Marcel Aloy
email: marcel.aloy@univ-amu.fr

Gilles De Truchis
email: gilles.de-truchis@univ-amu.fr

Gilles Dufrénot
email: gilles.dufrenot@univ-amu.fr

Benjamin Keddad
email: benjamin.keddad@univ-amu.fr

Shift-Volatility Transmission in East Asian Equity Markets

Keywords: Regime shifts, Equity Volatility, East Asia, TVPMS JEL: R31, G15, C32

published or not. The documents may come

Introduction

This paper attempts to provide evidence of "shift-volatility" transmission in the East Asian equity markets. By "shift-volatility", we mean that in these markets, the volatility shifts from a low level to a high level, usually corresponding respectively to tranquil and crisis periods. We investigate which of the markets, regional or global, are the most influential in driving the volatility into different regimes. The optimal equity portfolio is likely to be different in calm and turbulent periods, thereby implying that exploiting regime switching may lead to shift assets from one market to another when a bull or a bear market is expected. The presence of different states in the volatility can be explained by a diversity of factors, among which the heterogeneity of arbitrageurs, the rigidity of price adjustment, jumps in the asset allocation reflected for instance by time-varying betas [START_REF] Ang | Regime Switches in Interest Rates[END_REF], and asymmetric correlation phenomena [START_REF] Ang | Asymmetric Correlations in Equity Portfolios[END_REF][START_REF] Longin | Correlation structure of international equity markets during extremely volatile periods[END_REF].

Most empirical papers considering such possibilities focus on the asymmetric nature of volatility, and cross-market second-order moment correlations, using nonlinear models. These models can be divided into two categories, depending on whether they identify a priori the different regimes, or whether they are based on an endogenous identification from the data. The first category of models include nonlinear VARs, threshold autoregressive models (TAR), nonlinear multivariate GARCH models (see e.g. [START_REF] Chou | Cointegration in international stock market indices[END_REF][START_REF] Hamao | Correlation in price changes and volatility across international stock markets[END_REF]. The second type of models is based on Markov-switching models where a hidden state variable follows a Markov chain. These models have been designed as a way of doing "identification through heteroskedasticity" (see [START_REF] Rigobon | Identification through heteroscedasticity[END_REF] where the timing of changes from low to higher volatility regimes is endogenously determined. Examples of papers are [START_REF] Ang | Regime Switches in Interest Rates[END_REF], [START_REF] Billio | Contagion detection with switching regimes: a short and long run analysis[END_REF], [START_REF] Fratzscher | On currency crises and contagion[END_REF] and [START_REF] Gravelle | Detecting shift-contagion in currency and bond markets[END_REF]. The present paper is based on the second approach. Unlike many previous papers, we do not consider standard Markov-switching models [START_REF] Hamilton | A new approach to the economic analysis of nonstationary time series and the business cycle[END_REF][START_REF] Hamilton | Analysis of time series subject to changes in regime[END_REF] where the probabilities of moving from one regime to another are fixed over time. Instead, we assume that the regime-shifts are conditioned by the dynamics of the volatility in another influent market, which is a good way of capturing volatility spillover effects. We accordingly refer to the framework originally proposed by [START_REF] Filardo | Business-cycle phases and their transitional dynamics[END_REF] and explore time-varying probability Markov-switching models (TVPMS).

We propose to examine the interdependence of equity volatilities between Hong Kong, Indonesia, Japan, Malaysia, the Philippines, Singapore, Thailand and the United States. In doing this exercise, our main issue is whether shiftvolatility needs to be considered as a regional phenomenon, or from a more global perspective. It has been shown that the US market is more influential than the Japanese market in transmitting the volatility to the other Asian markets [START_REF] Liu | Mean and volatility spillover effects in the U.S. and Pacific-Basin stock markets[END_REF], but that during the 1997 and 1998 Asian crisis, the regional transmission played a significant role due to an increased co-movement of the Asian markets in turbulent periods [START_REF] In | Dynamic interdependence and volatility transmission of Asian stock markets evidence from the Asian crisis[END_REF][START_REF] Jang | The Asian financial crisis and the co-movement of Asian stock markets[END_REF].

Moreover, the impact of the Japanese market is found to be important whatever the state of the volatility, low or high [START_REF] Chuang | Interdependence of international equity variances: evidence from East Asian markets[END_REF]. In this paper, we investigate these issues by considering data that include both the years of the Asian crisis (at the end 1990s) and those of the more recent financial crisis (after 2008).Within this framework we analyze to what extent the influence of regional markets volatility is more significant than that of international markets (notably the US market). This sheds some light on the extent to which investors could benefit or not from portfolio diversification within the East Asian region. We find that international influence is not necessarily more important than regional influence in certain countries, although convincing evidence of different volatility regimes is obtained for all markets. The main contributions of the paper are the following. Firstly, the timing/spans of high volatility regimes identified by our model correspond adequately to years historically documented as those of crises (the Asian crisis and the years following the 2008 crisis). Secondly, our paper goes beyond finding just volatility regimes, since we suggest different indicators that could be useful to guide the investors in their arbitrage behavior in the different regimes: the duration of each state, the sensitivity of the volatility in a market following a change in the volatility in another market. Finally, we are able to identify which market can be considered as leading markets in terms of volatility.

The remainder of the paper is as follows. In section 2, we briefly review some previous empirical studies on volatility spillovers and contagion in East Asian stock markets. In section 3 we describe the empirical model. Section 4 presents and discusses the results. Section 5 concludes.

Volatility spillovers and contagion in East Asian stock markets

Over the last decades, East Asian economies have emerged as a pole of economic growth. The key factors underlying the catching-up process relate to greater economic openness and globalization. These economies have indeed undertaken substantial reforms aiming at liberalizing their capital account and their domestic financial system, opening up additional channels for cross-border linkages. Indeed, with the removal of restrictions, markets participants have found considerable opportunities to reduce the risk of their portfolio through international diversification strategy (see e.g. [START_REF] Grubel | Internationally diversified portfolio: Welfare gains and capital flows[END_REF][START_REF] Levy | International diversification of investment portfolios[END_REF]. Strictly speaking, Asian countries attracted about 57% of total financial inflows, such as FDI and FPI, to emerging market economies since 1990. Financial liberalization has contributed to more diversified and deepened linkages at both global and regional levels, which in turn, has given rise to a higher degree of stock market co-movement. For instance, [START_REF] Bekaert | Emerging equity market volatility[END_REF] examine the effects of financial liberalization and find that capital market reforms in emerging markets often increase the correlation between local stock returns and the world market without drive up local market volatility. This view is shared by some researchers for whom financial liberalization has either a reducing impact or no impact on volatility (see e.g. De Santis and [START_REF] De Santis | Stock returns and volatility in emerging financial markets[END_REF][START_REF] Hargis | Forms of foreign investment liberalization and risk in emerging markets[END_REF][START_REF] Umutlu | The degree of financial liberalization and aggregated stock-return volatility in emerging markets[END_REF].

On the other hand, intensified financial linkages in a world of high capital mobility may also harbor the risk of volatility spillovers, especially during episodes of financial stress. Accordingly, the tighter linkages among East Asian countries, and with the rest of the world, have enlarged the scope of spillover from global and regional shocks. This weakness was confirmed by the fact that these countries have experienced huge variations of their stock market volatility during both the 1997/1998 crisis and the 2008 global financial turmoil. The intrinsic vulnerability of these countries confronted to external shocks finds its causes in the lack of financial market depth which limit their capacity to absorb capital inflows but also in the heavy dependence on global financial intermediation conducted in international reserve currencies. Consequently, an abrupt swing in market sentiment and a sudden change in liquidity conditions could provoke a currency crisis (see e.g. [START_REF] Radelet | The East Asian financial crisis: diagnosis, remedies, prospects[END_REF][START_REF] Chang | The Asian liquidity crisis[END_REF]. One important feature of financial crisis is that market turbulence can spread rapidly to other countries when the markets are highly integrated. The recurrence as well as the magnitude of financial crisis during the 1990s has spurred numerous empirical studies which attempt to identify whether the spillovers are the results of contagion effects or rather the expression of strong linkages and interdependencies among markets (see e.g. [START_REF] Forbes | No contagion, only interdependence: measuring stock market comovements[END_REF].

The interaction between stock market volatilities in East Asia has been widely studied in the literature. Several econometric approaches have been applied such as, among other, VAR and cointegration analysis, GARCH class of models or Granger-causality methodology. In a recent study, [START_REF] Engle | Volatility spillovers in East Asian financial markets: a MEM-based approach[END_REF] use a MEM-Based approach to model the volatility spillovers in eight East Asian stock markets before, during, and after the Asian currency crisis.

