ(ξ t , y) m(t, dy)|F t0 = R d y 2 ϕ(ξ t0 , y) m(t 0 , dy).
The second requirement, which may be viewed as a weak (right) continuity of m(t, dy) along the paths of ξ, is satisfied for instance if m(t, dy) is absolutely continuous with a density which is right-continuous in t at t 0 . Assumption 2.2 (Integrability condition). ∃T > t 0 ,

E T t0 β s ds F t0 < ∞, E T t0 δ s 2 ds F t0 < ∞, E T t0 R d y 2 m(s, dy) ds F t0 < ∞.
Under these assumptions, the following result describes the asymptotic be-

havior of E [f (ξ t )|F t0 ] when t → t 0 : Theorem 2.1. Under Assumptions 2.1 and 2.2, for all f ∈ C 2 b (R d , R), lim t↓t0 1 t -t 0 (E [f (ξ t )|F t0 ] -f (ξ t0 )) = L t0 f (ξ t0 ). ( 4
)
where L t0 is the (random) integro-differential operator given by

∀f ∈ C 2 b (R d , R), L t0 f (x) = β t0 .∇f (x) + 1 2 tr t δ t0 δ t0 ∇ 2 f (x) (5)
+ R d [f (x + y) -f (x) - 1 1 + y 2 y.∇f (x)]m(t 0 , dy).
Before proving Theorem 2.1, we recall a useful lemma: Lemma 2.1. Let f : R + → R be right-continuous at 0, then

lim t→0 1 t t 0 f (s) ds = f (0). (6)
Proof. Let F denote the primitive of f , then

1 t t 0 f (s) ds = 1 t (F (t) -F (0)) .
Letting t → 0 + , this is nothing but the right derivative at 0 of F , which is f (0) by right continuity of f .

We can now prove Theorem 2.1.

Proof. of Theorem 2.1

We first note that, by replacing P by the conditional measure P |Ft 0 given F t0 , we may replace the conditional expectation in (4) by an expectation with respect to the marginal distribution of ξ t under P |Ft 0 . Thus, without loss of generality, we put t 0 = 0 in the sequel and consider the case where F 0 is the σ-algebra generated by all P-null sets.

Let f ∈ C 2 b (R d , R). Itô's formula yields f (ξ t ) = f (ξ 0 ) + t 0 ∇f (ξ s -)dξ i s + 1 2 t 0 tr ∇ 2 f (ξ s -) t δ s δ s ds + s≤t f (ξ s -+ ∆ξ s ) -f (ξ s -) - d i=1 ∂f ∂x i (ξ s -)∆ξ i s = f (ξ 0 ) + t 0 ∇f (ξ s -).β s ds + t 0 ∇f (ξ s -).δ s dW s + 1 2 t 0 tr ∇ 2 f (ξ s -) t δ s δ s ds + t 0 R d ∇f (ξ s -).κ(y) M (ds dy) + t 0 R d (f (ξ s -+ y) -f (ξ s -) -κ(y).∇f (ξ s -)) M (ds dy).
We note that

• since ∇f is bounded and given Assumption 2.2,

t 0 R d ∇f (ξ s -).κ(y) M (ds dy) is a square-integrable martingale.
• since ∇f is bounded and given Assumption 2.2, t 0 ∇f (ξ s -).δ s dW s is a martingale.

Hence, taking expectations, we obtain

E [f (ξ t )] = E [f (ξ 0 )] + E t 0 ∇f (ξ s -).β s ds + E 1 2 t 0 tr ∇ 2 f (ξ s -) t δ s δ s ds + E t 0 R d (f (ξ s -+ y) -f (ξ s -) -κ(y).∇f (ξ s -)) M (ds dy) = E [f (ξ 0 )] + E t 0 ∇f (ξ s -).β s ds + E 1 2 t 0 tr ∇ 2 f (ξ s -) t δ s δ s ds + E t 0 R d (f (ξ s -+ y) -f (ξ s -) -κ(y).∇f (ξ s -)) m(s, dy) ds ,
that is

E [f (ξ t )] = E [f (ξ 0 )] + E t 0 L s f (ξ s ) ds . ( 7
)
where L denote the integro-differential operator given, for all t ∈ [0, T ] and for

all f ∈ C 2 b (R d , R), by L t f (x) = β t .∇f (x) + 1 2 tr t δ t δ t ∇ 2 f (x) + R d [f (x + y) -f (x) - 1 1 + y 2 y.∇f (x)]m(t, dy), (8)
Equation ( 7) yields

1 t E [f (ξ t )] - 1 t f (ξ 0 ) -L 0 f (ξ 0 ) = E 1 t t 0 ds (∇f (ξ s ).β s -∇f (ξ 0 ).β 0 ) + 1 2 E 1 t t 0 ds tr ∇ 2 f (ξ s ) t δ s δ s -∇ 2 f (ξ 0 ) t δ 0 δ 0 + E R d 1 t t 0 ds m(s, dy) (f (ξ s + y) -f (ξ s ) -κ(y).∇f (ξ s )) -m(0, dy), (f (ξ 0 + y) -f (ξ 0 ) -κ(y).∇f (ξ 0 )) . Define ∆ 1 (t) = E 1 t t 0 ds (∇f (ξ s ).β s -∇f (ξ 0 ).β 0 ) , ∆ 2 (t) = 1 2 E 1 t t 0 ds tr ∇ 2 f (ξ s -) t δ s δ s -∇ 2 f (ξ 0 ) t δ 0 δ 0 , ∆ 3 (t) = E R d 1 t t 0 ds m(s, dy) (f (ξ s + y) -f (ξ s ) -κ(y).∇f (ξ s -)) -m(0, dy) (f (ξ 0 + y) -f (ξ 0 ) -κ(y).∇.f (ξ 0 )) .
Thanks to Assumptions 2.1 and 2.2,

E t 0 ds |∇f (ξ s ).β s -∇f (ξ 0 ).β 0 | ≤ E t 0 ds ∇f ( β s + β 0 ) < ∞.
Fubini's theorem then applies:

∆ 1 (t) = 1 t t 0 ds E [∇f (ξ s ).β s -∇f (ξ 0 ).β 0 ] .
Let us prove that

g 1 : [0, T [ → R t → E [∇f (ξ t ).β t -∇f (ξ 0 ).β 0 ] ,
is right-continuous at 0 with g 1 (0) = 0, yielding ∆ 1 (t) → 0 when t → 0 + if one applies Lemma 2.1.

|g 1 (t)| = |E [∇f (ξ t ).β t -∇f (ξ 0 ).β 0 ]| = |E [(∇f (ξ t ) -∇f (ξ 0 )) .β 0 + ∇f (ξ t ). (β t -β 0 )]| ≤ ∇f ∞ E [ β t -β 0 ] + β 0 ∇ 2 f ∞ E [ ξ t -ξ 0 ] , (9)
where ∞ denotes the supremum norm on

C 2 b (R d , R). Assumption 2.1 implies that: lim t→0 + E [ β t -β 0 ] = 0.
Thanks to Assumption 2.2, one may decompose ξ t as

ξ t = ξ 0 + A t + M t , A t = t 0 β s ds + R d (y -κ(y)) m(s, dy) ds, M t = t 0 δ s dW s + t 0 R d y M (ds dy), (10)
where A t is of finite variation and M t is a local martingale. First, applying Fubini's theorem (using Assumption 2.2),

E [ A t ] ≤ E t 0 β s ds + E t 0 R d y -κ(y) m(s, dy) ds = t 0 ds E [ β s ] + t 0 ds E R d y -κ(y) m(s, dy) . Thanks to Assumption 2.1, one observes that if s ∈ [0, T [→ E [ β s -β 0 ] is right-continuous at 0 so is s ∈ [0, T [→ E [ β s ]. Furthermore, Assumption 2.1 yields that s ∈ [0, T [→ E R d y -κ(y) m(s, dy)
is right-continuous at 0 and Lemma 2.1 implies that lim

t→0 + E [ A t ] = 0. Furthermore, writing M t = (M 1 t , • • • , M d t ), E M t 2 = 1≤i≤d E |M i t | 2 .
Burkholder's inequality [START_REF] Protter | Stochastic integration and differential equations[END_REF]Theorem IV.73] implies that there exists

C > 0 such that sup s∈[0,t] E |M i s | 2 ≤ C E [M i , M i ] t = C E t 0 ds |δ i s | 2 + t 0 ds R d |y i | 2 m(s, dy) .
Using Assumption 2.2 we may apply Fubini's theorem to obtain

sup s∈[0,t] E M s 2 ≤ C 1≤i≤d E t 0 ds |δ i s | 2 + E t 0 ds R d |y i | 2 m(s, dy) = C E t 0 ds δ s 2 + E t 0 ds R d y 2 m(s, dy) = C t 0 ds E δ s 2 + t 0 ds E R d y 2 m(s, dy) .
Thanks to Assumption 2.1, Lemma 2.1 yields

lim t→0 + E M t 2 = 0.
Using the Jensen inequality, one obtains

E [ M t ] = E   1≤i≤d |M i t | 2   ≤ E   1≤i≤d |M i t | 2   = E M t 2 .
Hence, lim

t→0 + E [ M t ] = 0, and lim t→0 + E [ ξ t -ξ 0 ] ≤ lim t→0 + E [ A t ] + lim t→0 + E [ M t ] = 0.
Going back to the inequalities (9), one obtains lim t→0 + g 1 (t) = 0. Similarly, ∆ 2 (t) → 0 and ∆ 3 (t) → 0 as t → 0 + . This ends the proof.

Remark 2.1. In applications where a process is constructed as the solution to a stochastic differential equation driven by a Brownian motion and a Poisson random measure, one usually starts from a representation of the form

ζ t = ζ 0 + t 0 β s ds + t 0 δ s dW s + t 0 ψ s (y) Ñ (ds dy), (11)
where ξ 0 ∈ R d , W is a standard R n -valued Wiener process, β and δ are nonanticipative càdlàg processes, N is a Poisson random measure on [0, T ] × R d with intensity ν(dy) dt where ν is a Lévy measure In particular, if one rewrites Assumption 2.1 in the framework of equation [START_REF]From implied to spot volatilities[END_REF], one recovers the Assumptions of [START_REF] Muhle-Karbe | Small-Time Asymptotics of Option Prices and First Absolute Moments[END_REF] as a special case.

