Equivalence for nonparametric drift estimation of a diffusion process and its Euler scheme

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Equivalence for nonparametric drift estimation of a

diffusion process and its Euler scheme

Catherine Laredo, Valentine Genon-Catalot

To cite this version:

Catherine Laredo, Valentine Genon-Catalot. Equivalence for nonparametric drift estimation of a

diffusion process and its Euler scheme. DYNSTOCH 2013, Apr 2013, Copenhagen, Denmark. pp.43.

�hal-02748689�

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DYNSTOCH 2013

University of Copenhagen

April 17-19

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Organizers:

Susanne Ditlevsen Michael Sørensen Niels Richard Hansen

Web:

http://www.math.ku.dk/english/research/conferences/dynstoch2013/dynstoch2013

Sponsors:

Program of Excellence: Statistical Methods for Complex and High Dimensional Models. http://statistics.ku.dk

Department of Mathematical Sciences http://www.math.ku.dk

Dynamical Systems Interdisciplinary Network, University of Copenhagen http://www.dsin.ku.dk

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General Information

Venue

All talks take place in Auditorium 4, HCØ Universitetsparken 5 2100 København Ø.

Lunch is served in a nearby lunch restaurent, see map below.

The reception Wednesday at 18.00 and the poster session Thursday 15.30-17.30 take place in the main hall of HCØ outside of Auditorium 4.

The conference dinner on Thursday takes place at Nørrebro Bryghus

Ryesgade 3

2200 København NV.

Wireless network

If you have an Eduroam account from your home university, you can use the Eduroam network. Alternatively, access codes for the Conference network will be made available during the conference.

Maps

Rutevejledning til Nørrebro Bryghus

Ryesgade 3, 2200 København N

1,5 km – ca. 18 minutter Rutevejledning til fods er i beta.

Vær forsigtig – Denne rute har muligvis ikke fortov eller gangsti.

Universitetsparken 5, 2100 København til Nørrebro Bryghus - Google Maps http://maps.google.dk/maps?f=d&source=s_d&saddr=Universitetspark...

Vi kan ikke angive rutevejledning mellem 55.700420, 12.561094 og Universitetsparken 13, 2100 København. 55.700420, 12.561094 til Universitetsparken 13, 2100 København - Go... http://maps.google.dk/maps?f=d&source=s_d&saddr=55.70042,12.561...

1 of 1 4/8/2013 8:15 PM

Both maps: A is lecture hall, Auditorium 4, HCØ, Universitetsparken 5

Left map: B is the restaurent for the conference dinner, Nørrebro Bryghus, Ryesgade 3. Right map: B is the lunch restaurent.

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DYNSTOCH WORKSHOP 2013

PROGRAMME

Venue: Auditorium 4

The H.C. Ørsted Institute, Universitetsparken 5

Directions

Wednesday, April 17

12:30 - 14:00 LUNCH

14:00 - 14:30

Ernst Eberlein (University of Freiburg): TWO PRICE ECONOMIES

IN CONTINUOUS TIME

14:30 - 15:00

Markus Bibinger (Humboldt-Universität zu Berlin): VOLATITLITY

MATRIX ESTIMATION FROM NOISY OBSERVATIONS

15:00 - 15:30

Ernst August v. Hammerstein (University of Freiburg): OPTIMALITY

OF PAYOFFS IN LEVY MODELS.

15:30 - 16:00 BREAK

16:00 - 16:30

Alexander Schnurr (TU Dortmund): AN ORDINAL PATTERN

APPROACH TO DETECT AND TO MODEL DEPENDENCE

STRUCTURES BETWEEN FINANCIAL TIME SERIES

16:30 - 17:00

Hilmar Mai (Weierstrass Institute Berlin): MINIMAX ESTIMATION OF A

SUBORDINATOR'S DENSITY: UNIFORM RATES OF CONVERGENCE

17:00 - 17:30

Olivier Bouaziz (University of Paris 5): A LASSO ESTIMATOR FOR

EVENT-SPECIFIC RATE MODELS FOR RECURRENT EVENTS

17:30 - 18:00 Li Zhou (Université du Maine): ON APPROXIMATION OF BSDE

18:00 - WELCOME RECEPTION

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Thursday, April 18

9:00 - 9:30

Nakahiro Yoshida (University of Tokyo): MARTINGALE EXPANSION

AND STATISTICS OF VOLATILITY

9:30 - 10:00

Kleptsyna Marina (Université du Maine): INFERENCE IN SYSTEMS

WITH MIXED FRACTIONAL BROWNIAN MOTION NOISES

10:00 - 10:30

Catherine Larédo (INRA & Université Paris Diderot): EQUIVALENCE

FOR NON PARAMETRIC DRIFT ESTIMATION OF A DIFFUSION

PROCESS AND ITS EULER SCHEME

10:30 - 11:00 BREAK

11:00 - 11:30

Adeline Samson (University Paris Descartes): PARAMETER ESTIMATION

IN THE STOCHASTIC MORRIS-LECAR NEURONAL MODEL

11:30 - 12:00

Masayuki Uchida (Osaka University): DISCRIMINANT ANALYSIS FOR

DISCRETELY OBSERVED STOCHASTIC DIFFERENTIAL EQUATIONS

12:00 - 12:30

Romain Guy (INRA & Université Paris Diderot): INFERENCE FOR PARTIALLY

AND DISCRETELY OBSERVED DIFFUSIONS WITH SMALL DIFFUSION

COEFFICIENT - APPLICATION TO EPIDEMICS

12:30 - 13:30 LUNCH

13:30 - 14:00 NETWORK MEETING

14:00 - 14:30

Valentine Genon-Catalot (University Paris Descartes): STATIONARY

DISTRIBUTIONS FOR STOCHASTIC DIFFERENTIAL EQUATIONS

WITH RANDOM EFFECTS AND STATISTICAL APPLICATIONS

14:30 - 14:45

PRESENTATION OF POSTERS: Laura Sacerdote (University of Torino),

Ana Prior (University of Porto) and Maud Delattre (AgroParisTech).

14:45 - 15:15

Helle Sørensen (University of Copenhagen): PARAMETER ESTIMATION

FOR DIFFUSION PROCESSES WITH RANDOM EFFECTS

15:15 - 15:30

PRESENTATION OF POSTERS: Émeline Schmisser (Université Lille 1),

Imma Valentina Curato (University of Pisa)