The authors find significant interdependencies within the region where Hong Kong has a crucial role in influencing other markets. Using a spillover index methodology (see e.g. [START_REF] Diebold | Measuring Financial Asset Return and Volatility Spillovers, with Application to Global Equity Markets[END_REF] for ten East Asian countries, Yilmaz (2010) shows that the volatility spillover index experiences significant bursts during major episodes of financial stress, including both the East Asian and the recent global financial crises. Worthington and Higgs (2004) use a Multivariate GARCH model to identify the source and magnitude of spillovers during the period 1988 to 2000 for a panel of nine Asian countries, including both developed and emerging equity markets. While they find evidences that these markets are highly integrated, the authors suggest that volatility spillovers from changes in domestic market conditions are generally higher than cross-volatility spillovers for all markets, but especially for the emerging markets. [START_REF] Otranto | The multi-chain Markov switching model[END_REF] proposes another approach in which the dynamics of the regime volatility of a variable depends on the dynamics of the regime volatility of the other variables. Gallo andOtranto (2007, 2008) apply this multi-chain Markov switching (MCMS) model to volatilities of several Asian stock markets and found that such model is also appropriate to estimate both contagion and co-movements. Nonetheless, their methodology implies to determinate the market from which the shock originates. Accordingly, this assumption excludes the possibility, for the dominated market, to influence the dominant market. Empirical studies on the extent of contagion and interdependence across East Asian countries during the 1997-98 crisis has also been conducted by [START_REF] Khan | Contagion in the stock markets: The Asian financial crisis revisited[END_REF], [START_REF] Ratanapakorn | Interrelationships among regional stock indices[END_REF] and [START_REF] Chiang | Empirical analysis of stock returns and volatility: Evidence from seven Asian stock markets based on TAR-GARCH model[END_REF], among other.

Other studies have focused on the interactions between the US and East Asian equity markets to trace the influence of world and regional factors on market volatility. For instance, [START_REF] Cheung | East Asian equity markets, financial crises, and the Japanese currency[END_REF] show that the US market leads the daily equity returns on Hong Kong, Korea Singapore and Taiwan markets before, during and after the 1997 Asian financial crisis and the opposite is true only during the crisis period. In a comparative analysis, [START_REF] Miyakoshi | Spillovers of stock return volatility to Asian equity markets from Japan and the US[END_REF] estimates a volatility spillover model to examine whether Asian stock markets are influenced by the world factor (namely the US market) or by the regional factor (the Japanese market). The author shows that the volatility of the Asian market is more influenced by the Japanese market than by the US one. Conversely, [START_REF] Ng | Volatility spillover effects from Japan and the US to the Pacific-Basin[END_REF] finds significant volatility spillovers inside the region, e.g. a regional shock originating from Japan is transmitted to most of the Pacific-Basin countries. In the context of the recent US financial crisis, the analysis of volatility spillovers has been conducted by [START_REF] Morales | The current global financial crisis: Do Asian stock markets show contagion or interdependence effects[END_REF]. According to these authors, the US crisis does not generate contagious effects in East Asia although they find strong evidence of volatility transmission suggesting that these markets are highly integrated. Another recent study conducted by [START_REF] Samarakoon | Stock market interdependence, contagion, and the U.S. financial crisis: The case of emerging and frontier markets[END_REF] shows that interdependence is driven more by US shocks, while contagion is driven more by emerging market shocks.

In this paper, we focus on both emerging and developed East Asian stock markets that are the Stock Exchange of Thailand (SET), the Straits Time Index of Singapore (STI), the Philippines Stock Exchange Index (PSEI), the Kuala Lumpur Composite Index (KLCI), the Jakarta Composite Index (JCI), the Hang Seng Index (HSI) and the Nikkei 225 (NKY). Since we are interested in the transmission of stock market volatility from developed markets and US market to emerging markets, we also consider the Standard & Poor's 500 (SPX). In line with many studies, we use the range-based volatility as a proxy for the true volatility (see e.g. [START_REF] Christensen | Realized range-based estimation of integrated variance[END_REF][START_REF] Martens | Measuring volatility with the realized range[END_REF][START_REF] Engle | Volatility spillovers in East Asian financial markets: a MEM-based approach[END_REF]. Therefore, the integrated volatility is constructed based on daily range and is calculated according to the following formula:

r t,i = ln h t,i l t,i (1)
with r t,i , the daily range measure defined as the difference between the highest (h t,i ) and the lowest (l t,i ) log security prices over a day, with i = {S ET, S T I, PS EI, KLCI, JCI, HS I, NKY, S PX}. The sample covers 18 July 1995 to 29

November 2012.

The empirical model

In order to investigate shift-volatility transmission, we adopt the framework initially proposed by [START_REF] Filardo | Business-cycle phases and their transitional dynamics[END_REF].

We assume that there are two volatility regimes with a given variable driving the data into either one or the other regimes. The transition between the two regimes is governed by a time-varying transition probability matrix. Though the choice of two regimes is made here for simplicity's sake, another argument for not considering a higher number of states in exploring the volatility dynamics is that, in general, they provide little improvement in the likelihood (see e.g. [START_REF] Ardia | Bayesian estimation of a Markov-switching threshold asymmetric GARCH model with student-t innovations[END_REF][START_REF] Henneke | MCMC-based estimation of Markov switching ARMA-GARCH models[END_REF]. The model we explore has the following parametric form:

y t = α s t + β s t-1 (y t-1 -α s t-1 ) + σ 2 s t ε t , s t = {1, 2}, ε t ∼ i.i.d. N(0, 1), t = 1, . . . , T (2)
The endogenous variable, y t , is assumed to visit the two states of a hidden variable s t that follows a first-order Markov chain, over the T observations. s t is assumed to satisfy regularity conditions such as aperiodicity, irreducibility and ergodicity. Denoting z t the driving (or transition) variable, we define the following transition probability functions,

p 11 (z t-1 ) = exp(γ 11 + γ 12 z t-1 ) 1 + exp(γ 11 + γ 12 z t-1 ) , p 22 (z t-1 ) = exp(γ 21 + γ 22 z t-1 ) 1 + exp(γ 12 + γ 22 z t-1 ) (3)
where p 11 (z t-1 ) and p 22 (z t-1 ) are elements of the following transition probability matrix

P(z t-1 ) =           p 11 (z t-1 ) p 12 (z t-1 ) p 21 (z t-1 ) p 22 (z t-1 )           (4) with p 21 (z t-1 ) = 1 -p 11 (z t-1 ), p 12 (z t-1 ) = 1 -p 22 (z t-1 ). p i j (z t-1
) is the probability of moving from state j to state i at time t, one day ahead following a change in the transition variable. For purpose of comparison we also consider the fixed probability model [START_REF] Hamilton | Analysis of time series subject to changes in regime[END_REF]

L(y t ; Θ) = T t=1 2 i=1 2 j=1 ln f (y t |s t = i, s t-1 = j, Ω t , ξ t-1 ; Θ) × P (s t = i, s t-1 = j|Θ t , ξ t-1 ; Θ) (5) = T t=1 ln f (y t |Ω t , ξ t-1 ; Θ) (6)
where ξ t = (y t , y t-1 , . . . , y t ) and Ω t = (Y t-1 , Z t-1 ) denotes the vector of observations of y and z up to t -1. Considering the normality assumed in Eq. ( 2), the regime-dependent densities are defined as

f (y t |Ω t , ξ t-1 ; Θ) = 1 √ 2πσ i exp           -y t -α i -β j (y t-1 -α j ) 2 2σ 2 i           (7)
The estimation of such a model is not trivial and can be achieved by employing three different methodologies. The first combines the gradient method and the Hamilton's filter and relies on the maximum likelihood (ML). The second relies on the so-called expectation-maximization (EM) algorithm. The third relies on a Bayesian approach and uses the Gibbs sampler. In this paper, we estimate the model applying the ML method for mixtures of Gaussian distributions, which provides efficient and consistent estimates under the normality assumption (see e.g. [START_REF] Kim | Estimation of Markov regime-switching regression models with endogenous switching[END_REF]. Applying the Bayes' rule, the weighting probabilities are computed recursively. We start from the following relation,

P(s t = i, s t-1 = j|Ω t , ξ t-1 ; Θ) = P(s t = i, s t-1 = j|z t ; Θ)P(s t-1 = j|Ω t , ξ t-1 ; Θ) (8) = p i j (z t )P(s t-1 = j|Ω t , ξ t-1 ; Θ) (9)
Then, we deduce the unconditional density of y t in each regime and finally obtain the following unconditional probability:

P(s t = i, s t-1 = j|Ω t , ξ t-1 ; Θ) = j f (y t |s t = i, s t-1 = j, Ω t , ξ t-1 ; Θ)P(s t = i, s t-1 = j|Ω t , ξ t-1 ; Θ) f (y t |Ω t , ξ t-1 ; Θ) (10)

Standard analysis of results

The transition variable (z t ) is chosen to be the volatility in some leading markets: we take alternatively the Standard and Poor's (SPX), the Nikkei (NKY), the Hang Seng (HSI) and the STI of Singapore indices. These time series 19904,4 19916,3 19912,9 19921,3 20334,2 20408,4 20348,8 20381,0 19962,2 20028,2 19974,3 19981,8 LR signif. 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 are standardized in the transition functions in order to get an easiest interpretation and comparison of the estimated parameters. Tables 1, 2 and 3 contain our estimation results. The models very often dichotomize between two regimes of respectively low and high volatility (compare the intercepts in each regime). For instance, in the case of the HSI index, any change in the standardized volatility of the SPX index is expected to yield a high volatile regime characterized by a mean volatility of 1.1% per day, or a low volatile regime with a mean volatility equal to only 0.5% per day. Therefore shift in volatility is caused by "jumps" in the volatility of the leading market equities. The Hang Seng Index (HSI) is chosen to illustrate graphically the relationships between the volatility of Asian markets and the volatility related to alternative leading markets: the United States and Singapore3 .