R d 1 ∧ y 2 ν(dy) < ∞, Ñ = N -ν(dy)dt,
and ψ : [0, T ] × Ω × R d → R d is

Examples

If one has further information on the behavior of f in the neighborhood of ξ 0 , then the quantity L 0 f (ξ 0 ) may be computed more explicitly. We summarize some commonly encountered situations in the following Proposition.

Proposition 2.1. Under Assumptions 2.1 and 2.2,

1. If f (ξ 0 ) = 0 and ∇f (ξ 0 ) = 0, then lim t→0 + 1 t E [f (ξ t )] = 1 2 tr t δ 0 δ 0 ∇ 2 f (ξ 0 ) + R d f (ξ 0 + y) m(0, dy). (12) 2. If furthermore ∇ 2 f (ξ 0 ) = 0, then lim t→0 + 1 t E [f (ξ t )] = R d f (ξ 0 + y) m(0, dy). ( 13
)
Proof. Applying Theorem 2.1, L 0 f (ξ 0 ) writes

L 0 f (ξ 0 ) = β 0 .∇f (ξ 0 ) + 1 2 tr ∇ 2 f (ξ 0 ) t δ 0 δ 0 (ξ 0 ) + R d [f (ξ 0 + y) -f (ξ 0 ) - 1 1 + y 2 y.∇f (ξ 0 )]m(0, dy).
The proposition follows immediately.

Remark 2.2. As observed by Jacod [START_REF] Jacod | Asymptotic properties of power variations of Lévy processes[END_REF]Section 5.8] in the setting of Lévy processes, if f (ξ 0 ) = 0 and ∇f

(ξ 0 ) = 0, then f (x) = O( x -ξ 0 2 ). If furthermore ∇ 2 f (ξ 0 ) = 0, then f (x) = o( x -ξ 0 2 ).
Let us now compute in a more explicit manner the asymptotics of (1) for specific semimartingales.

Functions of a Markov process

An important situations which often arises in applications is when a stochastic processe ξ is driven by an underlying Markov process, i.e.

ξ t = f (Z t ) f ∈ C 2 (R d , R), (14)
where Z t is a Markov process, defined as the weak solution on [0, T ] of a stochastic differential equation

Z t = Z 0 + t 0 b(u, Z u-) du + t 0 Σ(u, Z u-) dW u + t 0 ψ(u, Z u-, y) Ñ (du dy), (15)
where (W t ) is an n-dimensional Brownian motion, N is a Poisson random measure on [0, T ] × R d with Lévy measure ν(y) dy, Ñ the associated compensated random measure, Σ :

[0, T ] × R d → M d×d (R), b : [0, T ] × R d → R d and ψ : [0, T ] × R d × R d are measurable functions such that ψ(., ., 0) = 0 ψ(t, z, .) is a C 1 (R d , R d ) -diffeomorphism ∀t ∈ [0, T ], E t 0 { y ≥1} sup z∈R d 1 ∧ ψ(s, z, y) 2 ν(y) dy ds < ∞. (16)
In this setting, as shown in Proposition [6, Proposition 3], one may verify the regularity of Assumption 2.1 and Assumption 2.2 by requiring easy-to-check assumptions on the coefficients:

Assumption 2.3. b(., .), Σ(., .) and ψ(., ., y) are continuous in a neighborhood of (0, Z 0 ) Assumption 2.4. There exist T > 0, R > 0 such that

Either ∀t ∈ [0, T ] inf z-Z0 ≤R inf x∈R d , x =1 t x.Σ(t, z).x > 0 or Σ ≡ 0.
We then obtain the following result:

Proposition 2.2. Let f ∈ C 2 b (R d , R) such that ∀(z 1 , • • • , z d-1 ) ∈ R d-1 , u → f (z 1 , . . . , z d-1 , u) is a C 1 (R, R)-diffeomorphism. Define          β 0 = ∇f (Z 0 ).b(0, Z 0 ) + 1 2 tr ∇ 2 f (Z 0 ) t Σ(0, Z 0 )Σ(0, Z 0 ) + R d (f (Z 0 + ψ(0, Z 0 , y)) -f (Z 0 ) -ψ(0, Z 0 , y).∇f (Z 0 )) ν(y) dy, δ 0 = ∇f (Z 0 )Σ(0, Z 0 ) ,
and the measure m(0, .) via

m(0, [u, ∞[) = R d 1 {f (Z0+ψ(0,Z0,y))-f (Z0)≥u} ν(y) dy u > 0, m(0, [-∞, u]) = R d 1 {f (Z0+ψ(0,Z0,y))-f (Z0)≤u} ν(y) dy u < 0. ( 17
)
Under Assumptions 2.3 and 2.4, ∀g ∈ C 2 b (R d , R), lim t→0 + E [g(ξ t )] -g(ξ 0 ) t = β 0 g (ξ 0 )+ δ 2 0 2 g (ξ 0 )+ R d [g(ξ 0 +u)-g(ξ 0 )-ug (ξ 0 )] m(0, du). (18)
Proof. Under the conditions ( 16) and the Assumption 2.4, [6, Proposition 3] shows that ξ t admits the semimartingale decomposition

ξ t = ξ 0 + t 0 β s ds + t 0 δ s dB s + t 0 u K(ds du),
where

         β t = ∇f (Z t -).b(t, Z t-) + 1 2 tr ∇ 2 f (Z t-) t Σ(t, Z t-)Σ(t, Z t-) + R d (f (Z t -+ ψ(t, Z t-, y)) -f (Z t -) -ψ(t, Z t-, y).∇f (Z t -)) ν(y) dy, δ t = ∇f (Z t-)Σ(t, Z t-) ,
and K is an integer-valued random measure on [0, T ] × R with compensator k(t, Z t-, u) du dt defined via

k(t, Z t-, u) = R d-1 |det∇ y Φ(t, Z t-, (y 1 , • • • , y d-1 , u))| ν(Φ(t, Z t-, (y 1 , • • • , y d-1 , u))) dy 1 • • • dy d-1 , with Φ(t, z, y) = φ(t, z, κ -1 z (y)) κ -1 z (y) = (y 1 , • • • , y d-1 , F z (y)), F z (y) : R d → R f (z + (y 1 , • • • , y d-1 , F z (y))) -f (z) = y d .
From Assumption 2.3 it follows that Assumptions 2.1 and 2.2 hold for β t , δ t and k(t, Z t -, .) on [0, T ]. Applying Theorem 2.1, the result follows immediately.

Remark 2.3. Benhamou et al. [START_REF] Benhamou | Smart expansion and fast calibration for jump diffusions[END_REF] studied the case where Z t is the solution of a 'Markovian' SDE whose jumps are given by a compound Poisson Process. The above results generalizes their result to the (general) case where the jumps are driven by an arbitrary integer-valued random measure.

Time-changed Lévy processes

Models based on time-changed Lévy processes provide another class of examples of non-Markovian models which have generated recent interest in mathematical finance. Let L t be a real-valued Lévy process, (b, σ 2 , ν) be its characteristic triplet, N its jump measure. Define

ξ t = L Θt Θ t = t 0 θ s ds, (19)
where (θ t ) is a locally bounded F t -adapted positive càdlàg process, interpreted as the rate of time change.

Proposition 2.3. If R |y| 2 ν(dy) < ∞ and lim t→0, t>0 E [|θ t -θ 0 |] = 0 ( 20
)
then ∀f ∈ C 2 b (R, R), lim t→0 + E [f (ξ t )] -f (ξ 0 ) t = θ 0 L 0 f (ξ 0 ) ( 21
)
where L 0 is the infinitesimal generator of the L:

L 0 f (x) = b f (x) + σ 2 2 f (x) + R d [f (x + y) -f (x) - 1 1 + |y| 2 yf (x)]ν(dy). ( 22
)
Proof. Considering the Lévy-Itô decomposition of L:

L t = b - {|y|≤1} (y -κ(y)) ν(dy) t + σW t + t 0 R κ(z) Ñ (dsdz) + t 0 R (z -κ(z)) N (dsdz),
then ξ has the representation

ξ t = ξ 0 + t 0 σ θ s dZ s + t 0 b - {|y|≤1} (y -κ(y)) ν(dy) θ s ds + t 0 R κ(z)θ s Ñ (ds dz) + t 0 R (z -κ(z)) θ s N (ds dz).
where Z is a Brownian motion. With the notation of equation ( 2), one identifies

β t = b - {|y|≤1} (y -κ(y)) ν(dy) θ t , δ t = σ θ t , m(t, dy) = θ t ν(dy).
If [START_REF] Lee | The moment formula for implied volatility at extreme strikes[END_REF] holds, then Assumptions 2.1 and 2.2 hold for (β, δ, m) and Theorem 2.1 may be applied to obtain the result.

3 Short-maturity asymptotics for call options Consider a (strictly positive) price process S whose dynamics under the pricing measure P is given by a stochastic volatility model with jumps:

S t = S 0 + t 0 r(s)S s -ds + t 0 S s -δ s dW s + t 0 +∞ -∞ S s -(e y -1) M (ds dy), (23)
where r(t) > 0 represents a (deterministic) bounded discount rate. For convenience, we shall assume that r ∈ C b 0 (R + , R + ). δ t represents the volatility process and M is an integer-valued random measure with compensator µ(ω; dt dy) = m(ω; t, dy) dt, representing jumps in the log-price, and M = M -µ its compensated random measure. We make the following assumptions on the characteristics of S: Assumption 3.1 (Right-continuity at t 0 ). lim t→t0, t>t0 We recall that the value C t0 (t, K) at time t 0 of a call option with expiry t > t 0 and strike K > 0 is given by

E |δ t -δ t0 | 2 |F t0 = 0. For all ϕ ∈ C b 0 (R + × R, R), lim t→t0, t>t0 E R e 2y ∧
C t0 (t, K) = e -t t 0 r(s) ds E[max(S t -K, 0)|F t0 ]. (24)
The discounted asset price Ŝt = e

-t t 0 r(u) du S t ,
is the stochastic exponential of the martingale ξ defined by

ξ t = t 0 δ s dW s + t 0
(e y -1) M (ds dy).

Under Assumption 3.2, we have

E exp 1 2 ξ, ξ d T + ξ, ξ c T < ∞,
where ξ, ξ c and ξ, ξ d denote the continuous and purely discontinuous parts of [ξ, ξ] and [START_REF] Protter | Markov processes and related topics: A Festschrift for Thomas G[END_REF]Theorem 9] implies that ( Ŝt ) t∈[t0,T ] is a P-martingale. In particular the expectation in ( 24) is finite.