15:30 - 17:30 POSTER SESSION WITH WINE

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Friday, April 19

9:00 - 9:30

Yury A. Kutoyants (Université du Maine): GOODNESS OF FIT TESTS FOR

STOCHASTIC PROCESSES WITH PARAMETRIC BASIC HYPOTHESIS

9:30 - 10:00

Dominique Dehay (University of Rennes): PARAMETRIC AND NON-

PARAMETRIC ESTIMATION PROBLEMS FOR SOME TIME-PERIODIC-

DRIFT LANGEVIN TYPE STOCHASTIC DIFFERENTIAL EQUATION

10:00 - 10:30

Hiroki Masuda (Kyushu University): ON OPTIMAL ESTIMATION OF

STABLE ORNSTEIN-UHLENBECK PROCESSES

10:30 - 11:00 Break

11:00 - 11:30

Petra Posedel (Zagreb School of Economics and Management): ASYMPTOTIC

ANALYSIS FOR OPTIMAL ESTIMATING FUNCTIONS FOR A CLASS OF

STOCHASTIC VOLATILITY MODELS WITH JUMPS

11:30 - 12:00

Benedikt Funke (TU Dortmund): ADAPTIVE NADARAYA-WATSON LIKE

ESTIMATORS FOR THE ESTIMATION OF THE DRIFT IN A JUMP

DIFFUSION MODEL

12:00 - 12:30

Mathias Trabs (Humboldt-Universität zu Berlin): EFFICIENCY IN

DECONVOLUTION AND LÉVY MODELS

12:30 - 14:00 LUNCH

14:00 - 14:30

Christian Schmidt (Heidelberg University): LIMIT THEOREMS FOR

NON-DEGENERATE U-STATISTICS OF CONTINUOUS

SEMIMARTINGALES

14:30 - 15:00

Gang Huang (University of Amsterdam): LIMIT THEOREMS FOR

REFLECTED ORNSTEIN-UHLENBECK PROCESSES

15:00 - 15:30

Mark Podolskij (Heidelberg University): VARIOUS LIMIT THEOREMS FOR

AMBIT PROCESSES

15:30 - 16:00 BREAK

16:00 - 16:30

Jakob Söhl (University of Cambridge): CONFIDENCE SETS IN

NONPARAMETRIC CALIBRATION OF EXPONENTIAL LÉVY MODELS

16:30 - 17:00

Moritz Schauer (Delft University of Technology & EURANDOM): GUIDED

PROPOSALS FOR SIMULATING DIFFUSION BRIDGES

17:00 - 17:30

John Schoenmakers (WIAS Berlin): SIMULATION OF CONDITIONAL

DIFFUSIONS VIA FORWARD-REVERSE STOCHASTIC

REPRESENTATIONS

This page was last updated on April 9, 2013.

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Abstracts

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PARAMETER ESTIMATION IN THE STOCHASTIC

MORRIS-LECAR NEURONAL MODEL

Adeline Samson

University Paris Descartes, France, [email protected]

Susanne Ditlevsen

, University of Copenhagen, Denmark

Diffusions; partial observations; sequential Monte Carlo; membrane potential.

Parameter estimation in two-dimensional diffusion models with only one coordinate observed is highly relevant in many biological applications, but a statistically difficult problem.

The membrane potential evolution in single neurons can be measured at high frequency, but bio-physical realistic models have to include the unobserved dynamics of ion channels. One such model is the stochastic Morris-Lecar model, where random fluctuations in conductance and synaptic input are specifically accounted for by the diffusion terms. It is defined through a non-linear two-dimensional stochastic differential equation with only one coordinate observed. We aim at estimating the parameters of this stochastic Morris-Lecar model. We propose a sequential Monte Carlo particle filter algorithm to impute the unobserved coordinate, and then estimate parame-ters maximizing a pseudo-likelihood through a stochastic version of the Expectation-Maximization algorithm. Performance on simulated data and real data are very encouraging.

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AN ORDINAL PATTERN APPROACH TO DETECT

AND TO MODEL DEPENDENCE STRUCTURES

BETWEEN FINANCIAL TIME SERIES

Alexander Schnurr

TU Dortmund, Germany, [email protected]

Ordinal patterns; stationarity; leverage effect; VIX; model free data exploration; econometrics. We introduce the concept of ordinal pattern dependence between time series and show in an explorative study that both types of this dependence show up in real world financial data. The classical way to capture the leverage effect in models for stock markets is to assume a negative correlation between the two datasets which is constant in time (e.g. Barndorff-Nielsen and Shepard (2002)). However, there is strong evidence that this effect is not constant, but itself evolves in time. It seems that there are periods where the effect is weaker and sometimes it even seems to be ‘turned around’, i.e., there is a positive correlation between the two datasets (Carr and Wu (2007)). Taking these empirical findings into account, more sophisticated models where suggested. The correlation structure was modeled by a deterministic function or was made state space dependent. In Veraart and Veraart (2010) the correlation structure is itself a stochastic process on [−1, 1].

Instead of proposing even more complicated models we introduce a rather simple approach to analyze whether there is a dependence structure between two datasets. In order to capture the zik-zak of the datasets we use so called ordinal patterns. This method was developed by Bandt (2005) and Emonds et al. (2007) in order to handle time-series with several thousands of data points which appear in medicine, meteorology and finance (cf. Keller and Sinn (2005)). We compare the two datasets from this point-of-view. On some occasions, as an example we will consider the S&P 500 and the corresponding volatility index VIX, a dependence structure of this kind seems to be more likely to be found in real data than the dependence modeled by the classical approach via correlation (cf. Whaley (2008)).

References

[1] Bandt, C. (2005): Ordinal time series analysis. Ecological Modelling, 182 : 229–238.

[2] Barndorff-Nielsen, O.E. and N. Shephard (2002): Econometric analysis of realised volatility and its use in estimating stochastic volatility models. J. Royal Sta. Soc. B64: 253–280.

[3] Carr, P. and L. Wu (2007): Stochastic skew in currency options. J. of Fin. Econ. 86(1): 213–247. [4] Emonds, J., K. Keller and M. Sinn (2007): Time Series from the Ordinal Viewpoint. Stochastics and Dynamics, 2 : 247–272.

[5] Keller, K. and M. Sinn (2005): Ordinal Analysis of Time Series. Physica A, 356 : 114–120. [6] Schnurr, A. (2012): An Ordinal Pattern Approach to Detect and to Model Leverage Effects and Dependence Structures Between Financial Time Series. submitted.

[7] Veraart, A.E.D. and Veraart A.M.V. (2010): Stochastic Volatility and Stochastic Leverage. Ann. Finance, online first.

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ADAPTIVE NADARAYA-WATSON LIKE ESTIMATORS

FOR THE ESTIMATION OF THE DRIFT IN A

JUMP DIFFUSION MODEL

Benedikt Funke

TU Dortmund, Germany, [email protected]

Diffusion model; Nadaraya-Waton estimator; α-stable Lévy motion; bias reduction.

At first consider a stationary and ergodic stochastic process which fulfills a nonparametric diffusion model driven by a Brownian motion. Our aim is the estimation of the unknown drift and volatility functions using a discrete high-frequency sample. Motivated by Bandi and Phillips, [1], we use Nadaraya-Watson like estimators and extend their results to bandwidths which depend on the available sample. Using a specific bandwidth, we can reach a faster rate of convergence of the bias term of our estimators. Furthermore, we prove asymptotic properties like consistency and asymptotic normality.

Afterwards we consider a more general model, which is driven by an α-stable Lévy motion. Again we make use of a Nadaraya-Watson like estimator for the unknown drift function. Under comparable assumptions we prove the corresponding asymptotic properties. A short simulation study illustrates our results.

References

[1] Bandi, F.M., Phillips, P.C.B. (2003) Fully nonparametric estimation of scalar diffusion models, Econometrica 71, 241–283.

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EQUIVALENCE FOR NON PARAMETRIC DRIFT

ESTIMATION OF A DIFFUSION PROCESS AND

ITS EULER SCHEME

Catherine Larédo

INRA& LPMA, Université Paris Diderot, FRANCE, [email protected]

Valentine Genon-Catalot

, MAP5, Université Paris Descartes, FRANCE

Diffusion process; discrete observations; Euler scheme; nonparametric experiments; deficiency distance; Le Cam equivalence.