These relationships are respectively depicted in the first graphic of the Figure 1 where the shaded areas refer to the high volatility regime of the transition variables. The HSI clearly exhibits volatility clustering and important episodes of extreme volatility, particularly during the Asian financial crisis and the recent global financial turmoil.

Figures 1 and 2 display the smoothed posterior probability of regime 2 (high volatility) for the stock market volatility of HSI according to the transition variables. Obviously, regime 2 appears to be quite persistent over the sample LR signif. 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 and periods characterized by a high probability of being in a turbulent stock market regime are recurrent and longlived. When examining jointly the integrated volatility of HSI and the smoothed probabilities, the results offer fairly convincing evidence that clusters of extreme values in the volatility can be precisely identified by the TVTP model.

For instance, the smoothed probabilities indicate high volatility regimes during the major events of financial stress, such as the financial crises in emerging markets in the late 1990s, the global economic downturn in 2001, the subprime mortgage crisis of 2007, the collapse of Lehman brother in 2008 and the EU sovereign debt crisis in 2011. Further, these graphs also show that volatility transmission from United States to Hong Kong is clearly identified since the smoothed probabilities are highly correlated with the shaded areas corresponding to shocks emanating from United

States. We obtain similar conclusion for STI. This suggests that the volatility in the Hong Kong and Singapore equity markets evolves in line with the volatility in the US market. We also observe that the time-varying probabilities are function of the volatility in the leading markets. Figures 3 and4 plot the values of p 11 and p 22 implied by the changes in the volatility of SPX, and STI, respectively. Figure 3 shows that when the volatility of SPX is maintained at his sample mean value (i.e. z t-1 = 0) 4 the probability of staying in the calm regime is approximately 0.95. We find similar results with STI. However, as the volatility in the leading markets increases above his average level, the probability p 11 decreases, thereby suggesting that a positive increase in z t-1 makes a feedback to a high volatility regime more likely. This effect is particularly rapid for STI but less than SPX. 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 Turning to the estimated coefficients in the Tables 1, 2 and 3, we see that they are very often statistically significant at the 1% level of significance,and that the LR tests (which test the null hypothesis of a fixed transition probability matrix against the alternative hypothesis of a time-varying one) yield to reject the Hamilton constant probability model. In both cases of constant and time varying probabilities models, the transition probabilities p 11 and p 22 show that the expected duration of "calm" periods -associated with regime 1 -is longer than the duration of "turbulent" periods or high volatility regime associated with regime 2. Inspection of the model's log-likelihood value reveals some "preferred models", in the sense that their log-likelihood is superior to other specifications: (1) HSI and STI regime- HSI 1995HSI 1996HSI 1997HSI 1998HSI 1999HSI 2000HSI 2001HSI 2002HSI 2003HSI 2004HSI 2005HSI 2006HSI 2007HSI 2008 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 19361,2 19374,0 19377,7 19391,1 19397,5 19200,5 19246,9 19212,9 19244,6 19238,5 LR signif. 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000

HSI

State-dependent indicators

The expected duration of each state (ED)

Assume that the transition variable is permanently maintained at his sample mean, that is z t-1 = 0 since z t-1 is the standardized volatility in the leading market. From Eqs.

(3), we can infer the following transition probabilities

p ii (z = 0) = exp(γ i1 ) 1 + exp(γ i1 ) , i = {1, 2}. (11)
and thus, the expected duration (ED) of each state, which is expressed in number of days, can be easily computed as :

ED i (0) = 1 1 -p ii (z = 0) , i = {1, 2}. (12)
Since regime 1 corresponds to the calm period and regime 2 represents the turbulent period -as shown by the estimated constant in each regime -the first indicator measures the expected number of days of staying in the low volatility regime and the second one the expected number of days of staying in the high volatility regime. As shown in Tables 4, 5 and 6, the HSI index is characterized by the longer ED during the calm periods (33 days if the STI is 10 taken as the leading market and 41 days if the SPX is the leading market), followed by the KLCI (about 25 days) and the SET (about 20 days). At the opposite, the shortest ED in low volatility regime is found in PSEI and NKY markets (about 7 to 8 days).The longer ED related to turbulent periods is observed in the HSI market (about 20 days). In the other markets, the ED during regime 2 is found to be very short: it varies between 1.5 days in the NKY market and about 7 days in SET and KLCI markets. -0,016 -0,018 -0,040 -0,080 -0,028 -0,065 -0,081 -0,039 -0,044 MP 2 (0) 0,025 -0,002 0,024 0,129 0,007 0,105 0,168 0,095 0,109

The variation of expected duration of each state (VED)

Assume that, from the situation where z t-1 = 0, the volatility in the leading market is permanently increased by one standard-deviation. Thus, due to the standardization, it results that z t-1 = 0 and we have the following transition probabilities

p ii (z = 1) = exp(γ i1 + γ i2 ) 1 + exp(γ i1 + γ i1 ) , i = {1, 2}. (13)
Following this permanent shock of one standard deviation in the volatility of the leading market we can compute a new set of two expected durations (ED):

ED i (1) = 1 1 -p ii (z = 1) , i = {1, 2}. (14)
These indicators enable us to calculate the variation of the ED's resulting from a permanent increase (or decrease, because of local symmetry) of one standard deviation in the volatility of the leading market:

V ED i = ED i (1) -ED i (0), i = {1, 2}. (15)
These indicators capture the impact of a change in volatility occurring in the leading market in terms of expected duration of each regime. In the cases of our preferred specifications (see Tables 4, 5 and6), we obtain the following conclusions.

-The HSI and STI markets. A permanent increase of one standard deviation in the volatility of the SPX index decreases the expected duration of the low volatility regime by 20.5 days in the HSI market and 9 days in the STI market. Conversely, a decrease of one standard deviation in the volatility of the SPX index decreases the duration of the high volatility regime by 14 days in the HSI market and only 2 days in the STI market. The VED indicator makes clear an asymmetrical influence of the SPX index, since the HSI market appears to be much exposed to its variations than the STI market. Regarding the interactions between these two Asian markets, we can note that the index STI is influenced quite similarly by the two indices HSI and SPX. Conversely, at least during periods of high volatility (regime 2), the HSI market is less sensitive to changes in volatility on the STI market in comparison to those which affect the SPX market. -0,049 -0,029 -0,062 -0,042 -0,015 -0,019 -0,032 -0,028 MP 2 (0) 0,064 0,025 0,064 0,119 -0,001 0,004 0,005 0,050 -The JCI market. This market is identically influenced by the SPX and STI markets. A permanent increase of one standard deviation in the volatility of the SPX or the STI index decreases the expected duration of the low volatility regime by about 6 days in the JCI market. A decrease of one standard deviation in the volatility of the SPX or the STI index decreases the expected duration of the high volatility regime by less than 1 day.

-The KLCI, PSEI and SET markets. The three markets are influenced similarly by STI and HSI markets. In the case of KLCI and SET indices, we can notice a strong effect of the leading markets upon the expected duration in regime 1, but not in regime 2. In the case of the PSEI market, the influence of the leading markets is weak in both regimes in terms of VED. Nevertheless, since the PSEI market is characterized by a short expected duration in both regimes, we could not expect to get ahigh value of the VED.

The marginal probability (MP)

A limitation of the VED measure is that the expected duration in the high volatility regime is often very low, and thus it is also necessarily the case of the VED in this regime. We can thus suggest a related measure of the impact of an increase of one standard-deviation in the volatility of the leading market based on the first derivative of the probability function with respect to z t-1 :

∂p ii (z t-1 ) ∂z t-1 = γ i2 exp(γ i1 + γ i2 z t-1 ) 1 + exp(γ i1 + γ i1 ) 2 , i = {1, 2}. (16)
These expressions give a measure of the marginal transition probabilities (MP) associated with a variation in z t-1 and the value taken by these derivatives depends on the specific value of the transition variable. It is thus easy to compute the MP at z t-1 = 0, i.e. at the sample mean of the transition variable, which we denote MP i (0) for i = 1, 2:

MP i (0) = γ i2 exp(γ i1 ) 1 + exp(γ i1 ) 2 , i = {1, 2}. (17)
In this case, MP i (0) measures the variation in the transition probability induced by an increase (or a decrease because of local symmetry) of one standard deviation from the sample mean in the volatility of leading market. As shown in Tables 4, 5 and 6, in most cases (JCI, PSEI, SET) the MP i (0) indicator is about -0.05 in regime 1, which means that, starting from its sample mean, an increase of one standard deviation of the volatility in the leading market (SPX, HSI, or JCI) leads to a 0.05 decrease in the probability to stay within the same regime of low volatility at the next period. In some cases, this marginal probability reaches higher values, for instance -0.08 in the STI and NKY market when SPX is used as leading market. Finally, KLCI and HSI markets appear to be less sensitive in terms of marginal probability (less than -0.03 in the first case and less than -0.02 in the second one when SPX is taken as leading market).