Out-of-the money call options

We first study the asymptotics of out-of-the money call options i.e. the case where K > S t0 .

Theorem 3.1 (Short-maturity behavior of out-of-the money options). Under Assumption 3.1 and Assumption 3.2, if S t0 < K then

1 t -t 0 C t0 (t, K) -→ t→t + 0 ∞ 0 (S t0 e y -K) + m(t 0 , dy). (25)
This limit can also be expressed using the exponential double tail ψ t0 of the compensator, defined as

ψ t0 (z) = +∞ z dx e x ∞ x m(t 0 , du) z > 0. ( 26
)
Then, as shown in [5, Lemma 1],

∞ 0 (S t0 e y -K) + m(t 0 , dy) = S t0 ψ t0 ln K S t0 .

Proof. The idea is to apply Theorem 2.1 to smooth approximations f n of the function x → (x -K) + and conclude using a dominated convergence argument. First, as argued in the proof of Theorem 2.1, we put t 0 = 0 in the sequel and consider the case where F 0 is the σ-algebra generated by all P-null sets. Applying the Itô formula to X t ≡ ln (S t ), we obtain Note that there exists C > 0 such that

X t = ln (S 0 ) + t 0 1 S s - dS s + 1 2 t 0 -1 S 2 s - (S s-δ s ) 2 ds + s≤t ln (S s -+ ∆S s ) -ln (S s -) - 1 S s - ∆S s = ln (S 0 ) + t 0 r(s) - 1
|e y -1 -y 1 1 + |y| 2 | ≤ C (e y -1) 2 .
Thanks to Jensen's inequality, Assumption 3.2 implies that this quantity is finite, allowing us to write

+ t 0 +∞ -∞ y -y 1 1 + |y| 2 M (ds dy) = t 0 +∞ -∞ (e y -1) M (ds dy) - t 0 +∞ -∞ e y -1 -y 1 1 + |y| 2
M (ds dy)

- t 0 +∞ -∞ e y -1 -y 1 1 + |y| 2 m(s, y) ds dy + t 0 +∞ -∞ y -y 1 1 + |y| 2 M (ds dy) = t 0 +∞ -∞ y 1 1 + |y| 2 M (ds dy) - t 0 +∞ -∞ e y -1 -y 1 1 + |y| 2 m(s, y) ds dy + t 0 +∞ -∞ y -y 1 1 + |y| 2 M (ds dy).
We can thus represent X t as in ( 2)):

X t = X 0 + t 0 β s dt + t 0 δ s dW s + t 0 +∞ -∞ y 1 1 + |y| 2 M (ds dy) + t 0 +∞ -∞ y -y 1 1 + |y| 2 M (ds dy), (27)
with

β t = r(t) - 1 2 δ 2 t - ∞ -∞ e y -1 -y 1 1 + |y| 2 m(t, y) dt dy.
Hence, if δ and m(., dy) satisfy Assumption 3.1 then β, δ and m(., dy) satisfy Assumption 2.1. Thanks to Jensen's inequality, Assumption 3.2 implies that β, δ and m satisfy Assumption 2.2. One may apply Theorem 2.1 to X t for any function of the form f

• exp, f ∈ C 2 b (R, R). Let us introduce a family f n ∈ C 2 b (R, R) such that f n (x) = (x -K) + |x -K| > 1 n (x -K) + ≤ f n (x) ≤ 1 n |x -K| ≤ 1 n . Then for x = K, f n (x) -→ n→∞ (x -K) + . Define, for f ∈ C ∞ 0 (R + , R), L 0 f (x) = r(0)xf (x) + x 2 δ 2 0 2 f (x) + R [f (xe y ) -f (x) -
x(e y -1).f (x)]m(0, dy).

(28) First, observe that if N 1 ≥ 1/|S 0 -K|, ∀n ≥ N 1 , f n (S 0 ) = (S 0 -K) + = 0, so 1 t E (S t -K) + ≤ 1 t E [f n (S t )] = 1 t (E [f n (S t )] -f n (S 0 )) .
Letting t → 0 + yields lim sup

t→0 + 1 t e -t 0 r(s) ds E (S t -K) + ≤ L 0 f n (S 0 ). ( 29
)
Furthermore,

E (S t -K) + ≥ E f n (S t )1 {|St-K|> 1 n } = E [f n (S t )] -E f n (S t )1 {|St-K|≤ 1 n } ≥ E [f n (S t )] -f n (S 0 ) - 1 n E 1 {|St-K|≤ 1 n } . But E 1 {|St-K|≤ 1 n } ≤ P S t -K ≥ - 1 n ≤ P S t -S 0 ≥ K -S 0 - 1 n .
There exists N 2 ≥ 0 such that for all n ≥ N 2 ,

P S t -S 0 ≥ K -S 0 - 1 n ≤ P S t -S 0 ≥ K -S 0 2 ≤ 2 K -S 0 2 E (S t -S 0 ) 2 ,
by the Bienaymé-Chebyshev inequality. Hence,

1 t E (S t -K) + ≥ 1 t (E [f n (S t )] -f n (S 0 ))- 1 n 2 K -S 0 2 1 t E [φ(S t ) -φ(S 0 )] , with φ(x) = (x -S 0 ) 2 . Applying Theorem 2.1 yields lim inf t→0 + 1 t e -t 0 r(s) ds E (S t -K) + ≥ L 0 f n (S 0 ) - 1 n 2 K -S 0 2 L 0 φ(S 0 ). Letting n → +∞, lim t→0 + 1 t e -t 0 r(s) ds E (S t -K) + = lim n→∞ L 0 f n (S 0 ). Since S 0 < K, f n = 0 in a neighborhood of S 0 for n ≥ N 1 so f n (S 0 ) = f n (S 0 ) = f n (S 0 ) = 0 and L 0 f n (S 0 ) reduces to L 0 f n (S 0 ) = R [f n (S 0 e y ) -f n (S 0 )]m(0, dy).
A dominated convergence argument then yields

lim n→∞ L 0 f n (S 0 ) = R [(S 0 e y -K) + -(S 0 -K) + ]m(0, dy).
Using integration by parts, this last expression may be rewritten [5, Lemma 1] as

S 0 ψ 0 ln K S 0
where ψ 0 is given by ( 26). This ends the proof.

Remark 3.1. Theorem 3.1 also applies to in-the-money options, with a slight modification: for K < S t0 ,

1 t -t 0 (C t0 (t, K) -(S t0 -K)) -→ t→t + 0 r(t 0 ) S t0 + S t0 ψ t0 ln K S t0 , (30)
where

ψ t0 (z) = z -∞ dx e x x -∞ m(t 0 , du), for z < 0 (31)
denotes the exponential double tail of m(0, .).

At-the-money call options

When S t0 = K, Theorem 3.1 does not apply. Indeed, as already noted in the case of Lévy processes by Tankov [START_REF] Tankov | Pricing and Hedging in Exponential Lévy Models: Review of Recent Results[END_REF] and Figueroa-Lopez and Forde [START_REF] Figueroa-López | The small-maturity smile for exponential Lévy models[END_REF], the short maturity behavior of at-the-money options depends on whether a continuous martingale component is present and, in absence of such a component, on the degree of activity of small jumps, measured by the Blumenthal-Getoor index of the Lévy measure which measures its singularity at zero [START_REF] Jacod | Asymptotic properties of power variations of Lévy processes[END_REF]. We will show here that similar results hold in the semimartingale case. We distinguish three cases:

1. S is a pure jump process of finite variation: in this case at-the-money call options behave linearly in t -t 0 (Proposition 3.1).

2. S is a pure jump process of infinite variation and its small jumps resemble those of an α-stable process: in this case at-the-money call options have an asymptotic behavior of order |t -t 0 | 1/α as t -t 0 → 0 + (Proposition 3.2).

3. S has a continuous martingale component which is non-degenerate in the neighborhood of t 0 : in this case at-the-money call options are of order √ t -t 0 as t → t + 0 , whether or not jumps are present (Theorem 3.2).

These statements are made precise in the sequel. We observe that, contrarily to the case of out-of-the money options where the presence of jumps dominates the asymptotic behavior, for at-the-money options the presence or absence of a continuous martingale (Brownian) component dominates the asymptotic behavior.

For the finite variation case, we use a slightly modified version of Assumption 3.1:

Assumption 3.3 (Weak right-continuity of jump compensator). For all ϕ ∈ C b 0 (R + × R, R), lim t→t0, t>t0 E R e 2y ∧ |y| ϕ(S t , y) m(t, dy)|F t0 = R e 2y ∧ |y| ϕ(S t0 , y) m(t 0 , dy).
Proposition 3.1 (Asymptotic for ATM call options for pure jump processes of finite variation). Consider the process

S t = S 0 + t 0 r(s)S s-ds + t 0 +∞ -∞ S s -(e y -1) M (ds dy). (32)
Under the Assumptions 3.3 and 3.2 and the condition,

∀t ∈ [t 0 , T ], R |y| m(t, dy) < ∞, 1 t -t 0 C t0 (t, S t0 ) -→ t→t + 0 1 {r(t 0 ) > R (e y -1)m(t 0 , dy)} S t0 r(t 0 ) + R (1 -e y ) + m(t 0 , dy) + 1 {r(t 0 ) ≤ R (e y -1)m(t 0 , dy)} S t0 R (e y -1) + m(t 0 , dy). (33)
Proof. Replacing P by the conditional probability P Ft 0 , we may set t 0 = 0 in the sequel and consider the case where F 0 is the σ-algebra generated by all P-null sets. The Tanaka-Meyer formula applied to (S t -S 0 ) + gives

(S t -S 0 ) + = t 0 ds 1 {Ss->S0} S s-r(s) - R (e y -1) m(s, dy) + 0<s≤t (S s -S 0 ) + -(S s--S 0 ) + .
Hence, applying Fubini's theorem,

E (S t -S 0 ) + = E t 0 ds 1 {Ss->S0} S s-r(s) - R (e y -1)m(s, dy) + E t 0 R (S s-e y -S 0 ) + -(S s--S 0 ) + m(s, dy) ds = t 0 ds E 1 {Ss->S0} S s-r(s) - R (e y -1)m(s, dy) + t 0 ds E R (S s e y -S 0 ) + -(S s -S 0 ) + m(s, dy) .
Furthermore, under the Assumptions 2.1 and 2.2 for X t = log(S t ) (see equation ( 27)), one may apply Theorem 2.1 to the function

f : x ∈ R → exp(x), yielding lim t→0 + 1 t E [S t -S 0 ] = S 0 r(0) - R (e y -1)m(0, dy) .
Furthermore, observing that

S t 1 {St>S0} = (S t -S 0 ) + + S 0 1 {St>S0} ,
we write

E S t 1 {St>S0} = E (S t -S 0 ) + + S 0 1 {St>S0} ≤ E [|S t -S 0 |] + S 0 P (S t > S 0 ) ,
using the Lipschitz continuity of x → (x -S 0 ) + . Since t → E [S t ] is rightcontinuous at 0, for t small enough :

0 ≤ E S t 1 {St>S0} ≤ 1 2 + S 0 . Thus, lim t→0 + 1 t E t 0 ds 1 {Ss->S0} S s-r(s) - R (e y -1)m(s, dy) = S 0 r(0) - R (e y -1)m(0, dy) 1 {r(0) - R (e y -1)m(0, dy) > 0}.
Let us now focus on the jump term and show that

t ∈ [0, T [ → E R (S t e y -S 0 ) + -(S t -S 0 ) + m(t, dy) ,
is right-continuous at 0 with right-limit S 0 R (e y -1) + m(0, dy).