The main goal of the asymptotic equivalence theory of Le Cam (1986) is to approximate general statistical models by simple ones. We develop here a global asymptotic equivalence result for nonparametric drift estimation of a discretely observed diffusion process and its Euler scheme. The asymptotic equivalences are established by constructing explicit equivalence mappings. The impact of such asymptotic equivalence results is that it justifies the use in many applications of the Euler scheme instead of the diffusion process. We especially investigate the case of diffusions with non constant diffusion coefficient. To obtain asymptotic equivalence, experiments obtained by random change of times are introduced. This result ([2]) extends the asymptotic equivalence obtained by Milstein and Nussbaum ([3]) for small diffusion coefficient and the results of Dalalyan and Reiss ([4]) for constant diffusion coefficient.

References

[1] Le Cam, L.( 1986) Asymptotic Methods in Statistical Decision Theory. Springer, New York. [2] Genon-Catalot V. and Larédo C. (2012) Equivalence for nonparametric drift estimation of a diffusion process and its Euler scheme Prépublication hal 00738115.

[3] Milstein, G. and Nussbaum, M. (1998) Diffusion approximation for nonparametric autoregres-sion. Probab. Theory Related Fields 112 535-543.

[4] Dalalyan, A. and Reiss, M. (2006) Asymptotic statistical equivalence for scalar ergodic diffu-sions. Probab. Theory Related Fields 134 248-282.

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LIMIT THEOREMS FOR NON-DEGENERATE

U-STATISTICS OF CONTINUOUS SEMIMARTINGALES

Christian Schmidt

Heidelberg University, Germany, [email protected]

Mark Podolskij

, Heidelberg University, Germany

Johanna Fasciati Ziegel

, University of Bern, Switzerland U-statistics; high frequency data; semimartingales.

We study the asymptotic theory for non-degenerate U-statistics of high frequency observations of continuous Ito semimartingales. Based on empirical process methods we prove uniform convergence in probability and show a functional stable central limit theorem for the standardized version of the U-statistic. Finally we give a test for homoscedasticity as an application of the theory.

References

[1] Barndorff-Nielsen, O.E., Graversen, S.E., Jacod, J., Podolskij, M., Shephard, N. (2006) A central limit theorem for realised power and bipower variations of continuous semimartingales.

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PARAMETRIC AND NONPARAMETRIC ESTIMATION

PROBLEMS FOR SOME TIME-PERIODIC-DRIFT

LANGEVIN TYPE STOCHASTIC DIFFERENTIAL

EQUATION

Dominique Dehay

IRMAR University of Rennes, France, [email protected]

K. El Waled

, IRMAR University of Rennes, France

Estimation; parametric; nonparametric; Ornstein-Uhlenbech process.

In this work we investigate estimation problems for a diffusion process following the model

dξt= f (t)ξtdt + dBt

where f : R → R is a periodic continuous function with period P > 0, when the process is observed through continuous time [0, T ] as T → ∞.

First we assume that the drift function f (·) depends linearly on an unknown parameter θ ∈ R : f (t) = θb(t), b : R → R being a known periodic continuous function. Then the maximum likelihood estimator (MLE) of θ is consistent and we point out its asymptotic minimax efficiency. These results comply with the well-established case when the function f (·) is constant non null.

However the case whenRP

0 b(t)dt = 0 and b(·) is not identically null presents some particularities.

For instance in this case whatever is the value of θ, the rate of convergence of the MLE is T as

in the case when θ = 0 and R₀Pb(t)dt 6= 0. Futhermore when R₀Pb(t)dt = 0, the MLE is locally

efficient for the quadratic risk.

Next we deal with the problem of nonparametric estimation of the drift function f (·), its period P being known. We construct a kernel estimator of f (·) which is consistent. Its rate of convergence

which depends on the signum ofR₀Pf (t)dt as in the previous parametric problem (joint work with

Khalil El Waled).

References

[1] Dehay, D., (2013) Parameter maximum likelihood estimation problem for time-periodic-drift Langevin type stochastic differential equations, (submitted).

[2] Dehay, D., El Waled, K. (2013) Nonparametric estimation problem for a time-periodic signal in a periodic noise, Statistics and Probability Letters 83, 608–615.

[3] Dehay, D., El Waled, K. (2013) Nonparametric estimation problem for a time-periodic-drift Langevin type stochastic différential equations (work in progress).

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OPTIMALITY OF PAYOFFS IN LÉVY MODELS

Ernst August v. Hammerstein

University of Freiburg, Germany, [email protected]

Eva Lütkebohmert

, University of Freiburg, Germany

Ludger Rüschendorf

Viktor Wolf

Cost-efficient strategies; optimal payoff; Lévy model; Esscher transform.

In this talk we study optimal investment choices in incomplete markets where the prices of risky assets are driven by Lévy processes, and the pricing of derivatives is based on the Esscher martingale measure. In particular, we solve for the investment strategy with minimal costs that achieves a given payoff distribution. This strategy is called cost-efficient with respect to the given distribution and can be obtained using the well-known upper and lower Fréchet bounds. Our work builds on the paper of Bernard, Boyle, and Vanduffel [1] and extends their approach to a more general and incomplete Lévy market setting.

For a variety of relevant financial derivatives we explicitly derive the cost-efficient strategies, that

is, we improve the payoffs in the sense of the stochastic order ≤st for agents with increasing

preferences. We also show that a cost-efficient version of the put-call parity exists. The theoretical results are illustrated by a practical example where we calibrate three different Lévy models (NIG, Variance Gamma, and Brownian motion as benchmark) to German stock price data and calculate the potential savings that could be achieved by using the efficient strategies.

Further we derive explicit hedging strategies for cost-efficient payoffs. More specifically, we compute the Greek Delta, that is, the derivative of the cost of a strategy with respect to the underlying, for cost-efficient strategies corresponding to European call and put options. Using the aforementioned data set, we finally compare the magnitude of the Delta of a cost-efficient put with that of a plain vanilla put.

References

[1] Bernard, C., Boyle, P., Vanduffel, S. (2012) Explicit representation of cost-efficient strategies, working paper, available at http://ssrn.com/abstract=1561272

[2] v. Hammerstein, E.A., Lütkebohmert, E., Rüschendorf, L., Wolf, V. (2013) Optimality of payoffs in Lévy models, working paper, University of Freiburg

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TWO PRICE ECONOMIES IN CONTINUOUS TIME

Ernst Eberlein

University of Freiburg, Germany, [email protected]

Dilip Madan

, University of Maryland, Maryland, USA

Martijn Pistorius

, Imperial College London, Great Britain

Wim Schoutens

, K.U. Leuven, The Netherlands

Marc Yor

, Université Paris VI, France

Two price economy; nonlinear expectation operator; distortion.

In classical economic theory the law of one price prevails and market participants trade freely in both directions at the same price. This approach is appropriate for highly liquid markets. In the absence of perfect liquidity, the law of one price has to be replaced by a two price economy where market participants continue to trade freely with the market but the terms of trade now depend on the direction of the trade.

Such an approach has been considered in a static environment. It is the purpose of this paper to develop the continuous time theory for two price economies. The two prices are termed bid and ask or lower and upper price but they should not be confused with the literature relating bid-ask spreads to transaction costs or other frictions involved in modeling financial markets. The two prices are determined in a non marketclearing equilibrium with a view to make loss exposures acceptable. Acceptability is defined via a positive expectation under a family of test measures or scenarios. As a result the bid price is the infimum of test valuations whereas the ask price is the supremum of such valuations. The two prices are related to nonlinear expectations seen as G-expectations. Probability distortions are used to formulate G-G-expectations. We consider examples where the uncertainty is given by purely discontinuous Lévy processes. The approach is illustrated by explicit pricing of a book of derivatives.