In these cases, results are somewhat at odds with those obtained using the VED indicator. Moreover, one interesting point concerns the potential asymmetry between the MP i (0) over the two regimes. It appears clearly in most cases (PSEI, STI, NKY and -depending on the chosen leading market -JCI, SET and KLCI) that the marginal probability is greater in regime 2 (high volatility) than in regime 1. In those cases, a decrease of one standard deviation in the leading market's volatility (starting from its sample mean) leads to a decrease of about 0.1 of the probability to stay in the same regime of high volatility at the next period. Although this point is not confirmed in the case of the HSI market, these results leads us to conclude that, overall, a variation of the volatility in the leading market has greater effects in terms of marginal probability during high volatility periods than during calm periods.

The maximum marginal probability (MMP)

It is known that the curves of the transition probability functions have an inflection point which corresponds to p 11 = 0.5 and p 22 = 0.5. This specific point is associated to the maximum marginal probability, which means that a given variation of the transition variable will result in a maximum change in the transition probability. The maximum marginal probability (MMP) is the slope of the tangent at the inflexion point. In the case of the logistic function used as the transition probability function, we have:

MMP i = γ i2 1 4 , i = {1, 2}. (18)
Accordingly, the value of the transition variable corresponding to this MMP is given by

z i M = -γ i1 1 γ i2 , i = {1, 2}. ( 19
)
The interpretation is as follows: if the transition variable increases of one standard deviation from the specific value z i M , then the probability to stay in the regime 1 increases by the value of MMP i Reg.1. Since p ii = 0.5 when z t-1 = z i M it results that p ii will be equal to 0.5+MMP i after the shock occurs.Inspection of empirical results (Tables 4, 5 and6) reveals some interesting features: in all cases except the PSEI and NKY indices, one can find a maximal marginal probability measure heavily asymmetric and greater (in absolute value) during low volatility regimes. This result is in sharp contrast with the MP i (0) indicator. However, if we refer to the value z i M at which the MMP i is calculated, we must admit that there is little chance that the value of MMP i is actually reached (besides the negative values of the indicator are so strong that they correspond to the negative volatility). As a consequence, we have to admit that the MMP i measure is not relevant in this case.

Final comments

The results in this paper suggest several features regarding shift-volatility caused by cross-market interdependence.

Firstly, based on the model's likelihood, the Asian emerging markets can be segmented into two groups: the first one composed by HSI, STI, and JSI, appears to be positively influenced by the volatility in the U.S. market (SPX), while the second group composed by KLCI, PSEI and SET are primarily influenced by the others Asian emerging markets (namely HSI and STI). This suggests that, in Hong Kong, Singapore, and Japan, the existing linkage of the second moments between these markets and the US market, should provide less incentive towards an international equity diversification because of volatility's co-movements. On the contrary, the equities in Indonesia, Malaysia and the Philippines could be viewed as safer assets since the risk they incorporate is immune from the US equity risks.

Secondly, though very different from one market to another, the expected duration is much longer in the low volatility regime than in the turbulent one. Thirdly, the VED measure appears to be directly related to the expected duration in each regime: a variation in the volatility of the leading market will induce a reduction of the expected duration of a given regime as much stronger than the initial duration of this regime is high. Accordingly, an increase in the volatility of the leading market will thus reduce heavily the expected duration of the calm regime. However, the observed heterogeneity of expected durations between markets makes it difficult to compare the results related to the VED measure.In this respect, the marginal probability (MP) indicator provides useful additional information. In the specific case of the HSI market, a variation of the volatility in the leading market starting from the sample mean, is found to have greater effects in terms of marginal probability during low volatility periods (regime 1) than during turbulent periods (regime 2), but this conclusion is reversed in all of the others markets. Thirdly, it appears finally that the influence of the leading market on the transition probabilities depends on the initial starting value of the volatility in the leading market. The maximum marginal probability (MMP) indicator shows that, in most cases, an increase in volatility of the leading market starting from a high initial value implies a sharp reduction in the probability to stay in a low volatility regime, and thus provides some evidences of the financial contagion hypothesis. Nevertheless in the case of high volatility regime the inflexion point of the transition function corresponds to implausible values of the

Figure 1 :

 1 Figure 1: Range-based volatility of HSI market and the TVTP of being in regime 1 with SPX as information market.

  shifts in volatility are both influenced by the SPX market and mutually interrelated; (2) JCI regime-shifts in volatility is influenced by SPX and STI; (3) KLCI, PSEI and SET are influenced by HSI and STI; (4) NKY is influenced by SPX.

Figure 2 :

 2 Figure 2: TVTP of being in regime 1 with STI as information market.

Figure 3 :

 3 Figure 3: Logistic transition function of HSI regime switches with information market SPX.

Figure 4 :

 4 Figure 4: Logistic transition function of HSI regime switches with information market STI.

  If there is no statistically meaningful impact of the volatility in a leading market on the volatility in other markets, then the TVPMS model converges to the Hamilton fixed probability model. In this case we may have shift-volatility due to other factors than cross-market correlation. The unconditional log likelihood function of y t is defined as

as a benchmark. In this second model, the transition probability functions are defined as p 11 (θ 11 ) = exp(θ 11 /(1 + exp(θ 11 )) and p 22 (θ 22 ) = exp(θ 22 /(1 + exp(θ 22 )). In Eqs. (3), the coefficients γ 12 and γ 22 make explicit the link between the transition variable and the probability of remaining in or leaving regime 1 and regime 2.

Table 1 :

 1 Estimation results of FTP and TVTP models for dependent variables HSI, STI and NKY

	y t		HSI				STI				NKY
	z t	FTP	SPX	Nikkei	STI	FTP	SPX	Nikkei	HSI	FTP	SPX	STI	HSI
	α 1	0,005	0,005	0,005	0,005	0,004	0,005	0,004	0,004	0,005	0,006	0,006	0,006
	t-stat	(72,30)	(75,17)	(75,80)	(77,50)	(67,76)	(68,57)	(68,04)	(62,47)	(76,01)	(72,71)	(71,84)	(67,38)
	α 2	0,011	0,011	0,011	0,011	0,010	0,010	0,010	0,010	0,011	0,010	0,010	0,010
	t-stat	(41,77)	(44,02)	(44,23)	(48,33)	(37,35)	(36,43)	(37,49)	(37,12)	(36,64)	(42,11)	(36,43)	(31,83)
	β 11	0,242	0,248	0,250	0,263	0,445	0,489	0,453	0,459	0,426	0,569	0,565	0,558
	t-stat	(9,72)	(10,50)	(10,86)	(11,47)	(20,63)	(21,26)	(20,90)	(18,92)	(20,11)	(25,18)	(22,42)	(17,53)
	β 21	0,403	0,398	0,398	0,396	0,463	0,480	0,467	0,461	0,344	0,419	0,411	0,400
	t-stat	(17,25)	(17,57)	(22,42)	(17,28)	(19,32)	(20,15)	(19,47)	(18,60)	(12,04)	(21,14)	(17,49)	(12,78)
	σ 1	0,002	0,002	0,002	0,002	0,002	0,002	0,002	0,002	0,002	0,002	0,002	0,002
	t-stat	(52,64)	(54,35)	(55,83)	(54,58)	(47,23)	(48,21)	(48,12)	(50,38)	(50,21)	(39,47)	(36,34)	(32,07)
	σ 2	0,005	0,005	0,005	0,005	0,005	0,006	0,005	0,005	0,006	0,005	0,006	0,006
	t-stat	(47,47)	(50,05)	(47,90)	(45,50)	(42,77)	(39,56)	(41,76)	(39,91)	(35,22)	(34,59)	(29,79)	(26,80)
	θ 11	3,963				3,142				2,970
	t-stat	(22,22)				(23,99)				(20,15)
	θ 12	-3,328				-1,987				-1,482
	t-stat	-(15,41)				-12,765				-8,018
	γ 11		3,712	3,837	3,474		2,553	3,025	2,629		1,669	1,807	1,867
	t-stat		(20,83)	(21,62)	(20,69)		(18,34)	(21,62)	(20,54)		(13,89)	(11,58)	(9,09)
	γ 21		2,969	3,175	2,731		0,899	1,779	1,145		-0,825	-0,649	-0,588
	t-stat		(15,95)	(14,93)	(13,94)		(4,58)	(9,36)	(5,81)		-(4,96)	-(2,72)	-(1,76)
	γ 12		-0,696	-0,849	-1,369		-1,188	-0,626	-1,033		-0,607	-0,321	-0,378
	t-stat		-(4,77)	-(5,30)	-(6,27)		-(10,18)	-(6,26)	-(7,47)		-(7,35)	-(3,50)	-(4,45)
	γ 22		0,544	-0,060	0,415		0,628	0,053	0,571		0,793	0,420	0,476
	t-stat		(2,35)	-(0,45)	(1,66)		(4,19)	(0,49)	(2,54)		(5,47)	(4,15)	(3,49)
	p 11	0,981	0,971	0,986	0,991	0,959	0,899	0,966	0,973	0,951	0,815	0,890	0,901
	p 11	0,965	0,958	0,961	0,914	0,879	0,749	0,852	0,651	0,815	0,359	0,264	0,264
	LL