One shall simply observes that

(xe y -S 0 ) + -(x -S 0 ) + -(S 0 e y -S 0 ) + ≤ (x + S 0 ) |e y -1|,
using the Lipschitz continuity of x → (x -S 0 ) + and apply Assumption 3.3. To finish the proof, one shall gather both limits together :

= S 0 r(0) - R (e y -1)m(0, dy) 1 {r(0) - R (e y -1)m(0, dy) > 0} + S 0 R (e y -1) + m(0, dy).
This ends the proof.

Proposition 3.2 (Asymptotics of ATM call options for pure-jump martingales of infinite variation). Consider a semimartingale whose continuous martingale part is zero:

S t = S 0 + t 0 r(s)S s-ds + t 0 +∞ -∞ S s -(e y -1) M (ds dy). (34)
Under the Assumptions 3.1 and 3.2, if there exists α ∈]1, 2[ and a family m α (t, dy) of positive measures such that

∀t ∈ [t 0 , T ], m(ω, t, dy) = m α (ω, t, dy) + 1 |y|≤1 c(y) |y| 1+α dy a.s., (35)
where c(.) > 0 is continuous at 0 and

∀t ∈ [t 0 , T ] R |y| m α (t, dy) < ∞, (36) then 1
(t -t 0 ) 1/α C t0 (t, S t0 ) -→ t→t + 0 S t0 1 2π ∞ -∞ e -c(0) |z| α -1 z 2 dz. (37)
Proof. Without loss of generality, we set t 0 = 0 in the sequel and consider the case where F 0 is the σ-algebra generated by all P-null sets. The at-the-money call price can be expressed as

C 0 (t, S 0 ) = E (S t -S 0 ) + = S 0 E S t S 0 -1 + . ( 38
) Define, for f ∈ C 2 b (]0, ∞[, R) L 0 f (x) = r(0)xf (x) + R [f (xe y ) -f (x) -x(e y -1).f (x)]m(0, dy). (39)
We decompose L 0 as the sum L 0 = K 0 + J 0 where

K 0 f (x) = r(0)xf (x) + R [f (xe y ) -f (x) -
x(e y -1).f (x)] m α (0, dy),

J 0 f (x) = 1 -1 [f (xe y ) -f (x) -x(e y -1).f (x)] c(y) |y| 1+α dy.
The term K 0 may be be interpreted in terms of Theorem 2.1: if (Z t ) [0,T ] is a finite variation semimartingale of the form (34) starting from Z 0 = S 0 with jump compensator m α (t, dy), then by Theorem 2.1,

∀f ∈ C 2 b (]0, ∞[, R), lim t→0 + 1 t e -t 0 r(s) ds E [f (Z t )] = K 0 f (S 0 ). ( 40
)
The idea is now to interpret

L 0 = K 0 + J 0 in
1 t 1/α E (Y t -1) + t→0 + → 1 2π ∞ -∞ e -c(0)|z| α -1 z 2 dz. (41)
We will show that the term (41) is the dominant term which gives the asymptotic behavior of C 0 (T, S 0 ). Indeed, by the Lipschitz continuity of x → (x -S 0 ) + ,

|(S t -S 0 ) + -S 0 (Y t -1) + | ≤ Y t |Z t -S 0 |,
so, taking expectations and using that Y is independent from Z, we get

E[e -t 0 r(s) ds |(S t -S 0 ) + C0(t,S0) -S 0 (Y t -1) + |] ≤ E(Y t ) =1 E[e -t 0 r(s) ds |Z t -S 0 |].
To estimate the right hand side of this inequality note that |Z t -S 0 | = (Z t -S 0 ) + + (S 0 -Z t ) + . Since Z has finite variation, from Proposition 3.1

E[e -t 0 r(s) ds (Z t -S 0 ) + ] t→0 + ∼ tS 0 ∞ 0 dx e x m([x, +∞[).
Using the martingale property of e -t 0 r(s) ds Z t ) yields

E[e -t 0 r(s) ds (S 0 -Z t ) + ] t→0 + ∼ tS 0 ∞ 0 dx e x m([x, +∞[).
Hence, dividing by t 1/α and taking t → 0 + we obtain

1 t 1/α e -t 0 r(s) ds E |Z t -S 0 | + t→0 + → 0.
Thus, dividing by t 1/α the above inequality and using (41) yields

1 t 1/α e -t 0 r(s) ds E [(S t -S 0 ) + ] t→0 + → S 0 1 2π ∞ -∞ e -c(0)|z| α -1 z 2 dz.
We now focus on a third case, when S is a continuous semimartingale, i.e. an Ito process. From known results in the diffusion case [START_REF] Berestycki | Asymptotics and calibration of local volatility models[END_REF], we expect in this case a short-maturity behavior ins O( √ t). We propose here a proof of this behavior in a semimartingale setting using the notion of semimartingale local time. (42)

Under the Assumptions 3.1 and 3.2 and the following non-degeneracy condition in the neighborhood of t 0 ,

∃ > 0, P (∀t ∈ [t 0 , T ], δ t ≥ ) = 1,
we have

1 √ t -t 0 C t0 (t, S t0 ) -→ t→t + 0 S t0 √ 2π δ t0 . (43)
Proof. Set t 0 = 0 and consider, without loss of generality, the case where F 0 is the σ-algebra generated by all P-null sets. Applying the Tanaka-Meyer formula to (S t -S 0 ) + , we have

(S t -S 0 ) + = t 0 1 {Ss>S0} dS s + 1 2 L S0 t (S).
where L S0 t (S) corresponds to the semimartingale local time of S t at level S 0 under P. As noted in Section 3.

: x ∈ R → (exp(x) -S 0 ) 2 , yielding lim t→0 + 1 t E (S t -S 0 ) 2 = L 0 f (X 0 ),
where L 0 is defined via equation ( 5) with m ≡ 0. Since L 0 f (X 0 ) < ∞, then in particular, t → E (S t -S 0 ) 2 is right-continuous at 0 with right limit 0. Observing that

S t 1 {St>S0} = (S t -S 0 ) + + S 0 1 {St>S0} ,
we write

E S t 1 {St>S0} = E (S t -S 0 ) + + S 0 1 {St>S0} ≤ E [|S t -S 0 |] + S 0 P (S t > S 0 ) , using the Lipschitz continuity of x → (x -S 0 ) + . Since t → E [S t ] is right- continuous at 0, for t small enough : 0 ≤ E S t 1 {St>S0} ≤ 1 2 + S 0 .
Thus, applying Fubini, Hence (if the limit exists)

lim t→0 1 √ t C(t, S 0 ) = lim t→0 1 √ t e -t 0 r(s) ds E 1 2 L S0 t (S) = lim t→0 1 √ t E 1 2 L S0 t (S) . ( 46
) By the Dubins-Schwarz theorem [24, Theorem 1.5], there exists a Brownian motion B such that

∀t < [U ] ∞ , U t = t 0 δ s dW s = B [U ]t = B t 0 δ 2 s ds . So ∀t < [U ] ∞ S t = S 0 exp t 0 r(s) - 1 2 δ 2 s ds + B [U ]t = S 0 exp t 0 r(s) - 1 2 δ 2 s ds + B t 0 δ 2 s ds .
The occupation time formula then yields, for φ

∈ C ∞ 0 (R, R), ∞ 0 φ(K)L K t S 0 exp B [U ] dK = t 0 φ S 0 exp B [U ]u S 2 0 exp B [U ]u 2 δ 2 u du = ∞ -∞ φ(S 0 exp(y))S 2 0 exp(y) 2 L y t B [U ]
t S 0 exp B [U ] dy = ∞ -∞ φ(S 0 exp(y))S 2 0 exp(y) 2 L y t B [U ] . Hence L S0 t S 0 exp B [U ] = S 0 L 0 t B [U ] . We also have L 0 t B [U ] = L 0 t 0 δ 2 s ds (
E L S0 t S 0 exp B [U ] = S 0 E L 0 t 0 δ 2 s ds (B) = S 0 E   t 0 δ 2 s ds L 0 1 (B)   . Hence lim t→0 + 1 √ t E L S0 t S 0 exp B [U ] = lim t→0 + 1 √ t S 0 E   t 0 δ 2 s ds L 0 1 (B)   = lim t→0 + S 0 E   1 t t 0 δ 2 s ds L 0 1 (B)   .
Let us show that

lim t→0 + S 0 E   1 t t 0 δ 2 s ds L 0 1 (B)   = S 0 δ 0 E L 0 1 (B) . (47)
Using the Cauchy-Schwarz inequality,

E     1 t t 0 δ 2 s ds -δ 0   L 0 1 (B)   ≤ E L 0 1 (B) 2 1/2 E      1 t t 0 δ 2 s ds -δ 0   2    1/2 . The Lipschitz property of x → ( √ x -δ 0 ) 2 on [ , +∞[ yields E      1 t t 0 δ 2 s ds -δ 0   2    ≤ c( )E 1 t t 0 δ 2 s -δ 2 0 ds ≤ c( ) t t 0 ds E δ 2 s -δ 2 0 .
where c( ) is the Lipschitz constant of x → ( √ x -δ 0 )

E L 0 1 (B) = 2 π .
Clearly, since

L K t (S) = L K t S 0 exp B [U ] , lim t→0 1 √ t E 1 2 L S0 t (S) = S 0 √ 2π δ 0 . (48)
This ends the proof.