References

[1] Eberlein, E., Madan, D., Pistorius, M., Schoutens, W., Yor, M. (2012) Two price economies in continuous time. Preprint University of Freiburg.

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LIMIT THEOREMS FOR REFLECTED

ORNSTEIN-UHLENBECK PROCESSES

Gang Huang

University of Amsterdam, the Netherlands, [email protected]

Michel Mandjes

, University of Amsterdam, the Netherlands

Peter Spreij

, University of Amsterdam, the Netherlands

Ornstein-Uhlenbeck processes; reflection; large deviations; central-limit theorems.

We study one-dimensional Ornstein-Uhlenbeck processes, with the distinguishing feature that they are reflected on a single boundary (put at level 0) or two boundaries (put at levels 0 and d > 0). In the literature they are referred to as reflected OU (ROU) and doubly-reflected OU (DROU) respectively. For both cases, we explicitly determine the decay rates of the (transient) probability to reach a given extreme level. The methodology relies on sample-path large deviations, so that we also identify the associated most likely paths. For DROU, we also consider the loss process

Ut, that is, the local time at upper boundary d. We derive a central limit theorem (CLT) for

Ut, using techniques from stochastic integration and the martingale CLT. In addition, with

large-deviations techniques we prove that the tail of Utdecays essentially exponentially, and we identify

the corresponding decay rate.

References

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PARAMETER ESTIMATION FOR DIFFUSION

PROCESSES WITH RANDOM EFFECTS

Helle Sørensen

University of Copenhagen, Denmark, [email protected]

Susanne Ditlevsen

, University of Copenhagen, Denmark

discrete observations; likelihood approximation: stochastic differential equations; square-root process.

We consider data consisting of samples of discretely observed diffusion processes. The model set-up is hierarchical: (1) For each sample (subject), the diffusion process is defined by a parametric stochastic differential equation; and (2) the parameters — or at least some of them — are random. The talk is about estimation of the parameters, including those in the distribution of the random effects. We suggest to replace the correct one-step-ahead transitions densities in the likelihood function with Gaussian approximations and maximize the corresponding pseudo-likelihood. In the talk, emphasis will be on the square-root (or Cox-Ingersoll-Ross) process with random drift parameters. We present simulation results and apply the methods to data on pig growth.

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MINIMAX ESTIMATION OF A SUBORDINATOR’S

DENSITY: UNIFORM RATES OF CONVERGENCE

Hilmar Mai

Weierstrass Institute Berlin, Germany, [email protected]

Denise Belomestny

, Universität Duisburg-Essen, Germany

John Schoenmakers

, Weierstrass Institute Berlin, Germany

Density estimation; subordinated Brownian motion; inverse Laplace transform; minimax rates of convergence .

We consider the problem of estimating the density of the time change of a time-changed Brownian motion based on low-frequency observations. This class of processes contains many well-known models from mathematical finance and the construction via a time-change makes them feasible in application if the distribution of the time change is accessible.

Our approach is based on a relation between the Fourier transform of the observed process and the Laplace transform of the target density. Hence, a main step in our estimation procedure is the inversion of the realized Laplace transform of the target density. This inversion step is an interesting statistical problem on its own and we will derive uniform rates of convergence for it in order to obtain convergence rates for the density estimator and study their optimality.

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ON OPTIMAL ESTIMATION OF STABLE

ORNSTEIN-UHLENBECK PROCESSES

Hiroki Masuda

Kyushu University, Japan, [email protected]

Infill sampling; optimal estimation; Ornstein-Uhlenbeck jump process.

Ornstein-Uhlenbeck (OU) processes driven by a Lévy process form a particular tractable class of Markovian stochastic differential equations with jumps. Among them, the non-Gaussian stable driven ones, the study of which dates back to Doob’s work [1], are known to have a pretty inherent character. Especially, a special property of stable integrals allows us to exactly generate the discrete-time sample from the process, and more importantly, to study in a transparent way the likelihood ratio associated with discrete-time sampling.

We are concerned with optimal estimation of the stable OU processes observed at high-frequency. We clarify that, due to the infinite-variance character of the model, the likelihood ratio exhibits entirely different asymptotic behaviors according to whether or not the terminal sampling time

Tn → ∞. If in particular Tn ≡ T , a fixed time, we present the LAMN (Local Asymptotic Mixed

Normality) structure of the statistical model, entailing the notion of asymptotic efficiency of a regular estimator. Also presented is how to construct some simple rate-efficient estimators having asymptotic mixed normality, together with numerical experiments.

References

[1] Doob, J. L. (1942) The Brownian movement and stochastic equations, Ann. of Math. (2) 43, 351–369.

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CONFIDENCE SETS IN NONPARAMETRIC

CALIBRATION OF EXPONENTIAL LÉVY MODELS

Jakob Söhl

University of Cambridge, United Kingdom, [email protected]

European option; jump diffusion; confidence sets; asymptotic normality; nonlinear inverse problem.

We construct confidence intervals for the spectral calibration method of exponential Lévy models. In these models the log price of an asset is described by a Lévy process. The observations are given by prices of European put and call options with fixed maturity and different strike prices. Belomestny and Reiß [1] introduced the spectral calibration method and showed that its estimators are minimax optimal. The method is designed for finite intensity Lévy process with absolutely continuous jump measures. First, the volatility, the drift and the intensity are estimated and then nonparametrically the jump density. The estimators are based on a cut–off scheme in the spectral domain.

We show that the estimators of the volatility, the drift and the intensity are asymptotically normally distributed. We also derive asymptotic normality for the pointwise estimation of the jump density at finitely many points and study the joint distribution of these estimators. The calibration of exponential Lévy models is a nonlinear inverse problem and our results cover both the mildly ill– posed case of volatility zero and the severely ill–posed case of positive volatility. Finally, the results on the asymptotic distribution of the estimators allow us to construct confidence intervals and confidence sets, see Söhl [2]. We will also present results from the joint work Söhl and Trabs [3], where we assess the finite sample performance of the confidence intervals by simulations and apply them to empirical data.

References

[1] Belomestny, D., Reiß, M. (2006) Spectral calibration of exponential Lévy models, Finance and Stochastics 10, 449–474.

[2] Söhl, J.(2012) Confidence sets in nonparametric calibration of exponential Lévy models, arXiv: 1202.6611.

[3] Söhl, J., Trabs, M. (2013) Option calibration of exponential Lévy models: Confidence intervals and empirical results, Journal of Computational Finance, to appear.