Table 2 :

 2 Estimation results of FTP and TVTP models for dependent variables KLCI and PSEI

	y t			KLCI					PSEI
	z t	FTP	SPX	Nikkei	STI	HSI	FTP	SPX	Nikkei	STI	HSI
	α 1	0,004	0,004	0,004	0,004	0,004	0,005	0,005	0,005	0,005	0,005
	t-stat	(58,83)	(57,63)	(55,07)	(56,01)	(52,18)	(82,11)	(83,48)	(81,48)	(84,80)	(90,47)
	α 2	0,009	0,009	0,009	0,009	0,009	0,010	0,010	0,010	0,010	0,010
	t-stat	(28,47)	(29,34)	(30,27)	(29,33)	(29,53)	(40,51)	(42,33)	(43,98)	(42,12)	(47,26)
	β 11	0,558	0,559	0,572	0,567	0,575	0,356	0,363	0,353	0,368	0,365
	t-stat	(27,54)	(27,23)	(29,43)	(30,78)	(29,73)	(14,63)	(14,54)	(14,90)	(15,64)	(15,74)
	β 21	0,622	0,623	0,625	0,628	0,623	0,257	0,265	0,260	0,270	0,254
	t-stat	(28,71)	(30,13)	(29,56)	(32,37)	(32,25)	(11,89)	(12,89)	(12,42)	(12,78)	(15,32)
	σ 1	0,002	0,002	0,002	0,001	0,001	0,002	0,002	0,002	0,002	0,002
	t-stat	(58,69)	(59,03)	(57,31)	(60,61)	(58,18)	(45,66)	(44,51)	(46,05)	(46,26)	(46,95)
	σ 2	0,007	0,007	0,007	0,007	0,007	0,005	0,005	0,005	0,005	0,005
	t-stat	(34,96)	(38,39)	(37,14)	(35,32)	(36,78)	(40,12)	(41,88)	(41,71)	(42,37)	(43,36)
	θ 11	3,362					2,200
	t-stat	(26,04)					(20,26)
	θ 12	-1,848					-0,694
	t-stat	-(11,32)					-(4,58)
	γ 11		3,351	3,254	3,165	3,171		2,107	2,164	1,961	1,962
	t-stat		(25,48)	(26,07)	(27,28)	(26,01)		(19,50)	(20,23)	(19,23)	(20,19)
	γ 21		1,766	1,619	1,546	1,376		0,487	0,631	0,274	0,246
	t-stat		(10,72)	(9,70)	(9,00)	(8,03)		(3,50)	(4,18)	(1,84)	(1,88)
	γ 12		-0,466	-0,526	-0,835	-0,713		-0,531	-0,157	-0,518	-0,423
	t-stat		-(6,38)	-(2,94)	-(7,40)	-(8,36)		-(6,77)	-(2,13)	-(6,03)	-(4,88)
	γ 22		-0,006	0,031	0,037	0,310		0,149	0,360	0,528	0,815
	t-stat		-(0,06)	(0,16)	(0,48)	(2,51)		(1,46)	(3,40)	(4,57)	(6,30)
	p 11	0,966	0,961	0,971	0,980	0,979	0,900	0,875	0,904	0,919	0,913
	p 11	0,864	0,854	0,832	0,819	0,749	0,667	0,630	0,611	0,451	0,378
	LL	20775,9	20790,5	20788,5	20802,5	20806,5	20259,5	20290,1	20269,0	20298,3	20306,5

Table 3 :

 3 Estimation results of FTP and TVTP models for dependent variables SET and JCI

	y t			SET					JCI
	z t	FTP	SPX	Nikkei	STI	HSI	FTP	SPX	Nikkei	STI	HSI
	α 1	0,006	0,006	0,006	0,006	0,006	0,006	0,006	0,006	0,006	0,006
	t-stat	(67,64)	(77,96)	(72,89)	(71,42)	(71,55)	(71,77)	(71,10)	(68,48)	(70,31)	(72,00)
	α 2	0,012	0,012	0,012	0,012	0,012	0,013	0,012	0,012	0,012	0,012
	t-stat	(40,18)	(43,52)	(41,64)	(39,25)	(39,67)	(28,50)	(32,67)	(30,93)	(29,87)	(30,41)
	β 11	0,423	0,434	0,439	0,442	0,436	0,491	0,513	0,503	0,501	0,505
	t-stat	(18,79)	(20,53)	(19,60)	(19,70)	(19,77)	(22,76)	(25,41)	(24,66)	(22,96)	(24,60)
	β 21	0,330	0,328	0,334	0,333	0,333	0,380	0,422	0,405	0,408	0,406
	t-stat	(13,48)	(12,94)	(13,58)	(11,99)	(12,12)	(9,42)	(12,28)	(10,78)	(10,35)	(11,39)
	σ 1	0,002	0,002	0,002	0,002	0,002	0,002	0,002	0,002	0,002	0,002
	t-stat	(49,60)	(54,26)	(50,44)	(50,42)	(53,17)	(48,17)	(47,45)	(49,80)	(49,36)	(48,26)
	σ 2	0,006	0,006	0,006	0,006	0,006	0,008	0,008	0,008	0,008	0,008
	t-stat	(39,95)	(44,84)	(45,52)	(44,82)	(46,55)	(39,96)	(43,33)	(39,98)	(40,85)	(43,60)
	θ 11	3,141					2,651
	t-stat	(24,59)					(22,18)
	θ 12	-2,130					-1,270
	t-stat	-14,198					-(9,04)
	γ 11		3,071	2,990	2,929	2,974		2,477	2,538	2,407	2,430
	t-stat		(24,83)	(23,73)	(22,89)	(23,85)		(21,96)	(22,81)	(21,69)	(21,50)
	γ 21		1,952	1,830	1,734	1,732		0,891	1,078	0,827	0,812
	t-stat		(13,22)	(12,29)	(11,29)	(11,66)		(6,57)	(7,55)	(5,32)	(5,68)
	γ 12		-0,346	-0,603	-0,718	-0,884		-0,680	-0,430	-0,812	-0,562
	t-stat		-(3,38)	-(4,94)	-(5,67)	-(7,32)		-(7,64)	-(5,81)	-(8,05)	-(6,49)
	γ 22		0,456	0,280	0,584	0,364		0,308	0,130	0,302	0,557
	t-stat		(3,37)	(1,86)	(3,25)	(2,22)		(2,45)	(1,29)	(2,84)	(4,26)
	p 11	0,959	0,951	0,964	0,973	0,978	0,934	0,906	0,940	0,958	0,950
	p 11	0,894	0,890	0,844	0,771	0,802	0,781	0,728	0,734	0,636	0,575
	LL

Table 4 :

 4 State dependent indicators for dependent variables HSI, STI and NKY

	y t		HSI			STI			NKY
	z t	SPX	Nikkei	STI	SPX	Nikkei	HSI	SPX	STI	HSI
	MMP 1	-0,174	-0,212	-0,342	-0,297	-0,156	-0,258	-0,152	-0,080	-0,095
	MMP 2	0,136	-0,015	0,104	0,157	0,013	0,143	0,198	0,105	0,119
	z 1 M z 2 M	5,335 -5,454	4,522 53,024	2,538 -6,580	2,148 -1,431	4,833 -33,429	2,545 -2,004	2,748 1,041	5,628 1,546	4,934 1,235
	ED 1 (0)	41,917	47,393	33,268	13,842	21,594	14,864	6,308	7,094	7,467
	ED 2 (0)	20,464	24,936	16,353	3,456	6,923	4,141	1,438	1,523	1,555
	V ED 1	-20,510	-26,536	-24,060	-8,929	-9,581	-8,930	-2,416	-1,674	-2,037
	V ED 2	14,081	-1,391	7,900	2,147	0,324	2,420	0,530	0,273	0,339
	MP 1 (0)

Table 5 :