We can now treat the case of a general Itô semimartingale with both a continuous martingale component and a jump component. 1 t E (S t -S 0 ) 2 = L 0 f (X 0 ), where L 0 is defined via equation [START_REF] Bentata | Forward equations for option prices in semimartingales[END_REF]. Since L 0 f (X 0 ) < ∞, then in particular, t → E (S t -S 0 ) 2 is right-continuous at 0 with right limit 0. Observing that Since δ 0 ≥ , equation (48) yields the result.

S
Remark 3.2. As noted by Berestycki et al [START_REF] Berestycki | Asymptotics and calibration of local volatility models[END_REF][START_REF] Berestycki | Computing the implied volatility in stochastic volatility models[END_REF] in the diffusion case, the regularity of f at S t0 plays a crucial role in the asymptotics of E [f (S t )]. Theorem 2.1 shows that E [f (S t )] ∼ ct for smooth functions f , even if f (S t0 ) = 0, while for call option prices we have ∼ √ t asymptotics at-the-money where the function x → (x -S 0 ) + is not smooth.

(e y - 1 -

 1 y) M (ds dy)

e y - 1 -y 1 1 +

 11 |y| 2 M (ds dy)

  terms of a multiplicative decomposition S t = Y t Z t where Y = E(L) is the stochastic exponential of a pure-jump Lévy process with Lévy measure c(y)/|y| 1+α dy, which we can take independent from Z. Indeed, let Y = E(L) where L is a pure-jump Lévy martingale with Lévy measure 1 |y|≤1 c(y)/|y| 1+α dy, independent from Z, with infinitesimal generator J 0 . Then Y is a martingale and [Y, Z] = 0. Then S = Y Z and Y is an exponential Lévy martingale, independent from Z, with E[Y t ] = 1. A result of Tankov [26, Proposition 5, Proof 2] for exponential Lévy processes then implies that

Proposition 3 . 3 ( 0 S

 330 Asymptotic for at-the-money options for continuous semimartingales). Consider the process S t = S 0 + t 0 r(s)S s ds + t s δ s dW s .

  u) du r(s)S s 1 {Ss>S0} ds = O(t), u) du r(s)S s 1 {Ss>S0} ds = o √ t .

S 0

 0 exp B [U ] (resp. L y t B [U ] ) denotes the semimartingale local time of the process S 0 exp B [U ] at K and (resp. B [U ] at y). A change of variable leads to ∞ -∞ φ(S 0 exp(y))S 0 exp(y)L S0e y

B) , where L 0 t 0 δ 2 s

 2 ds (B) denotes the semimartingale local time of B at time t 0 δ 2 s ds and level 0. Using the scaling property of Brownian motion,

Theorem 3 . 2 ((

 32 Short-maturity asymptotics for at-the-money call options). Consider the price process S whose dynamics is given byS t = S 0 + t 0 r(s)S s -ds + t 0 S s -δ s dW s + s -(e y -1) M (ds dy).Under the Assumptions 3.1 and 3.2 and the folllowing non-degeneracy condition in the neighborhood of t 0∃ > 0, P (∀t ∈ [t 0 , T ], δ t ≥ ) = 1,we have1 √ t -t 0 C t0 (t, S t0) Applying the Tanaka-Meyer formula to (S t -S 0 ) + , we have(S t -S 0 ) + = -S 0 ) + -(S s--S 0 ) + -1 {Ss->S0} ∆S s .(50)As noted above, Assumption 3.2 implies that the discounted price Ŝt = e -t 0 r(s) ds S t is a martingale under P. So we can write dS t = e t 0 r(s) ds r(t)S t-dt + d Ŝt , and u) du r(s)S s-1 {Ss->S0} ds,where the first term is a martingale. Taking expectations, we gete t 0 r(s) ds C(t, S 0 ) = E t 0 e s 0 r(u) du r(s)S s 1 {Ss->S0} ds + S s -S 0 ) + -(S s--S 0 ) + -1 {Ss->S0} ∆S s   . Since Ŝ is a martingale, ∀t ∈ [0, T ] E [S t ] = e t 0 r(s) ds S 0 < ∞. (51)Hence t → E [S t ] is right-continuous at 0:lim t→0 + E [S t ] = S 0 .(52) Furthermore, under the Assumptions 2.1 and 2.2 for X t = log(S t ) (see equation (27)), one may apply Theorem 2.1 to the function f : x ∈ R → (exp(x) -S 0 ) 2 , yielding lim t→0 +

(

  u) du r(s)S s 1 {Ss>S0} ds = O(t), u) du r(s)S s 1 {Ss>S0} ds = o √ t .Let us now focus on the jump part, S s -S 0) + -(S s--S 0 ) + -1 {Ss->S0} ∆S s s, dx) (S s-e x -S 0 ) + -(S s--S 0 ) + -1 {Ss->S0} S s-(e xx -S 0 ) + -(z -S 0 ) + -1 {z>S0} z(e x -1) ≤ C (S 0 e x -z) 2 ,then, together with Assumption 3.1 and Lemma 2.1 implies,-S 0 ) + -(S s--S 0 ) + -1 {Ss->S0} ∆S s   = O(t) = o( √ t).

Remark 3 . 3 .

 33 In the particular case of a Lévy process, Proposition 3.1, Proposition 3.2 and Theorem 3.2 imply a recent result of Tankov [26, Proposition 5, Proof 2].

  |y| 2 ϕ(S t , y) m(t, dy)|F t0 =

						e 2y ∧ |y| 2 ϕ(S t0 , y) m(t 0 , dy).
						R
	Assumption 3.2 (Integrability condition).
	∃T > t 0 , E exp	1 2	T t0	δ 2 s ds +	T t0	ds

R

(e y -1) 2 m(s, dy) |F t0 < ∞ .

  t 1 {St>S0} = (S t -S 0 ) + + S 0 1 {St>S0} ,we writeE S t 1 {St>S0} = E (S t -S 0 ) + + S 0 1 {St>S0} ≤ E [|S t -S 0 |] + S 0 P (S t > S 0 ) , using the Lipschitz continuity of x → (x -S 0 ) + . Since t → E [S t ] is rightcontinuous at 0, for t small enough : 0 ≤ E S t 1 {St>S0} ≤ 1 2 + S 0 .

	Thus, applying Fubini,


K De Moerlooze
email: kris.demoerlooze@mech.kuleuven.be

F Al-Bender

H Van Brussel
email: hendrik.vanbrussel@mech.kuleuven.be

Modeling of the dynamic behavior of systems with rolling elements

Keywords: contact mechanics, Hertzian contact, longitudinal creepage, lateral creepage, spin, tractive rolling dynamics, dry friction

Bearings, friction wheels, cams etc. are widely employed elements in machine construction. The modeling of the dynamics of rolling friction plays therefore a crucial role in the simulation and optimization of such systems. This paper describes a recently developed transient-rolling-friction model with its application to different generic dynamical systems, to simulate the effects of rolling friction or traction on the dynamics of mechanical systems. The results, which are expressed in terms of the complex stiffness arising in the rolling-contact patches, show that the behavior depends on both the amplitude and the frequency of oscillation. For low amplitudes and frequencies, the behavior is quasi linear. For increasing amplitudes and frequencies, however, strong nonlinearities appear, leading to complex behavior.

Introduction

Rolling elements are widely used in machine construction for motion guidance and transmission, in various applications, including robotics and positioning systems, railway and vehicle applications. The growing demands for accu-racy, performance, and reliability in machines require an accurate modeling of friction in their moving parts.

Positioning and machining devices often employ rolling elements, such as ball bearings and plain bearings (see, e.g. [START_REF] Futami | Nanometer positioning and its micro-dynamics[END_REF][START_REF] Awabdy | Nanometer Positioning of a Linear Motion Stage Under Static Loads[END_REF][START_REF] Mekid | High precision linear slide. Part I: design and construction[END_REF][START_REF] Yi | Dynamic analysis of a linear motion guide having rolling elements for precision positioning devices[END_REF]). The study of rolling-element bearings was and still is an important research topic [START_REF] Harris | An Analytical Method to Predict Skidding in Thrust-Loaded, Alngular-Contact Ball Bearings[END_REF][START_REF] Harris | Ball Motion in Thrust-Loaded, Angular Contact Bearings With Coulomb Friction[END_REF][START_REF] Szuminski | Determination of the Dynamic Angle of the Bearings Rolling Element Location and of Its Influence on the Loads and Rolling Friction[END_REF][START_REF] Neng | A New Method for the Analysis of Deformation and Load in a Ball Bearing With Variable Contact Angle[END_REF][START_REF] Neng | Ball bearing skidding under radial and axial loads[END_REF][START_REF] Neng | An Analysis of Misaligned Single-Row Angular-Contact Ball Bearing[END_REF][START_REF] Szuminski | Determination of the stiffness of rolling kinematic pairs of manipulators[END_REF]. These elements introduce nonlinear frictional behavior, which often mortgages accurate and fast positioning. Furthermore, the replacement of failed or damaged elements constitutes a considerable part of the maintenance costs in industrial applications and maintenance schemes.

The study of traction in tyre-road contacts [START_REF] Heinrich | Rubber friction, tread deformation and tire traction[END_REF][START_REF] Schutte | A Contact Solver Suitable for Finite Elements[END_REF] can lead to significant improvements in the drive comfort, handling and reliability of vehicles. Similar research is performed and is still ongoing in the field of railway applications [START_REF] Garg | Dynamics of Railway Vehicle Systems[END_REF][START_REF] Ekberg | Rolling contact fatigue of railway wheels -a parametric study[END_REF][START_REF] Nilsson | On wear in rolling/sliding contacts[END_REF][START_REF] Nielsen | New Developments in the Theory of Wheel/Rail Contact Mechanics[END_REF][START_REF] Telliskivi | Simulation of wear in a rolling-sliding contact by a semi-Winkler model and the Archard's wear law[END_REF], where noise and vibrations such as squeaking and rattling while braking and taking bends, and associated wear, are of great concern [START_REF] Cataldi-Spinola | Influence of the Wheel Diameter on the Curve Squealing of Railway Vehicles[END_REF][START_REF] Brunel | Transient model for curve squeal noise[END_REF][START_REF] Hsu | Experimental and theoretical investigation or railway wheel squeal[END_REF].