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SIMULATION OF CONDITIONAL DIFFUSIONS VIA

FORWARD-REVERSE STOCHASTIC

REPRESENTATIONS

John Schoenmakers

WIAS Berlin, Germany

Christian Bayer

, WIAS Berlin, Germany, [email protected]

Forward-reverse representations; pinned or conditional diffusions; Monte Carlo simulation. In this paper we derive stochastic representations for the finite dimensional distributions of a multi-dimensional diffusion on a fixed time interval, conditioned on the terminal state. The conditioning can be with respect to a fixed point or more generally with respect to some subset. The repre-sentations rely on a reverse process connected with the given (forward) diffusion as introduced in Milstein et al. [1] in the context of a forward-reverse transition density estimator. In more detail,

given a grid s0< s1< · · · < sK = t∗< t1< · · · < tL= T , we prove that

E [g(X(s1), . . . , X(sK), X(t1), . . . X(tL−1)) | X(s0) = x, X(T ) = y] equals lim →0 Eg X(s1), . . . , X(sK), Y (btL−1), . . . , Y (bt1) K(Y (T ) − X(t∗)) Y(T ) E [K(Y (T ) − X(t∗)) Y(T )] , (1)

where X is the given diffusion process and the dynamics of the reverse process (Y, Y) are explicitly

given in terms of the dynamics of X and do not include any singular or exploding terms. K is a

kernel with bandwidth andbt1< · · · <btL = T is a certain (deterministic) re-arrangement of the

grid t1< · · · < tL.

The corresponding Monte Carlo estimators have root-N accuracy (ignoring possible discretization errors induced by the construction of the samples from X and (Y, Y)), hence they do not suffer from the curse of dimensionality. We provide a detailed convergence analysis and give a numerical example involving the realized variance in a stochastic volatility asset model conditioned on a fixed terminal value of the asset.

References

[1] Milstein, G. N., Schoenmakers, J., Spokoiny, V. (2004) Transition density estimation for stochas-tic differential equations via forward-reverse representations, Bernoulli 10(2), 281–312.

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ON APPROXIMATION OF BSDE

Li Zhou

Université du Maine, France, [email protected]

Yury Kutoyants

, Université du Maine, France

BSDE; diffusion processes; small noise asymptotics; expansions.

We consider the problem of the construction of the BSDE in the markovian case, when the forward equation depends on some unknown parameter. The proposed approximation of the solution of BSDE is based on the MLE of this parameter. We show (in the asymptotics of “small noise”) that our approximation is asymptotically efficient and present some results of numerical simulations.

References

[1] Kutoyants, Y., Zhou, L. (2013) On approximation of BSDE, submitted.

[2] Zhou, L. (2013) Problèmes Statistiques pour les EDS et les EDS Rétrogrades. PhD Thesis, Université du Maine, Le Mans.

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INFERENCE IN SYSTEMS WITH MIXED FRACTIONAL

BROWNIAN MOTION NOISES

Marina Kleptsyna

Chigansky Pavel

, University of Jerusalem, Israel

Cai Chunhao

, Université du Maine, France

Fractional Brownian motion; maximum likelihood estimator; large sample asymptotic.

This talk addresses the problem of parameter estimation in linear systems, driven by a sum of the fractional and the standard Brownian noises. The likelihood function in this problem can be defined, using the analogs of the usual reprsentation theorems with respect to the semimartingale, which generates the same filtration as the observed process. The consistency and asymptotic normality of the maximum likelihood estimator are established using the properties of the corresponding second type Fredholm equation with weakly singular kernel or as the Wiener-Hopf equation on the finite interval. Another contribution of this talk is a formula for the Radon–Nikodym derivative of the

probability, induced by the mixed fractional Brownian motion ξt= Bt+ BtH, H > 3/4 with respect

to the Wiener measure.

References

[1] Cheridito, P. (2001) Mixed fractional Brownian motion, Bernoulli 7(6), 913–934.

[2] Bender, C., Sottinen, T., Valkeila, E. (2011) Fractional processes as models in stochastic finance, Advanced mathematical methods for finance, 75—103.

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VARIOUS LIMIT THEOREMS FOR AMBIT PROCESSES

Mark Podolskij

Heidelberg University, Germany, [email protected] Ambit processes, high frequency data; limit theorems.

We present some new asymptotic results for high frequency statistics of ambit processes. We start with a one-dimensional framework of the so called Brownian semistationary processes and review some existing probabilistic results and their statistical applications. In the second step, we demonstrate some non-trivial extensions to (a) High dimensional fields driven by a Gaussian white noise, (b) One-dimensional semistationary processes driven by a pure jump Lévy process. We will see that the three scenarios are quite different in nature, i.e. they produce different limit theorems. The talk is based on ongoing research with O.E. Barndorff-Nielsen, A. Basse-O’Connor, J.M. Corcuera and Mikko Pakkanen.

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VOLATITLITY MATRIX ESTIMATION FROM NOISY

OBSERVATIONS

Markus Bibinger

Humboldt-Universität zu Berlin, Germany, [email protected]

Markus Reiß

, Humboldt-Universität zu Berlin, Germany

Nikolaus Hautsch

Peter Malec

Asymptotic equivalence; asynchronous observations; microstructure noise; semi-parametric estimation.

The talk is devoted to inference on the volatility matrix of a continuous martingale from asyn-chronous high-frequency observations perturbed by noise. Asymptotic equivalence in Le Cam’s sense to a continuous-time white noise model reveals that asynchronicity is asymptotically imma-terial in combination with noise. A local parametrization and a locally adaptive spectral estimation approach exploiting multivariate Fisher information calculus leads to an efficient semi-parametric estimator. Upper bounds on the variance are shown to coincide with lower bounds derived for the model and within a reasonable class of estimators.

References

[1] Bibinger, M., Reiß, M., Hautsch, N., Malec, P. (2013) Asymptotically efficient covolatility matrix estimation via local likelihoods, preprint.

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DISCRIMINANT ANALYSIS FOR DISCRETELY

OBSERVED STOCHASTIC DIFFERENTIAL EQUATIONS

Masayuki Uchida

Osaka University, Japan, [email protected]

Classification criterion; ergodic diffusion process; maximum likelihood type estimator; misclassification probability.

We deal with a discriminant analysis for stochastic differential equations based on discrete ob-servations. First we consider the situation where a discretely observed ergodic diffusion process

belongs to one of two diffusion models Π1 and Π2, and we present two kinds of classification

cri-teria based on discriminant functions and asymptotic distributions of the discriminant functions. The discriminant functions are derived from the quasi-likelihood functions and the adaptive max-imum likelihood type estimators (Kessler (1997) and Uchida and Yoshida (2012)) with training data. Next, we propose a discriminant rule for stochastic differential equations from sampled data observed on the fixed interval and show the asymptotic property of the discriminant function. We also prove that the misclassification probabilities based on the classification criteria converge to zero. This is a joint work with Nakahiro Yoshida.

References

[1] Kessler, M. (1997) Estimation of an ergodic diffusion from discrete observations , Scandinavian Journal of Statistics, 24, 211–229.

[2] Uchida, M., Yoshida, N. (2012) Adaptive estimation of an ergodic diffusion process based on sampled data , Stochastic Processes and their Applications, 122, 2885–2924.

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EFFICIENCY IN DECONVOLUTION AND

LÉVY MODELS

Mathias Trabs

Humboldt-Universität zu Berlin, Germany, [email protected] Lévy process; information bound; score operator; Fourier multiplier.

Considering first the classical nonparametric deconvolution setting, we derive information bounds in the sense of Hájek–Le Cam for the estimation of linear functionals of the unknown density. The technique can then be used in the more complicated Lévy models. Observing a Lévy process at low frequency, we study the information bound for linear functionals of the jump density. Of particular interest is the estimation of the (generalized) distribution function in both settings. The lower bounds coincide with recent results [1,2] on the asymptotic variance of estimators in both models. Essential in our proofs are estimates of Hellinger integrals of infinitely divisible distributions and the mapping properties of the underlying score operator, which can be described using Fourier multiplier theory.