 5 State dependent indicators for dependent variables JCI and KLCI

	y t		JCI				KLCI
	z t	SPX	Nikkei	STI	HSI	SPX	Nikkei	STI	HSI
	MMP 1	-0,170	-0,108	-0,203	-0,140	-0,116	-0,131	-0,209	-0,178
	MMP 2	0,077	0,032	0,076	0,139	-0,002	0,008	0,009	0,078
	z 1 M	3,640	5,900	2,964	4,327	7,194	6,188	3,793	4,445
	z 2 M	-2,890	-8,321	-2,738	-1,458	278,936	-51,648	-41,232	-4,436
	ED 1 (0)	12,900	13,657	12,101	12,363	29,525	26,897	24,692	24,840
	ED 2 (0)	3,438	3,939	3,286	3,253	6,850	6,046	5,692	4,958
	V ED 1	-5,874	-4,425	-6,172	-4,884	-10,621	-10,591	-13,408	-12,159
	V ED 2	0,881	0,407	0,806	1,681	-0,037	0,161	0,179	1,439
	MP 1 (0)

Table 6 :

 6 State dependent indicators for dependent variables PSEI and SET

	y t		PSEI				SET
	z t	SPX	Nikkei	STI	HSI	SPX	Nikkei	STI	HSI
	MMP 1	-0,133	-0,039	-0,130	-0,106	-0,086	-0,151	-0,179	-0,221
	MMP 2	0,037	0,090	0,132	0,204	0,114	0,070	0,146	0,091
	z 1 M	3,965	13,756	3,782	4,638	8,888	4,956	4,080	3,364
	z 2 M	-3,275	-1,752	-0,519	-0,302	-4,278	-6,540	-2,967	-4,761
	ED 1 (0)	9,223	9,707	8,108	8,113	22,570	20,878	19,707	20,564
	ED 2 (0)	2,627	2,879	2,316	2,279	8,046	7,233	6,663	6,650
	V ED 1	-3,390	-1,267	-2,876	-2,454	-6,302	-9,004	-9,583	-11,481
	V ED 2	0,261	0,814	0,916	1,612	4,076	2,012	4,497	2,478
	MP 1 (0)	-0,051	-0,015	-0,056	-0,046	-0,015	-0,028	-0,035	-0,041
	MP 2 (0)	0,035	0,082	0,130	0,201	0,050	0,033	0,075	0,046

To save place, other similar graphs for the other countries are not reported, but available upon request to the authors.

Note that z t are standardized in the transition functions.

Worthington, A., Higgs, H., 2004. Transmission of equity returns and volatility in Asian developed and emerging markets: a multivariate GARCH analysis. International Journal of Finance & Economics 80, 71-80. Yilmaz, K., 2010. Return and volatility spillovers among the East Asian equity markets. Journal of Asian Economics 21, 304-313.


Ricardo M Ribeiro

Vinicius N H Silva

Andrés P L Barbero

Murilo B Carvalho

Frédéric Lucarz
email: frederic.lucarz@telecom-bretagne.eu

Bruno Fracasso

An Optical Pulse Generator from a Sinusoidal Optical Signal Using Sagnac Loop -Self-Sampling

Keywords: laser, sampling, ultrafast-optics, optical signal processing, Sagnac interferometer, semiconductor optical amplifier I

It is experimentally shown an original optical pulse generator able to produce pulses at MHz rate with few ns timewidth at 1/3 of input period. It is based on the self-switched Semiconductor Laser Amplifier Loop Mirror (SLALOM) configuration. The input is a sinusoidal signal in the C-band and the output is amplified pulses. Numerical simulations are also carried out and shown that GHz & ps pulse generation is possible. Besides their usefulness as an optical pulse generator, application in self-sampling is also discussed.

INTRODUCTION

Many techniques have been described to generate a solitonic or non-solitonic optical pulses such as electro-optic modulation [START_REF] Veselka | Pulse Generation for Soliton Systems Using Lithium Niobate Modulators[END_REF], gain-switching [START_REF] Sayinc | Gain switched laser diode based all-fiber laser source emitting simultaneously at 8 different wavelengths in the NIR region[END_REF], mode-locking [START_REF] Chi | Properties of Mode-Locked Optical Pulses in a Dispersion-Managed Fiber-Ring Laser Using Semiconductor Optical Amplifier as Active Device[END_REF] and interferometric [START_REF] Wei | Delay-interferometer-based optical pulse generator, Optical Fiber Communication (OFC)[END_REF] schemes. In the latter, the light is phasemodulated before it traverses a Mach-Zehnder interferometer.

Short optical pulses are largely used on high-capacity fibreoptic digital communications [START_REF]Fiber-Optic Communication Systems[END_REF], but they have other applications, e.g. time-response characterization of ultra-fast devices and systems [START_REF] Zilkie | Characterization of the ultrafast carrier dynamics of an InAs/InGaAsP quantum dot semiconductor optical amplifier operating at 1.55 µm[END_REF], switching of short electrical pulses [START_REF] Motet | 1.4 ps rise-time highvoltage photoconductive switching[END_REF] and optical signal processing [START_REF] Khilo | Photonic ADC: overcoming the bottleneck of electronic jitter[END_REF].

Stable optical pulses as those generated by mode-locked lasers, are crucial to carry out GSa/s rate optical sampling [START_REF] Khilo | Photonic ADC: overcoming the bottleneck of electronic jitter[END_REF]. The authors have shown an optical circuit design and numeric simulations [START_REF] Ribeiro | An All-Optical Sampler for Digitising Radio-over-Fibre Transceivers[END_REF] of a real-time optical sampler based on the Sagnac loop using a semiconductor optical amplifier (SOA) as the nonlinear element. The device uses a train of short pulses to sample an analogue optical signal, e.g. a Radio-over-Fibre (RoF) signal. By using polarisation controllers inserted in the loop, the device was firstly adjusted to be 100% reflective. However, it was also shown [START_REF] Ribeiro | An All-Optical Sampler for Digitising Radio-over-Fibre Transceivers[END_REF] that the loop length should be carefully adjusted to almost contain an integer number of analogue signal periods. After such settings, the sampling pulses were released and sampled (and compressed) pulses were transmitted [START_REF] Ribeiro | An All-Optical Sampler for Digitising Radio-over-Fibre Transceivers[END_REF].

In this paper, we describe for the first time, in the best of our knowledge, experimental results on the use of an SLALOM configuration [START_REF] Eiselt | Optical Loop Mirror with Semiconductor Laser Amplifier[END_REF] in the self-switched regime, to generate an 118 MHz pulse train from an incoming sinusoidal optical signal around 1550 nm wavelength. The latter may be an analogue RoF carrying or not digital modulation on the RF carrier. Computer simulations for GHz & ps pulse generation on VPI Transmission Maker TM software are presented. The time compression of generated pulses and their possible use to perform self-sampling are also discussed.

II. THE EXPERIMENTAL SET-UP

Figure 1 depicts the experimental set-up based on a Sagnac loop using an SOA. The circuit operates on self-switching regime where the self-phase modulation (SPM) occurs inside the SOA. The circuit differs from those presented in [START_REF] Ribeiro | An All-Optical Sampler for Digitising Radio-over-Fibre Transceivers[END_REF] as to be an o-Sampler in that no sampling (control) pulses are used here.

A sinusoidal optical signal with P INPUT = 344 µW power is launched into the circuit. It is generated by direct modulation of a distributed feedback laser diode emitting 1550 nm wavelength, driven with 25 mA bias current and + 1 dBm Radio-Frequency (RF) voltage signal from a signal generator.

The loop is assembled by using one fibre-coupler (FC) featuring a measured coupling ratio of ~70/30. A polarisation controller (PC) is inserted in the loop in order to perform the polarisation alignment of the interfering light beams in the coupler junction. An SOA is asymmetrically inserted in the loop. The used SOA features 21-dB maximum gain, +6dBm saturated power and 45-nm bandwidth around 1552.2 nm wavelength. The bias current was set at 200 mA. The length L 1 + L 2 (L 1 < L 2 ) of the fibre loop is kept constant. The input frequency f INPUT of the sinusoidal probe signal is varied around f RES =1/T RES corresponding to an analogue signal period T RES given by [START_REF] Veselka | Pulse Generation for Soliton Systems Using Lithium Niobate Modulators[END_REF].

L 1 + L 2 = m c n eff T RES (1)
In [START_REF] Veselka | Pulse Generation for Soliton Systems Using Lithium Niobate Modulators[END_REF], m is an integer number, c = 3x10 8 m/s and n eff is the effective refractive index of the fibre. The output signal, that eventually may be a pulse train, is collected from the transmission arm (P T ). A tuneable bandpass filter (BPF) operating in the 1530-1561 nm wavelength range with 1.5 nm optical bandwidth centred at λ 1 = 1550 nm, was used in the output arm of the circuit to reduce the Amplified Spontaneous Emission (ASE) noise from the SOA. The output optical signal is detected by an 8 GHz bandwidth photodetector. The resulting electrical signal can be displayed on an oscilloscope. Figure 2a shows that for f INPUT = 98 MHz, the output signal is always sinusoidal shaped despite the PC adjustment. Only the amplitude varies when the PC is adjusted.