The preceding examples point to the exigency of simulating and optimizing rolling contacts. A profound study of the tractive forces in the contact between a rolling element and its counter surface is therefore a pertinent issue, together with the investigation of the reciprocal influence of friction and structural dynamics.

The traction stress field in the contact of a body of revolution and its mating surface, originates from the relative motion between points on the contacting bodies. When an elastic body of revolution rolls with traction over another, the traction field in the contact patch changes progressively with the traversed distance and the prevailing creepages. This traction field, which is generally dynamic, determines the traction forces developed in the system. During the past decades, there has been a significant amount of research into rolling elements, which remained, however, restricted to steady-state rolling. Consequently, nowadays, the steady-state gross rolling process is well understood, while transient effects still need a profound investigation.

The transition period, building up to steady, gross rolling, which is denominated the pre-rolling period, is characterized by rate independent hysteresis behavior with nonlocal-memory effect [START_REF] Al-Bender | Characterization of frictional hysteresis in ball-bearing guideways[END_REF]. In an earlier publication [START_REF] Al-Bender | A model of the transient behavior of tractive rolling contacts[END_REF], the authors present a transient model which is able to simulate the motion and traction stresses in a tractive rolling contact, subjected to a constant creepage. This model has been validated in Ref. [START_REF] De Moerlooze | Experimental investigation into the tractive prerolling behavior of balls in V-grooved tracks[END_REF] for the specific case of a ball rolling in a V-grooved track. In the present paper the application of the model is extended towards dynamical systems, in which the creepage is no longer constant over time, but depends on the dynamics of the constituent system.

In applications where transient conditions are expected, such as vibrations, accelerations or decelerations, the dynamics of the system play a part in the analysis. Gupta proposes an advanced approach to simulate the rolling element motion in a general way for both roller and ball bearings [START_REF] Gupta | Advanced Dynamics of Rolling Element Bearings[END_REF], based on a time integration of the general differential equations of motion, and a calculation of the external forces acting on the rolling element. The determination of the external forces relies on the applied bearing load, the roller-race and the roller-cage interactions, together with the lubrication effects. In his analysis, the traction forces are based on steady-state elastohydrodynamic lubrication theory. Abascal [START_REF]2D transient dynamic friction contact problems. I. Numerical analysis[END_REF] studies the transient dynamic friction in two-dimensional contacts, using the boundary element method. Spiegelberg et al. [START_REF] Spiegelberg | Simulation of transient friction of a cylinder between two planes[END_REF] study the transient friction effects in the contact between a cylinder and two flat plates, using a brush model. Overall, the knowledge of rolling friction under transient conditions is somewhat limited, while there is an obvious need for such knowledge, and for the possible models it can yield, for the purpose of characterizing the dynamics of systems comprising rolling elements.

The objective of this study is to simulate the transient rolling-friction behavior together with the dynamics of generic systems comprising rolling elements. The used rolling-friction model, based on a Winkler bedding and an appropriate choice of the flexibility parameter L, yields a linear approximation with a solu-tion close to the exact theory [START_REF] Al-Bender | A model of the transient behavior of tractive rolling contacts[END_REF]. The transparency and the easy application of the theory is advantageous for implementing the model in larger systems, i.e. including system dynamics. The dynamics of the system will affect the developed friction behavior, while the transient effects of the friction will, in their turn, influence the system's behavior. This interaction is the underlying theme of this paper.

The paper is structured as follows. Section 2 outlines the rolling friction model, which is used in the subsequent case studies. In section 3, the dynamics of a ball on a flat plane is investigated, where the rotation of the ball is affected by the imposed motion of the contacting flat. In section 4, a two degree-of-freedom model is presented, namely the motion of a ball in between two flat planes, where the system is excited through the motion of the upper plane. The last case study in section 5 comprises the motion of a ball in between two V-grooved tracks. In this case, the kinematic spin motion of the ball is added to the dynamical creepage in the contact. Finally, appropriate conclusions are drawn.

Transient rolling of a sphere on a plane surface

Based on [START_REF] Al-Bender | A model of the transient behavior of tractive rolling contacts[END_REF][START_REF] De Moerlooze | Experimental investigation into the tractive prerolling behavior of balls in V-grooved tracks[END_REF], this section delineates the basic theory for modeling the transient, tractive rolling of a sphere on a plane rigid surface. The rolling situation is depicted in Fig. 1. The sphere, of radius R, being loaded by a force normal to the plane surface, W , rolls in the x-direction such that its centre translates at a velocity V , while being subjected to creepages (see further).

Problem specification

The rolling problem is specified by determining (i) the contact patch size and pressure distribution and (ii) the rolling creepages, as follows.

Normal stresses and contact patch

When an elastic sphere, of radius R and equivalent modulus of elasticity E e , is pressed against a rigid plane surface with a load W , we have:

• The contact patch will be the circular region A = {(x, y) :

x 2 + y 2 ≤ a 2 },
where the radius is given by

a = 3W R 4E e 1/3 , ( 1
)
• The normal stress, p z , is given by: rigid. Referring to Fig. 1, if the sphere's geometrical centre translates with a velocity V , while the sphere is spinning around its x, y and z axes with angular speeds ψ, ω and φ respectively, the creepages are defined by:

p z (x, y) = p 0 1 -(x/a) 2 -(y/a) 2 with p 0 = 3W 2πa 2 (2)
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ δV x = -ωR + V the longitudinal creepage,
δV y = ψR the lateral creepage, φ the spin creepage.

(

) 3
Creepage can originate from (i) the kinematics of the system, or (ii) from the dynamics of the system. The case (i) can be illustrated by the rolling of a ball in a V-grooved track. Assuming that the velocity of the ball is low enough to avoid dynamic effects, or equivalently that the ball is massless, the creepage in the contact is that of pure spin, due to kinematics of the system. This is illustrated in Fig. 2. In case (ii), the creepage originates from the dynamics of the ball with respect to the track surface. Examples of this case can be found in high speed spindle bearings, in which the high accelerations applied by the electric motors result in slipping of the bearing balls in their raceways. Similar effects can occur in ball-screw nuts of dynamically driven systems.

The problem of an elastic sphere, tractively rolling on a rigid plane, can be reduced to that in which the contact patch is stationary in space and time. This is accomplished by assuming the rigid plane upon which the sphere is rolling to be moving in the opposite direction with a velocity V (see Fig. 1.)

Constitutive equations 2.2.1. The slip equation

In order to arrive at an equation to describe the kinematics of the surface points lying inside the contact patch A, one follows a point on the surface of the sphere, which enters the contact patch and mates, at the entrance, with a counterpoint on the rigid plane. Owing to the creepage, the point on the surface of the sphere will have to deform in the contact plane, by an amount u = (u, v) being called the displacement, which is generally function of space and time. The slip s is defined as the relative velocity between mating points in the interface. We obtain the following differential equation for the plane deformations:

s = c -V ∂u ∂x + ∂u ∂t , (4)
with c = (δV x -φy, δV y + φx).

Generally, V = V (t). If the creepage is scaled with V , then one can rewrite the previous equation in terms of the traversed rolling distance, q by making use of the substitution:

q = t 0 V (τ )dτ (5)
Substituting this into Eq. 4 and normalizing the creepage and the spin by the rolling velocity results in:

S = C - ∂u ∂x + ∂u ∂q ( 6
)
where S = s/V = (S x , S y ) is the relative slip and

C = c/V = (ξ x -φy a , ξ y + φx a ) is the relative creepage, with ξ x = δV x V , ξ y = δVy V , φ = φa V .

Traction-displacement relationship: A linearized theory

Here, the relationship between surface displacement (u, v) and the tangential traction field (p x , p y ) for the case of no slip, is determined using a "Winkler bedding" model. In this approximating approach, the surface of the elastic object is considered to be covered with elastic bristles in the normal direction with constant stiffness for tangential deformations, being characterized by the flexibility parameter L, such that the following relation holds for no-slip conditions:

(u, v) = L.(p x , p y ) (7)
The flexibility parameter L depends on the elasticity characteristics of the sphere and the magnitude of the creepages, and is given by:

L = L 1 |ξ x | + L 2 |ξ y | + L 3 |φ| ξ 2 x + ξ 2 y + φ 2 (8)
The formulas for the parameters L 1 , L 2 and L 3 are given in Appendix B.

The traction limit and slip conditions

Equation 7 is no longer valid when slip occurs between the mating points.

This will be the case when the condition

(p x , p y ) ≤ μp z (9)
is violated. Here, p z is given by Eq. 2 and μ is a local coefficient of friction.

This condition 9 is known as the traction bound, which results in a relationship for determining the tangential tractions: [START_REF] Neng | An Analysis of Misaligned Single-Row Angular-Contact Ball Bearing[END_REF] In this analysis, the local coefficient of friction is assumed to be constant in time and in contact condition. However, in real contacts, the coefficient of friction may vary owing to many factors such as variable contact pressure and/or wear caused by (partial) slip. In a more refined analysis, these effects can be taken into account. However, the error committed by neglecting them, appears to be acceptable [START_REF] De Moerlooze | Experimental investigation into the tractive prerolling behavior of balls in V-grooved tracks[END_REF], which justifies assuming a constant coefficient of friction for the time being.

(p x , p y ) = ⎧ ⎨ ⎩ (u,v) L ; (p x , p y ) ≤ μp z μp z (u,v) (u,v) elsewhere

Initial conditions

The final necessary ingredient for solving the rolling-contact problem is the specification of the initial traction field present in the contact prior to commencement of (pre)rolling motion. This traction field is induced by the Hertzian loading of the rolling bodies, in the normal direction, when friction is present between the contacting surfaces. The procedure for calculating this traction field is detailed in [START_REF] Al-Bender | A model of the transient behavior of tractive rolling contacts[END_REF]. Obviously, this initial traction field will determine the transition trajectory to steady-state rolling. However, the traction field pertaining to steady-state rolling is independent of the initial traction field (see [START_REF] Al-Bender | A model of the transient behavior of tractive rolling contacts[END_REF]). Thus, if periodic rolling involves passing through the steady-state traction field, it will, consequently, be independent of the initial conditions. In the simulation experiments presented in the remainder of this paper, which consider only sinusoidal motion, one may therefore assume any arbitrary initial traction field, e.g. a null field, without loss of generality.