References

[1] Nickl, R., Reiß, M. (2012) A Donsker theorem for Lévy measures, J. Funct. Anal. 263(10), 3306–3332.

[2] Söhl, J., Trabs, M. (2012) A uniform central limit theorem and efficiency for deconvolution estimators, Electron. J. Stat. 6, 2486–2518.

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GUIDED PROPOSALS FOR SIMULATING

DIFFUSION BRIDGES

Moritz Schauer

Delft University of Technology / EURANDOM, The Netherlands, [email protected]

Frank van der Meulen

, Delft University of Technology, The Netherlands

Diffusion bridge; change of measure; componentwise Metropolis–Hastings; data augmentation; nonparametric inference.

We are interested in simulating from a multidimensional diffusion process conditioned on hitting a fixed point at a future time. Proposals are obtained from the unconditioned process with its drift superimposed by an additional guiding term. The acceptance of a proposal is determined by a likelihood ratio, which is computed in closed form. The use of the technique in a componentwise Metropolis–Hastings data augmentation step is illustrated in the context of Bayesian inference on the drift function of a discretely observed diffusion process.

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MARTINGALE EXPANSION AND STATISTICS OF

VOLATILITY

Nakahiro Yoshida

University of Tokyo, Japan, [email protected]

Masayuki Uchida

, Osaka University, Japan

Martingale expansion; random symbol; quadratic form; quasi likelihood analysis.

In order to develop higher-order theory of statistical inference, asymptotic expansion of the dis-tribution of a statistic plays an essential role. Martingale expansion was presented in [1] and [2] in the central limit case, and applied to the Edgeworth expansion for an ergodic diffusion process and an estimator of a volatility parameter.

The quadratic form of a semimartingale is in general asymptotically mixed normal under finite time horizon. Thus, it is necessary to develop an asymptotic expansion technique in the mixed normal situation to construct a higher-order statistical theory for high-frequency data.

In the non-ergodic statistics, we need a new methodology since traditional methods of asymptotic expansion do not work. Recently, the asymptotic expansion of a perturbed martingale with mixed normal limit was derived, together with applications to a quadratic form ([3], [4]). The expansion formula is expressed by a random symbol that consists of the adaptive random symbol and the anticipative random symbol. The former corresponds to the correction term appearing in the central limit case, and the latter is essentially new and often described by the Malliavin calculus. The anticipative random symbol reflects torsion of a martingale under randomness of the characteristics of the limit distribution.

In this talk, we give overview of the martingale expansion and its applications to a quadratic form of a diffusion process ([5]) as well as the realized volatility. We will also discuss expansion of the power variation (a joint work with M. Podolskij). Other applications will be found in the quasi-likelihood analysis for volatility.

References

[1] Yoshida, N. (1997) Malliavin calculus and asymptotic expansion for martingales, Probability Thoery Related Fields. 109, 301–342.

[2] Yoshida, N. (2001) Malliavin calculus and martingale expansion, Bull. Sci. math. 125, 431–456. [3] Yoshida, N. (2010) Expansion of the asymptotically conditionally normal law, ISM Research Memorandum 1125.

[4] Yoshida, N. (2013) Martingale expansion in mixed normal limit, arXiv: 1210.3680v3.

[5] Yoshida, N. (2012) Asymptotic expansion for the quadratic form of the diffusion process, arXiv: 1212.5845.

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A LASSO ESTIMATOR FOR EVENT-SPECIFIC

RATE MODELS FOR RECURRENT EVENTS

Olivier Bouaziz

University of Paris 5, Laboratory MAP5, France, [email protected]

Agathe Guilloux

, University of Paris 6, LSTA, France

Counting process; recurrent events; censored data; Aalen model; Cox model.

This talk considers statistical inference for the rate function of a recurrent event process. Two semi-parametric models of event-specific types are studied. These kind of models are stratified with respect to each event which allows more flexibility to fit the data. The first model studied in this paper was introduced by Prentice et al. (1981) and has a multiplicative form. The second one is based on the Aalen’s additive model and was introduced by Scheike (2002) in the context of recurrent events. For reasonable sizes of sample in event-specific models the number of estimated parameters can be very large compared to the number of covariates. In order to remedy to this over-parametrization, a total-variation penalty is used which constrain some of the parameters to be constant. The asymptotic behavior of the penalized estimator is derived. Through a simulation study and analysis of a real dataset, the performance of our estimator is compared with the unconstrained estimator and the Andersen and Gill (1980) constant estimator.

References

[1] Aalen, O. O. (1980) A model for non-parametric regression analysis of counting processes, Lecture Notes in Statistics-2: Mathematical Statistics and Probability Theory, 1–25.

[2] Andersen, P. K., Gill, R. D. (1982) Cox regression model for counting processes: a large sample study, Ann. Statist. 10, 1100–1120.

[3] Prentice, R. L., Williams, B. J., Peterson, A. V. (1981) On the regression analysis of multivariate failure time data, Biometrika 68, 373–379.

[4] Scheike, T. H. (2002) The additive nonparametric and semiparametric Aalen model as the rate function for a counting process, Lifetime Data Anal. 8, 247–262.

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ASYMPTOTIC ANALYSIS FOR OPTIMAL ESTIMATING

FUNCTIONS FOR A CLASS OF STOCHASTIC

VOLATILITY MODELS WITH JUMPS

Petra Posedel

Zagreb School of Economics and Management, Croatia, [email protected]

Friedrich Hubalek

, Vienna University of Technology, Austria Stochastic volatility models; optimal estimating functions; efficiency.

In [1] Barndorff-Nielsen and Shephard introduced a class of stochastic volatility models in continu-ous time, where the instantanecontinu-ous variance follows an Ornstein-Uhlenbeck type process driven by an increasing Lévy process. Those models have been studied from various points of view in math-ematical finance and related fields. The martingale estimating function technique for parameter estimation in BNS-models was used in [2] to obtain estimators that are consistent and asymptot-ically normal. The obtained estimators are not efficient though. The aim of the present paper is to develop estimators for BNS-models that are also optimal. Furthermore, the derivation of the asymptotic results for optimal estimators constructed using martingale estimating functions are analyzed in a more general setting. We study the consistency and asymptotic normality of the optimal estimator. Even though the estimator must be solved numerically, the asymptotic analy-sis of the resulting estimator can be performed. Moreover, the variance-covariance matrix of the estimator is calculated explicitly. The well-known theory from [3] for finding optimal estimating functions is reviewed and extended to a case of a bivariate Markov process.

References

[1] Barndorff-Nielsen, O. E., Shephard, N. (2001) Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics, J. R. Stat. Soc. Ser. B Stat. Methodol. 63(2), 167–241. [2] Hubalek, F., Posedel, P. (2012) Asymptotic analysis and explicit estimation of a class of stochastic volatility models with jumps using the martingale estimating function approach, Glasnik Matematicki, to appear.

[3] Heyde, C. C. (1997) Quasi-Likelihood And Its Application: A General Approach to Optimal Parameter Estimation, Springer-Verlag, New York.

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INFERENCE FOR PARTIALLY AND DISCRETELY

OBSERVED DIFFUSIONS WITH SMALL DIFFUSION

COEFFICIENT. APPLICATION TO EPIDEMICS

Romain Guy

INRA, Jouy-en-Josas & Université Paris Diderot, France, [email protected]

Catherine Laredo

, INRA, Jouy-en-Josas & Université Paris Diderot, France

Elisabeta Vergu

, INRA, Jouy-en-Josas, France

Diffusion processes; small diffusion coefficient; partial observations; discrete observations; epidemics.