As is shown in Figure 2b, after increasing the input frequency as f INPUT = 107 MHz, the output signal is no longer sinusoidal. It starts to be distorted. Figure 2c shows that for f INPUT = 117 MHz there is a trend to double the frequency of the output signal. analogue signal period. This result may be compared with those similar as obtained in [START_REF] Veselka | Pulse Generation for Soliton Systems Using Lithium Niobate Modulators[END_REF] by using a Mach-Zehnder modulator operating in the nonlinear region of the interferometer transfer function, but using an electrical input signal and [START_REF] Chi | Properties of Mode-Locked Optical Pulses in a Dispersion-Managed Fiber-Ring Laser Using Semiconductor Optical Amplifier as Active Device[END_REF] using a different set-up. It should be observed that 118.3 MHz does not correspond to the f RES . The generation of pulses was also observed to be quite sensitive to the launched frequency f INPUT and to the polarisation setting through the PC adjustment.

Figure 2e shows that from f INPUT = 120 MHz is no longer possible to generate pulses. By increasing further the launched frequency, the output signal goes back to its sinusoidal shape as is shown in Figure 2f for f INPUT = 141 MHz.

Because the fibre-coupler features ~ 70/30 coupling ratio, it was not possible to set the Sagnac loop as to be 100% reflective (or transmissive). Therefore, it was not possible to erase the transmitted signal (P T ).

Although the results were reproducible, instabilities were observed mainly when pulses were generated as displayed in Figure 2d. The reason is firstly because the pulses can only be generated for a particular paddles adjustment of the PC. Secondly, the used PC does not present high mechanical stability. Thirdly, connectors were used in the optical circuit. The use of non-mechanical PC [START_REF] Zhang | Complete polarization controller based on magnetooptic crystals and fixed quarter wave plates[END_REF] and fibre splices instead of connectors may reduce the observed instabilities.

IV. DISCUSSIONS Originally [START_REF] Ribeiro | An All-Optical Sampler for Digitising Radio-over-Fibre Transceivers[END_REF], an optical sampler was designed to operate after adjustments of three optical delay lines. The first delay line denoted Delay #0 is out of the loop and is used to temporally align the sampling pulses with the analogue signal to be sampled. The Delay #0 it is not used here. The second and third delay lines denoted Delay #1 and Delay #2, respectively, are included in the circuit design of Figure 3 used for numerical simulations. By setting the PC, the device is also phase biased to be 100% reflective when there are no sampling pulses [START_REF] Ribeiro | An All-Optical Sampler for Digitising Radio-over-Fibre Transceivers[END_REF]. Now, the Delay #1 and Delay #2 lines are both inserted in the loop. They allow a fine mismatch of L 1 and L 2 from the exact fit of an integer number of analogue signal periods

T INPUT , i.e. L 1 = n(c/n eff )T INPUT ± ΔL 1 and L 2 = k(c/n eff )T INPUT  ΔL 2 ,
where n and k are integer numbers, and ΔL 1 and ΔL 2 are the amount of additional optical delay. In the present model, a symmetric adjustment of delay lines is assumed, i.e. ΔL 1 = ΔL 2 = ΔL so that the loop length is kept unaltered. Therefore, the symmetric adjustment of the Delay #1 and Delay #2 lines automatically ensures a fine placement deviation "x" of the SOA in reference of the midpoint of the loop (see Figure 3). The interferometric transfer function of the transmitted (P T ) signal of an ideal Sagnac interferometer is given by ( 2) after assuming all polarisations aligned [START_REF] Miyoshi | All-Optical Analog-to-Digital Conversion Using Splitand-Delay Technique[END_REF][START_REF]Applications in Nonlinear Fiber Optics[END_REF].

P T t ( ) = P INPUT t ( ) 1 2 1-cos γL SOA P INPUT t ( ) ( ) [ ] (2)
In (2), Δφ(t) = γL SOA P INPUT (t) represents the SPM-generated differential phase-shift between the clockwise (CW) and counter-clockwise (CCW) analogue signals, respectively. The Δφ(t) depends on the optical nonlinear strength γ and the length L SOA of the SOA inserted in the loop, respectively, and the input power P INPUT (t) of the analogue signal. The SOA inserted in the loop is an active element and the output signal generally will experiment an optical gain, i.e. P T > P INPUT .

When a single cycle of P INPUT (t) starts and ends, the power magnitude is very low (Δφ ≈ 0 rad) and the SLALOM acts as an almost perfect mirror since the modelled fibre-coupler is of 3 dB ratio coupling. The start and end of the analogue signal cycle is totally reflected back, i.e. P T = 0 mW according (2). When P INPUT (t) is increased, a differential phase-shift increased also due to the SPM effect that takes place into the SOA. The SPM causes spectral enlargement and distortion of input signal and is behind the mechanism of pulse generation.

Since the aforementioned ± ΔL setting of the loop delay lines is carried out, the Δφ phase-shift is imprinted in different parts of the CW and CCW analogue waveforms. By symmetrically adjusting the delay lines, it opens a switching window and the output signal P T may be generated as pulses. When the input power increases (and decreases), the increase (decrease) of P T (t) cycle is not linear thus leading the compression of the cycles thereby generating pulses. It implies that no complete switching is observed, i.e. the high power parts of the cycle switch rather than its wings for pulses [START_REF] Doran | Experimental investigation of all-optical switching in fibre loop mirror device[END_REF] and presently for sinusoidal modulation.

Another possible explanation may be outlined. It is shown in [START_REF] Chi | Properties of Mode-Locked Optical Pulses in a Dispersion-Managed Fiber-Ring Laser Using Semiconductor Optical Amplifier as Active Device[END_REF] an experimental set-up where actively mode-locked pulses were generated from a SOA-fibre ring laser excited with 10 GHz modulated optical signal. Therefore, present experimental set-up seems to generate non-dispersion compensated mode-locked pulses since the L 1 and L 2 lengths are properly adjusted. Both CW and CCW analogue signals modulates the SOA gain.

The VPI Transmission Maker TM software is used here to carry out some numerical simulations. The software solves the nonlinear Schrödinger equation (NLSE) taking into account the SPM effect in the bulk and isotropic SOA with low feedback as an active (amplifier) nonlinear waveguide. It uses the Transmission Line Model (TLM) to compute the differential nonlinear phase-shift induced by a change in carrier density in the semiconductor. As a result, the switching is driven by the own input signals due to the SPM effect occurring into the SOA. The circuit of Figure 3 was firstly probed with a f INPUT = 2.5 GHz (400 ps period) sinusoidal carrier frequency analogue signal at P INPUT = 344 µW power and 1550 nm wavelength. The loop length was set to be L 1 + L 2 ∝ 1.6 ns or 4 integer cycles of the analogue signal, where L 1 and L 2 correspond to 0.4 ns and 1.2 ns, respectively. Because of limited amount of available SOA's data, it is here assumed L SOA = 500 µm, linewidth enhancement factor = 6.0, null linear recombination, bimolecular recombination = 1.0 x 10 -16 m 3 /s, Auger recombination = 1.3 x 10 -41 m 6 /s and initial carrier density = 1.0 x 10 24 1/m 3 . The bias current was set at I BIAS = 130 mA.

Figure 4a and4b shows the plot of the calculated P T (t) for f INPUT = 2.5 GHz when 50 ps of delay (detuning) is symmetrically adjusted in both delay lines for 65/35 and 50/50 (3 dB fibre-coupler) coupling ratio, respectively. The relative optical phase-bias between the CW and CCW signals must be carefully adjusted through PC to optimise the output from the circuit thus shaping the pulses. It is clearly the generation of ~ 130 ps timewidth pulses, i.e. around 1/3 of the period. Figure 4a shows that the calculated pulse shape after assuming 65/35 coupling ratio reasonable fits the experimental measurement shown by Figure 1d. However, when the coupling ratio is assumed to be 50/50, more sharply pulses are calculated as is shown by Figure 4b. Therefore, the interferometer visibility contributes to increase the extinction ratio of the device. Simulations also have shown that it is not possible to generate pulses when the SOA is removed from the loop, but only sinusoidal signals exit.

Simulations at f INPUT = 2.5 GHz shows that when an "optical DC offset" of P OFFSET ≤120 µW is superimposed with 400 µW sinusoidal amplitude (≤ 120/400 = 30%), the optical DC component is blocked by the circuit of Figure 3. This means that the pulses remain to be generated. For P OFFSET > 120 µW the pulses start to turn out to be sinusoidal in shape. This optical blocking was also observed in the experiments.

When the SOA is placed at the middle of the loop (x = 0), it is not possible to generate pulses. However, after slightly detuning of delay lines (x ≠ 0), the optical pulses can again be generated.