Numerical implementation and calculation of the traction forces and moment

In order to solve the system of equations presented above, a numerical procedure was developed in which the contact patch was discretized into a set of nodes. Starting from any given initial traction field, the new displacement distribution (u, v) at every time step, is obtained by shifting the previous u and v vectors by the rolling increment Δq and augmenting it with the calculated extra displacement. Since the rolling increment is generally independent of the grid size, the shifting must be performed by 2D bilinear interpolation of the displacement distribution. Afterwards, the traction bound is calculated and the traction limit is verified. For details, the reader is referred to [START_REF] Al-Bender | A model of the transient behavior of tractive rolling contacts[END_REF]. When the traction fields have been determined, the resultant traction forces and traction moment are calculated by quadrature of the fields over the contact patch:

F x = a -a a(y)
-a(y) p x (x, y)dxdy [START_REF] Szuminski | Determination of the stiffness of rolling kinematic pairs of manipulators[END_REF])

F y = a -a a(y) -a(y) p y (x, y)dxdy (12) M z = a -a a(y) -a(y)
p(x, y)r(x, y)dxdy [START_REF] Schutte | A Contact Solver Suitable for Finite Elements[END_REF] with r(x, y) being the radial distance from the centre of the contact patch to the location of the node. Figure 3 shows an example of a transient simulation for the case of rolling with longitudinal creepage, where the friction force F x is calculated using Eq. 11.

-1 -0.5

0 0.5 1 -1 -0.5 0 0.5 1 x co-ordinate [-] y co-ordinate [-] contact contour slip region traction arrows A (a) -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x co-ordinate [-] y co-ordinate [-] B (b) -2 -1 0 1 2 -1 -0.5 0 0.5 1 Displacement x [A.U.] F x [A.U.] Longitudinal creepage v x = 0.5 A C D B (c) -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x co-ordinate [-] y co-ordinate [-] C (d) -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x co-ordinate [-] y co-ordinate [-] D (e)

Normalization of the model

Before going into the case studies, and in order to facilitate analysis and presentation, we consider it useful first to normalize the problem with respect to its parameters, using dimensional analysis, so as to permit an easier analysis of the results and provide better insight into the behavior. For this purpose, the lengths are rescaled by a, the masses by I/a 2 (I being the moment of inertia of the sphere about an axis through its centre), and the times by I/(p 0 a 3 ). The frequency of excitation is made dimensionless via:

f * t = f t I p 0 a 3 . ( 14
)
With

I = 8 15 πR 5 ρ, (15)
for a solid sphere, we obtain

f t = f * t 15E 4ρ a 2 πR 3 , (16)
which gives the conversion relationship between dimensional and dimensionless frequency.

In the following sections, all results are given in dimensionless form, where the * is dropped for convenience.

Case 1:

A sphere rolling on a plane

System description

As a first case to study the influence of dynamics on the frictional behavior of a system comprising a rolling element, the system depicted in Fig. 4 is treated.

The rolling sphere has one degree-of-freedom, namely its (frictionless) rotation around the z axis. A massless flat plate is pressed against the sphere, resulting in a circular contact patch. Obviously, the dynamic behavior of the sphere is affected by the traction between plane and sphere. A prescribed tangential displacement x t (t), is imposed on the upper plane, while the effect on the motion of the sphere and the friction force in the contact is determined. The rotation of the sphere is determined by the following equation:

I q y x z x (t) t
θ = F(θ, x t , his) ( 17
)
where F is the nonlinear frictional force, which depends on the position of the upper plate, x t , the rotation of the sphere θ, and the history of the motion, his.

The simulation is carried out as follows. At every time step, the friction force is calculated using the rolling friction model of section 2. For given geometrical, material and load properties of the system, the input to this model is the longitudinal creepage in the contact, given by:

ξ x = ẋt -R θ ẋt ( 18
)
The friction force acting on the sphere constitutes the input to the numerical scheme to solve Eq. 17 for lightly damped systems. When we apply a ramp input of the position x t , that is, a step input of the velocity ẋt , we may expect damped oscillatory behavior of the rotational velocity θ of the sphere, while the angular position of a point on the surface of the sphere will lag the reference signal. In Figure 6, which simulates this situation, the angular position and velocity are plotted together with their reference signal. One can clearly see the resonant behavior of the ball in the velocity signal.

Simulation results

The resonance originates from the ball inertia and the tangential stiffness in the contact patch. To perform a systematic analysis of the nonlinear behavior, a stepped-sine excitation is applied to the system. That is, the upper plane motion x t is made sinusoidal, for a set of discrete frequencies. This experiment is carried out for different excitation amplitudes, in order to study the influence of the amplitude on the behavior. In Figure 7(a), the frequency response function F fric /x t is plotted, which represents a measure of the complex stiffness of the ball-flat contact. For low frequencies, one can notice an increasing stiffness, up to a certain break frequency after which the stiffness becomes constant and a phase lag begins. For low frequencies, the motion of the plate is (almost) completely transmitted to the the ball, which can thus follow the imposed motion. For higher excitation frequencies, the velocity of the plate increases and as a result, the ball will no longer be able to follow the plane's motion, which results in a certain amount of slip between plane and sphere. The transition frequency from increasing stiffness to a constant amplitude complex stiffness is denoted by f s .

In Fig. the contact patch a, the maximum normal pressure p 0 , and the ball inertia I, these parameters fall out of the model equations, so that Eq. 1 becomes:

1 = 3W * R * 4E * h 1/3 ⇔ R * = 4E * h 3W * ( 19
)
In other words, R * increases if E * increases or W * decreases. In this way, R * contains all geometry, elasticity and load information. Since the ball inertia is also reduced by nondimensionalizing the model, we get

I * = 1 = R 2 g m * = f (R * )m * ⇒ R * = f (m * ) (20)
From Eq. 20, R * also holds the inertial or mass information. To study the influence of R * , this parameter is varied between reasonable bounds. Owing to geometrical limitation, a cannot be larger than about R/4 without violating the assumptions of the Hertz theory.

The resonance frequency f res is defined as:

f res = 1 2π R * 2 K * I * , ( 21
)
where K * is the tangential stiffness in the contact patch.

Dynamic behavior of a mass on a nonlinear element

In this section, the dynamic behavior is discussed when non-vanishing inertia, a (dimensionless) mass M , is assigned to the plate. In this case, the input force F x , needed to maintain a sinusoidal displacement x t (applied to the plate, as in the previous section), is function of the inertia of the plate. Figure 8

Case 2: A rolling sphere in between two planes

System description

The system of a rolling sphere in between two flat plates is depicted in Fig. 9. This is a simplified model representing various kinds of rolling, such as that found in ball bearings, linear guideways, cam-follower systems, etc. The system is excited by applying a tangential displacement to the upper plate, while the motion of the lower plate is obstructed. The friction force in the upper contact is denoted by F x1 , while the friction force in the lower contact is denoted by F x2 . The creepage in each contact is given by Eq. 22.

ξ 1 = ẋt-( ẋ+R θ) ẋt ξ 2 = -( ẋ-R θ) ẋ (22)

Simulation results

When a velocity step is applied to the system, the (exponentially windowed) spectrum of F x1 shows two peaks, originating from the natural frequencies of the translation and rotation oscillations, respectively, given by

ω 1 = K * up +K * lo m * ω 2 = R * 2 (K * up +K * lo ) I * , ( 23
)
x (t) t q y x z where, K * up and K * lo are the tangential contact stiffness in the upper and lower contact patches, respectively. These stiffnesses can be calculated from the tractive behaviour in each contact. Since for a solid sphere I * = 2/5m R * 2 and under the assumption that K * up equals K * lo , one can derive that ω 2 = 5/2ω 1 . This is confirmed by the simulation results of Fig. 10. v 2 = ẋ -θR, these particle velocities contain frequency information from both modes. (v 1ẋt ) and v 2 denote the slipping velocities in both contacts and account for the energy dissipation and consequently for damping and wear in both contacts.

The response to a velocity step input, is given in

In Figure 12, three sinusoidal excitations x t = A sin(2πf t t) with different excitation frequencies are compared. The initial velocities θ(0) and ẋ(0) are set equal to 2πf t A/(2R) and 2πf t A/2 respectively, to avoid transient effects at the start of the simulation. For the lowest excitation frequency f t1 = 1, Fig. 12(a), there is almost no sliding in the contacts. When increasing the frequency to f t2 = 6.5, the sliding increases, Fig. 12(c). For yet higher excitation frequency f t3 = 10, Fig. 12(e), high sliding speeds occur in the contacts, while the motion of the sphere, both for rotation and translation, is no longer sinusoidal, but approximates a rounded triangular signal. The friction forces in both contacts are visualized for these three cases in the right column of Fig. 12(b,d,f).

When applying a stepped-sine excitation to the system, identical results are obtained as for the system with one degree-of-freedom of section 3. We can therefore conclude that the translating motion does not influence the stiffness properties of the upper contact. In a significant part of the applications, the rolling element is subjected to a varying normal load, implying a variable contact patch. An obvious example is the loaded deep groove ball bearing, provided with a static radial load. Here, the load for each ball in the bearing alters with its changing radial position. Fig. 13 shows the evolution of the contact patch radius as function of time. In Fig. 14, the frictional forces are plotted as function of time, when the externally applied force W * changes progressively with time, from 2 up to 3.6. From Fig. 14, one can notice the beneficial effect of the increasing normal load on the damping of high frequency components.

0 1 2 -5 0 5 t [-] v t ω R v v 1 v 2 (a) 0 1 2 -0.02 0 0.02 t [-] F x [-] F x1 F x2 (

Case 3: A rolling sphere in between two V-grooved tracks

System description

Figure 15 shows the configuration of a sphere rolling in between two Vgrooves. To the upper V-groove, a displacement x t (t) is applied, while the lower V-groove stands still. From the right panel of Fig. 15, one can notice that the rolling motion of the ball ω can be resolved in the contact patches as a spin motion ω s combined with a pure rolling motion ω r . In a kinematic analysis, that is, where inertial effects are neglected, the spin motion of the ball is the main source for the frictional losses in the contact. For a low-frequency excitation of the upper V-groove, the spin motion is determined by the geometry of the system, i.e. the ball radius R, and the opening angle of the V-groove, α. Analysis of rolling in V-groove can be found in [START_REF] De Moerlooze | Experimental investigation into the tractive prerolling behavior of balls in V-grooved tracks[END_REF], where experimental evidence, pertaining to low frequency motion, is weighted against the theoretical model briefly presented in section 2. For higher dynamical excitation, the inertia of the sphere will cause an additional creepage in the contact patches and consequently extra losses.