Estimation of key parameters of epidemic models is still a challenging problem despite the progress made last years through computer intensive algorithms. In epidemiology, it is classical to use com-partmental SIR-like models (Susceptible-Infectious-Removed), where each individual is, at a given time, in one of these three mutually exclusive states. These models can be described in vari-ous mathematical frameworks and, amongst them, a natural representation is a multidimensional continuous-time Markov jump process. However, epidemic data are often partially observed and temporaly aggregated and the tractability in large populations of such processes is difficult. In this context, diffusion processes with small diffusion coefficient allow shedding new light on inference problems of epidemic data.

In [1], we considered a multidimensional diffusion (X

t)t≥0with drift coefficient b(α, X

t) and

diffu-sion coefficient σ(β, X

t), observed at times tk= k∆ on a fixed time interval [0, T ] with T = n∆.

We obtained consistent and asymptotically Gaussian estimators under the two asymptotics: n fixed and → 0; n → ∞ and → 0 simultaneously.

In this talk, we extend the results of [1] to study the inference problem in the presence of unobserved components in the diffusion model within the same asymptotic framework. For the SIR epidemic model, the susceptible component is usually not observed. The performances of our estimators are assessed on exact simulations from various epidemic models. Finally, we applied this method on a real data set of Influenza-like illness cases collected by the Sentinel surveillance network.

References

[1] GUY, R., Laredo, C., Vergu, E. (2012) Parametric inference for discretely observed multidi-mensionnal diffusions with small diffusion coefficient, ArXiv: 1206.0916

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STATIONARY DISTRIBUTIONS FOR STOCHASTIC

DIFFERENTIAL EQUATIONS WITH RANDOM

EFFECTS AND STATISTICAL APPLICATIONS

Valentine Genon-Catalot

MAP 5, University Paris Descartes, FRANCE, [email protected]

Catherine Larédo

, INRA and LPMA, University Paris Diderot, FRANCE

Stochastic differential equations; random effects; stationary distribution; statistical inference. Let (X(t), t ≥ 0) be defined by a stochastic differential equation including a random effect φ in the drift and diffusion coefficients. We characterize the stationary distributions of the joint process ((φ, X(t)), t ≥ 0) which are non unique and prove limit theorems and central limit theorems for functionals of the sample path (X(t), t ∈ [0, T ]) as T tends to infinity. This allows to build several estimators of the random variable φ which are consistent and asymptotically mixed normal with

rate√T . Examples are given fulfilling the assumptions of the limit theorems. Parametric estimation

of the distribution of the random effect from N i.i.d. processes (Xj(t), t ∈ [0, T ]), j = 1, . . . , N is

considered. Parametric estimators are built and proved to be √N -consistent and asymptotically

Gaussian as both N and T = T (N ) tend to infinity with T (N )/N tending to infinity. These results complete the parametric results obtained by Delattre et al. (2012) and the nonparametric results of Comte et al.(2013).

References

[1] Delattre M., Genon-Catalot V. and Samson A. (2012) Maximum likelihood estimation for stochastic differential equations with random effects, Scand. J. Statist., to appear.

[2] Comte F, Genon-Catalot V, Samson A. (2013) Nonparametric estimation for stochastic dif-ferential equations with random effects. Prepublication hal-00761394. To appear in Stoch. Proc. appl.

[3] Genon-Catalot V, Larédo C. (2013) Stationary distributions for stochastic differential equations with random effects and statistical applications. Prepublication MAP5 2013-09, hal-00807258.

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GOODNESS OF FIT TESTS FOR STOCHASTIC

PROCESSES WITH PARAMETRIC BASIC

HYPOTHESIS

Yury A. Kutoyants

Goodness of fit tests; diffusion processes; inhomogeneous Poisson processes; asymptotically distribution free tests.

We consider the problem of the construction of asymptotically distribution and parameter free tests by the observations of (continuous time) diffusion and inhomogeneous Poisson processes. We suppose that under the basic hypothesis the trend coefficients (for diffusion) and intensity function (for Poisson) depend on finite-dimensional parameter. The tests are of the Cramer-von Mises type. The asymptotics correspond to large samples (ergodic diffusion, periodic Poisson) and to small noise (dynamical systems). For each model of observations we propose linear transformations which allow to make the corresponding statistics to be asymptotically distribution or parameter free.

References

[1] Dachian, S. and Kutoyants, Yu.A. (2007) On the goodness-of-fit tests for some continuous time processes, in Statistical Models and Methods for Biomedical and Technical Systems, F.Vonta et al. (Eds), Birkhäuser, Boston, 395-413.

[2] Kleptsyna, M., Kutoyants, Yu.A. and Liptser, R. (2013) On asymptotically distribution free tests with parametric hypothesis for ergodic diffusion processes, submitted.

[3] Kutoyants Yu. A., (2011) Goodness-of-fit tests for perturbed dynamical systems,J. Statist. Planning Inference, 141, 1665-1666.

[4] Kutoyants Yu. A., (2012) On asymptotically distribution and parameter free goodness-of-fit tests for ergodic diffusion processes submitted.

[5] Negri, I., and Zhou, L. (2012) On goodness-of-fit testing for ergodic diffusion process with shift parameter, sumitted.

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Abstracts

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MAXIMUM LIKELIHOOD ESTIMATION OF 2N

DIMENSIONAL ORNSTEIN-UHLENBECK PROCESSES

WITH SINGULAR DIFFUSION MATRIX:

THE LAN PROPERTY

Ana Prior

University of Porto, Portugal, [email protected]

Paula Milheiro-Oliveira

, University of Porto, Portugal

Maximum likelihood estimator; Ornstein-Uhlenbeck process; LAN property; singular diffusion matrix.

Many real-life mechanical and structural systems respond dynamically to random environmental loads such as wind, wave or earthquake forces leading to stochastic estimation problems. The Ornstein-Uhlenbeck process appears in the engineering literature as a model for mechanical sys-tems subjected to random vibrations. A 2n dimension Ornstein-Uhlenbeck process for which the diffusion matrix is singular is considered. We study the problem of estimating the drift parameters of the stochastic differential equation that governs the Ornstein-Uhlenbeck process. The maximum likelihood estimator is proposed and explored in Koncz(1987). Unbaisedness and consistency of this estimator is a known result from the literature but little is known about the local asymptotic normality property since the process involved is not trivialy ergodic. In this case it is crucial to study the convergency of the covariance matrix of the estimator. We use the Yuima software to simulate the Ornstein-Uhlenbeck process, we study the assymptotics of the estimator and show that computations are in agreement with the obtained theoretical results.

References

[1] Arato, M. (1982). Linear Stochastic Systems with constant coefficients. A statistical approach. Lectures Notes in Control and Information Sciences, 45. Springer-Verlag, Berlin.

[2] Basak, G. and Lee, P., (2008) Asymptotic properties of an estimator of the drift coefficients of multidimensional Ornstein-Uhlenbeck processes that are not necessarily stable., Electronic Journal of Statistics, Vol.2, 1309-1344.

[3] Koncz, K., (1987) On the parameter estimation of diffusional type processes with constant coefficients (elementary Gaussian processes)., Journal of Anal. Math. 13. No.1, 75-91.

[4] Rao, B., and Basawa, I., (1980). Statistical Inference for Stochastic Processes. Academic Press, London.