Figure 5a and 5b show the plot of the calculated P T (t) at f INPUT = 2.5 GHz when 0 ps and 100 ps delay (detuning) is symmetrically adjusted in both delay lines for 50/50 coupling ratio, respectively. Sinusoidal-like outputs are calculated for 0 ps and 100 ps delays despite the PC adjustment. A clearly distorted sinusoidal signal is shown in Figure 5b for 0 ps delay. Now, the 0 ps symmetrical delays are kept unaltered and the input frequency f INPUT is varied. Figures 6a-6c show plots for f INPUT = 1.8, 2.3 and 2.7 GHz, respectively, after setting 50/50 coupling ratio.

From the simulation results displayed by the plots of Figures 456 V. SELF-SAMPLING Analogue-to-digital converters (ADCs) require the sampling of the signal to be digitised. Even an electronic ADC-quantizer may take the advantages of a real time optical sampling [START_REF] Khilo | Photonic ADC: overcoming the bottleneck of electronic jitter[END_REF][START_REF] Valley | Photonic analog-to-digital converters[END_REF] because of their high stability. Most of optical sampling circuits use very short pulses generated from a mode-locked laser that usually features few femtoseconds of time-jitter [START_REF] Khilo | Photonic ADC: overcoming the bottleneck of electronic jitter[END_REF][START_REF] Miyoshi | All-Optical Analog-to-Digital Conversion Using Splitand-Delay Technique[END_REF][START_REF] Ng | Ultra-Low Jitter Mode-Locking of Er-Fiber Laser at 10 GHz and its Application in Photonic Analog-to-Digital Conversion[END_REF].

Simulations from the circuit of Figure 3 were carried out for f INPUT = 2.5 GHz, 50/50 coupling-ratio and 50 ps symmetrical delays leading to generation of ~130 ps timewidth pulses as shown in Figure 4b. Numerical simulations also show the possibility of pulse compression. Figure 7 shows a temporal compression at -3.1 ps/km rate by using a Dispersion Compensating Fibre (DCF) featuring D = -160 ps/nm.km and 0.25 dB/km dispersion and attenuation coefficients, respectively. It is assumed that the DCF coil is placed outside the loop, i.e. the transmitted pulses traverse the DCF length before photo-detection. The shape of the compressed pulses were calculated to be quite similar of those presented in Figure 4b.

Figure 7 shows a slightly compression from ~130 ps to ~100 ps by using 10 km length of DCF coil. The spectral bandwidth of the BPF was set to be Δf FILTER = 100 GHz. After repeating the calculation with Δf FILTER = 400 GHz and L FIBRE = 1 km, the output pulses remain their timewidth around 117 ps. Both experiments and simulations used a DFB laser diode that may be assumed to present a null linewidth spectrum for practical purposes. By increasing the P INPUT it turn out possible to obtain further temporal compression. Figure 8 shows the calculated time-shape of the compressed pulses by setting P INPUT = 1376 µW using the same 10 km length DCF coil now leading pulses with ~ 75 ps timewidth, but presenting pedestals (background).

Other more suited to compact systems are available to obtain pulse compression, e.g. chirped Fibre Bragg Grating (FBG) [START_REF] Taketomi | Compression of picosecond pulses with a chirped volume Bragg grating[END_REF] or a Nonlinear Optical Loop Mirror (NOLM) [START_REF] Doran | Experimental investigation of all-optical switching in fibre loop mirror device[END_REF]. The latter may be also useful to suppress the pulse pedestals [START_REF] Pelusi | Pedestal Suppression from Compressed Femtosecond Pulses Using a Nonlinear Fiber Loop Mirror[END_REF].

The compression generates a pulse train with a reduced duty-cycle. By using one or more serially connected split-anddelay optical circuits [START_REF] Miyoshi | All-Optical Analog-to-Digital Conversion Using Splitand-Delay Technique[END_REF][START_REF] Ribeiro | Proposal and Design of an All-Optical Encoder for Digitising Radio-over-Fibre Transceivers[END_REF], it turns out possible to produce temporally interleaved pulses thus increasing the rate of the output pulse train by Nf INPUT , where N is an integer number. Since N ≥ 2, the train may be used to sample the original analogue waveform under the Nyquist theorem. Present calculations for 2.5 GHz (400 ps period) show that at least N ~ 5 are possible once the derived compressed pulses at ~75 ps (without pedestals) are used to sample the analogue signal.

The scheme described in [START_REF] Ribeiro | An All-Optical Sampler for Digitising Radio-over-Fibre Transceivers[END_REF] may be used for real-time self-sampling, but the wavelength should be converted or the polarisation between the signal to be sampled and the sampling pulses should be set at 90 o angle each other. Therefore, (amplified) sampling pulses may be derived from the analogue signal to be sampled. Since an analogue RoF (optical) periodic signal is already available, two advantages in the use of self-sampling for digitising RoF or an optical-ADC are pointed out here: i) It does not require the use of a local mode-locked laser.

ii) Similarly to digital signals, analogue signals also presents jitter. Considering the pulse generation mechanism, we are lead to believe that the time-jitter present in the input signal should be transferred to the derived sampling pulses. In that case, the sampling pulses would be naturally synchronized to the input. It should be noted however, that further investigations are required to validate this assumption. Additional measurements are also needed to assess the input to output amplitude relationship of the self-sampling process. This is essential to quantify the linearity, an important parameter of ADCs.

VI. CONCLUSIONS

In this paper, it is described a SLALOM-based device able to generate optical pulses with timewidth ~1/3 of the period of an analogue sinusoidal input optical signal. Neither microwave circuit nor input electrical signal is required. The pulses are generated when the period of the input sinusoidal signal is slightly detuned from the fundamental period that can be adjusted in the loop length and when the optical phase is carefully biased by using the polarisation controller. In this paper, the experiments were constrained to only ~100 MHz input analogue signal mainly to demonstrate the proof-ofprinciple of the proposed optical generator.

Numerical simulations in the (1.8-2.7) GHz frequency range have shown that it is equally possible to generate pulses by: i) Keeping unchanged the input period (or f INPUT ) matched to be an integer fraction of the loop length, and slightly adjusting the symmetrical delays (detuning) of the SOA relative to the fibre-junction. ii) Keeping unchanged the L 1 and L 2 and varying the f INPUT around f RES .

In both cases, one should also carefully to adjust the optical phase-bias by using the controller PC.

Although much work remains to be done, the possibility of time compression of the generated pulses and their use to implement a scheme for self-sampling is proposed and discussed. The sampling pulses may be extracted from the analogue signal to be sampled and the time-jitter of both might be automatically synchronized. This opens the possibility to eliminate the use of local mode-locked laser in an opticalsampler and the availability of sampling pulses free of relative jitter with the analogue signal to be sampled. Since the modelocking may be the mechanism behind pulse generation, the stability issue should also be addressed.

The proposed and tested device is potentially optically integrable, requires ~mW optical power amplitude for the input analogue signal and outputs amplified pulses.

Fig. 1 .

 1 Fig. 1. The experimental fibre-optic circuit of the pulse generator where L1 + L2 = Sagnac loop length.

FigureFig. 2 .

 2 Figure 2d shows that for f INPUT = 118.3 MHz and after a careful adjustment of PC, it turns out possible to generate optical pulses with ~ 8.4 ns period that exactly corresponds to the original 118.3 MHz input frequency. The pulse timewidth is ~ 2.6 ns, i.e. slightly smaller than 1/3 of the

Fig. 3 .

 3 Fig. 3. The fibre-optic circuit design of the pulse generator as used for numerical simulations using the VPI Transmission Maker TM software, where L1 + L2 = Sagnac loop length.

  , it becomes possible to conclude that the optical delay |Δτ| adjustment has analogous effect of variation of input frequency f INPUT . The experiments were carried out by varying the f INPUT merely due to the convenience, i.e. the optical delay lines were not available.

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig. 4. The calculated PT(t) for 50 ps symmetrical delay (350/1250 ps) at fINPUT = 2.5 GHz for (a) 65/35 and (b) 50/50 coupling ratio.

Fig. 7 .

 7 Fig. 7. The calculated timewidth of the compressed pulses after propagation along a DCF coil with PINPUT = 344 µW.

Fig. 8 .

 8 Fig. 8. The calculated time-shape of the compressed pulses after propagation along 10 km length of DCF coil with PINPUT = 1376 µW.

Ricardo M. Ribeiro thanks Foundation Capes/MEC-Brasil for the financial support through a Post-Doctoral fellowship under the grant BEX 9096/11-6.

ACKNOWLEDGMENT Ricardo M. Ribeiro thanks the team at Département d'Optique of Telecom Bretagne and the CapilR TM platform for hosting his Post-Doctoral study during which the present work was carried out. The authors also thank Profs. Ammar Sharaiha and Tierry Rampone of ENIB (Plouzané, France) and INCT-Fotonicom/CNPq (Brasil).