The spin component is given by [START_REF] De Moerlooze | Experimental investigation into the tractive prerolling behavior of balls in V-grooved tracks[END_REF]:

φ = ω s v t = ω cos( α 2 ) v t (24)

Simulation results

The motion response to a step input of the velocity ẋt is presented in Fig. 16, with groove angle α = π 3 . The transient is damped out much faster than in the case of systems 1 and 2 (cf. sections 3 and 4) because of the extra, permanent damping due to the spinning motion in the contact patches. The spin component, and consequently the extra damping, decrease with increasing groove angle α. Due to the geometry of the system, the natural frequencies of the translation and rotation oscillations are given by

ω 1 = K * up +K * lo m * ω 2 = R * 2 (K * up +K * lo ) I * = R * 2 sin 2 ( α 2 )(K * up +K * lo ) I * , ( 25
)
with K * lo and K * up the tangential contact stiffness in the lower and upper contact respectively, which results in ω 2 = 5/2 sin(α/2)ω 1 , assuming that K * up equals K * lo . Thus, when √ 2.5 sin(α/2) > 1, the resonance frequency of the rotational mode ω 2 is higher than the translational model ω 1 . The flipping point of the modes, that is, the point at which the two frequencies become equal, lies at

α = 78.46 • .
The frequency-response characterization of an externally excited nonlinear dynamic system can only be carried out in an approximating way, since analytical solutions are not available [START_REF] Al-Bender | Theoretical analysis of the dynamic behavior of hysteresis elements in mechanical systems[END_REF]. One of the methods is the describing function method. Here, the nonlinear element is replaced by an element that gives the fundamental component in Fourier terms, of the output of that non-linear element for a sinusoidal input. In [START_REF] Al-Bender | Characterization of frictional hysteresis in ball-bearing guideways[END_REF], this type of analysis is carried out to obtain three characterizing parameters of the hysteresis loop. The equivalent stiffness k e and the equivalent damping, c e , which give rise to an excitation dependent equivalent damping ratio ζ r . The equations obtained in [START_REF] Al-Bender | Characterization of frictional hysteresis in ball-bearing guideways[END_REF] are

k e ≈ F m A m c e f = loop area πA 2 m ζ r = loop area 2πA m F m ( 26
)
Since these three parameters comprise the hysteresis characteristic behavior, a study of these shows the dependence of the characteristics on the influence parameters. From the previous section, it is clear that the dimensionless ball radius R * comprises load information, geometrical and material properties. The local coefficient of friction μ, forms an additional influence parameter which, together with the V-groove angle of the system α, and the ball radius, influences the spin creepage φ * in the contact patches. The influence of these parameters on the equivalent stiffness and damping ratio of the hysteresis are depicted in For a sinusoidal excitation x t , at low frequencies, the system behaves quasi similar to the kinematic model described in [START_REF] De Moerlooze | Experimental investigation into the tractive prerolling behavior of balls in V-grooved tracks[END_REF], since the creepage in the contact originates almost only from the spin due to rolling in a V-groove. When the excitation frequency is increased, however, extra creepage will occur in the contacts due to the inertial effect of the ball. For very high frequencies, the latter effect will dominate. In Figure 18, the hysteresis curve is plotted for different excitation frequencies for a constant excitation amplitude. One can clearly see the increasing stiffness of the hysteresis loop with increasing excitation frequency f t . However, as the frequency increases further, the hysteresis behavior becomes distorted and shows chaotic behavior, as shown in Fig. 19, where the force-displacement plot is depicted for an amplitude A of 1 and a frequency f t of 5. The combination of sliding due to the inertial effect of the ball, together with the spinning motion of the ball, results in an additional oscillatory behavior.

Finally, in Figure 20, the equivalent stiffness and equivalent damping ratio are plotted as function of the excitation frequency and the excitation amplitude.

The equivalent stiffness k e increases with increasing excitation frequency f t , while the latter has no significant effect on the equivalent damping c e f . Thus, the damping ratio ζ r decreases with increasing excitation frequency. However, the results shown in Fig. 20 are no longer valid for high excitation frequency and large amplitudes, since regular hysteretic behavior is absent in that region of operation (see Fig. 19).

Conclusions

In this paper, the simulation of transient rolling friction is reported. The interplay between friction and the other system dynamics is investigated.

Firstly, a brief introduction is given to to rolling friction model, which enables one to simulate the transient traction stresses in the contact of a rolling body and upon a counter surface. Thereafter, that model is used in the dynamic simulation of systems comprising rolling elements. The basic characteristics of such systems, such as the traction forces in the contact and the resulting complex stiffness are examined. It turns out that systems comprising rolling elements can manifest complex dynamics that depends on the amplitude and frequency of excitation.

For the case of a ball rolling on a massless plate, an increasing complex stiffness as a function of excitation frequency is observed, up to a certain break frequency. For higher frequencies, the amplitude of the complex stiffness becomes constant. The influence of plate inertia, was also investigated in the time and the frequency domains.

Finally, owing to its relevance in industrial applications, a system of a ball rolling between two V-grooved tracks is investigated, where a combination of kinematic creepage and dynamic creepage occurs. The study of these dynamic effects in frictional contacts, can significantly help improve the position accuracy of systems comprising these elements. Furthermore, a profound knowledge of the frictional dissipation in the rolling contacts, such as given in this paper, permits a better estimation of lifetime and performance degradation due to fatigue and wear. With G being the equivalent modulus of rigidity of the sphere, L 1 , L 2 and L 3 are obtained from [START_REF] Kalker | Three-dimensional elastic bodies in rolling contact[END_REF] as:

L 1 =

Figure 1 :Figure 2 :

 12 Figure 1: Overview of the rolling motion. (reproduced with permission from [23])

Figure 3 :

 3 Figure 3: The results for transient (pre-)rolling with longitudinal creepage. In the middle figure, the hysteresis plot is depicted, the other figures depict the local traction field. (reproduced with permission from [23])

Figure 4 :

 4 Figure 4: Rolling dynamics case 1: A 1-DOF system.

.Figure 5 :

 5 Figure 5: Free response of the system. (a) Phase portrait, (b) friction-force vs. angular displacement.

Figure 5 (

 5 Figure 5(a) depicts the phase plot of the free response, with initial conditions θ0 = 0.3 and θ 0 = 0 and null initial traction field. In the nomenclature section, an additional table is added with the numerical values of the variables used in the simulations. The amplitude dependency of the damping is clearly visible. In Figure 5(b), the friction force is plotted as function of the ball rotation angle. During the oscillatory decay, the increasing stiffness of the hysteresis behavior is observed, as evidenced by the ever increasing slope of force-displacement curve.

Figure 6 :

 6 Figure 6: Ramp input to the position of the plate xt. (a) the ball lags the plate, (b) damped oscillations of the ball.

  7(b), f s is plotted as a function of the excitation amplitude. One can notice a decreasing exponential trend for f s as function of A. A profound study of this behavior in function of the different model parameters can result in a better insight into the problem and a consequently better design to avoid early wear and failure of the system due to excessive slip in the contacts. Because the model parameters are reduced by normalizing w.r.t. the radius of

Figure 7 :

 7 Figure 7: Frequency response function of the tangential contact stiffness. (a) complex stiffness Fx/xt as function of frequency for different excitation amplitudes A. (b) break frequency fs as function of excitation amplitude A.

  shows the absolute frequency response function of the simulated force/displacement for an indicated excitation amplitude A=1, for different masses of the plate. As can be seen, the shape of the transfer function changes drastically with changing plate mass. Obviously, the limit M =0 corresponds to Fig. 7(a) (A=1), while for M → ∞ , the mass-line limit is approached.

Figure 8 :

 8 Figure 8: The frequency response function for different plate masses (A * =1).

Figure 9 :

 9 Figure 9: Rolling dynamics case 2:A 2-DOF system: rotation and translation.

Figure 10 :

 10 Figure 10: Spectrum of F x1 . The two distinct peaks represent the translation and the rotation resonance frequencies.

Fig. 11 .Figure 11 :

 1111 Figure 11: Rolling dynamics case 2: A velocity step response.

Figure 12 :

 12 Figure 12: Sinusoidal excitation with different frequencies ft. (a) motion response for ft = 1, (b) friction force in the contacts for ft = 1, (c) motion response for ft = 6.5, (d) friction force in the contacts for f t = 6.5, (e) motion response for f t = 10, (f) friction force in the contacts for ft = 10.

Figure 13 :

 13 Figure 13: Rolling in between two plates with sinusoidal excitation of the top plate and increasing normal load: increasing dimensionless contact patch radius as function of time corresponding to the linearly increasing dimensionless external load.

Figure 14 :

 14 Figure 14: Rolling in between two plates with sinusoidal excitation of the top plate and increasing normal load: evolution of the friction force in the upper (F x1 ) and lower (F x2 ) contact.

Figure 15 :Figure 16 :

 1516 Figure 15: Rolling dynamics case 3: V-groove rolling (reproduced with permission from [24]).

Fig. 17

 17 Fig. 17. The equivalent stiffness k e and the equivalent damping c e f decrease

  Fig. 17. The equivalent stiffness k e and the equivalent damping c e f decrease with increasing ball radius R * and increasing groove angle α, and increase with increasing coefficient of friction μ. The damping ratio decreases with increasing

Figure 17 :

 17 Figure 17: Equivalent stiffness k e and equivalent damping ratio ζ r as function of excitation amplitude for (a) different dimensionless ball radii, (b) different coefficients of friction and (c) different V-groove opening angles.

Figure 18 :

 18 Figure 18: The hysteresis loop for different excitation frequencies f t (α = π 3 ).

Figure 19 :Figure 20 :

 1920 Figure 19: The displacement-force plot for ft = 5.

  pr : pre-rolling distance α : V-groove opening angle δV x : dimensional longitudinal creepage δV y : dimensional lateral creepage

Acknowledgements

This research is supported by the Fund for Scientific Research -Flanders (F.W.O.) under Grant FWO4283. The scientific responsibility is assumed by its authors.