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NON PARAMETRIC ESTIMATION OF THE DIFFUSION

COEFFICIENT OF A JUMP DIFFUSION

Émeline Schmisser

Université Lille 1, France, [email protected] Jump diffusions; non-parametric estimation.

We consider the stochastic differential equation with jumps:

dXt= b(Xt)dt + σ(Xt)dWt+ ξ(Xt−)dL_t X₀= η

where η is a random variable, (Wt)t≥0 is a Brownian motion independent of η and (Lt)t≥0 is a

Lévy process independent of (Wt)t≥0and η. This process is observed at discrete times 0, ∆, . . . , n∆

where ∆ → 0 and n∆ → ∞. The process (Xt)t≥0is assumed to be stationnary and β-mixing. Our

aim is to estimate the diffusion coefficient σ2 _{on a compact set A (A = [0, 1] for instance). To}

estimate σ2 _{for a diffusion process, we often consider}

Tk∆=

X(k+1)∆− Xk∆

2

∆ .

In the case of jump diffusions, Tk∆ = σ2(Xk∆) + ξ2(Xk∆) + centred terms + small terms. As a

consequence, in order to estimate σ2_{, we have to cut off most of the jumps, that’s why we introduce}

the random variables

Yk∆=

X(k+1)∆− Xk∆

2

∆ 1|X(k+1)∆−Xk∆|≤c ln(n)∆1/2.

To estimate σ2 _{non-parametrically, we consider a family of vectorial subspaces (S}

m)m≥0of L 2_(A)

and we construct a collection of estimators ˆσ2

mof σ2 by minimising a contrast function on Sm:

ˆ σ2_m= arg min t∈Sm γn(t) where γn(t) = 1 n n X k=1 (t(Xk∆) − Yk∆) 2 .

We prove that the risk of ˆσ2

mis bounded by Rn(ˆσm2) := E 1 n n X k=1 ˆ σm2(Xk∆) − σ2(Xk∆) 2 ! ≤ kσ2 m− σ2k2L2_(A)+ Dm n + ∆ 1−β/2

where Dm is the dimension of Sm, σm2 is the orthogonal projection of σ

2 _{on S}

m and β is the

Blumenthal-Getoor index of the Lévy process. We also construct an adaptive estimator of σ2 and

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ASYMPTOTICS FOR THE FOURIER ESTIMATORS OF

THE VOLATILITY OF VOLATILITY AND LEVERAGE

Imma Valentina Curato

University of Pisa, “Dipartimento di Economia e Management”, Italy , [email protected] Volatility of volatility; leverage; non-parametric estimation; semi-martingale; Fourier transform; high frequency data.

Inference for stochastic volatility models mainly focuses on parametric methods. In most cases, it is impossible to assess the distribution of the price process and to use the maximum likelihood theory. A non parametric approach based on the Fourier analysis can be developed to overcome this bottleneck defining a procedure that allows us to estimate all the latent variables in a stochastic volatility model. The Fourier methodology has been applied for the first time to the computation of the spot volatility and covariance in [1] and [2]. In this paper, we focus our attention on the estimations of the variance of volatility and the leverage component- covariance between the asset price and the volatility process. We extend the procedure developed in [1] in order to compute the Fourier coefficients of the volatility of volatility and of the leverage using n discrete observations of the price process and the Fourier estimation of the spot volatility. Then, we define spot and integrated estimators of the volatility of volatility and leverage processes. We prove consistency for the proposed estimators and we are able to attain both unfeasible and feasible central limit theorems for the integrated estimators.

References

[1] Malliavin, P., Mancino, M.E. (2002) Fourier series method for measurement of multivariate volatilities, Finance and Stochastics 4, 49–61.

[2] Malliavin, P., Mancino, M.E. (2009) A Fourier transform method for non parametric estimation of volatility , The Annals of Statistics, 37(4), 1983–2010.

[3] Clément, E., Gloter, A. (2010) Limit theorems in the Fourier transform method for the estima-tion of multivariate volatility , Stochastic Processes and their Applicaestima-tions 121, 1097–1124. [4] Jacod,J. and Protter, P. (1998) Asymptotic error distributions for the euler method for stochas-tic differential equations , The Annals of Probability 26(1), 267–307.

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SUPERPROCESSES AS A MODEL OF INFORMATION

DISSEMINATION BETWEEN MOBILE DEVICES

Laura Sacerdote

University of Torino, Italy, [email protected]

Federico Polito

, University of Torino, Italy

Matteo Sereno

Michele Garetto

Superprocess; continuos space branching process; future internet; device to device transmissiom. New technologies, services and applications of internet come into play at a high rate and there is an increasing need to update existing quantitative methods and performance evaluation tools to manage these new trends. Use of mobile devices to substitute or to support servers is a possible future direction to disseminate information [2]. Performance evaluations request answers on a set of topics such as their ability to achieve city level coverage, the estimation of the time delay to reach far away users, the presence of zones not attained by the signal. However the analysis implies a substantial change of perspective with respect to previous approaches, which are often based on Markov chain simulations. The number of involved devices prevents the use of Markov chains for the analysis of interest and suitable continuous limits become necessary Here we present some preliminary ideas to deal with these problems. We propose to model the information dissemination between mobile devices through superprocesses (cf.[1],[3]) and we discuss a possible approach to their simulation.

References

[1] Etheridge, A. (2000) An Introduction to Superprocesses, American Mathematical Society. [2] Klein, D.J., Hespanha, J., Madhow, U. (2004) A Reaction-Diffusion Model for Epidemic Routing in Sparsely Connected MANETs, INFOCOM 2010, IEEE Proceedings 1–9. [3] Le Gall, J. F. (1999) Spatial Branching Processes, Random Snakes and Partial Differential Equations, Birkhauser.

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POPULATION PHARMACOKINETICS AND

STOCHASTIC DIFFERENTIAL EQUATIONS:

MODELS AND METHODS

Maud Delattre

AgroParisTech, France, [email protected]

Marc Lavielle

, University Paris Sud - Inria Saclay Île-de-France, France

Stochastic differential equations; nonlinear mixed-effects models; SAEM algorithm; extended Kalman filter.

Models based on stochastic differential equations (SDEs) have an interest in population pharma-cokinetics (PK), since they allow a better consideration of the variability occuring within any individual kinetics than classical PK models based on ordinary differential equations (ODEs). The objective of this contribution is twofold. First, we present new SDE-based PK models. Second, we suggest some specific estimation procedure for these models.

An extension of the traditional PK models based on ODEs consists in adding a system noise to the ODEs (see [1], [2]). However, the resulting SDE systems do not comply with some constraints on the biological dynamics (sign, monotony,...) and therefore give an overly erratic description of the evolution of the drug concentrations within the compartments of the human body. We rather assume that the diffusion process randomly perturbs the transfer rate constants of the system. Some simulated kinetics show that this assumption is more realistic and allows a more accurate representation of the biological system.

Estimating the population parameters in such SDE models is however not straightforward. Indeed, in a population approach, the model needs to account for the between-subjects variability. Then, we consider that the parameters of the model are random variables, resulting in a mixed-effects models ([3]). Specific procedures are required to make inference in mixed-effects models. Here, we suggest estimating the population parameters by combining the SAEM algorithm ([4], [5]) with the extended Kalman filter. This methodology was implemented in a working version of MONOLIX ([6]) and tested on some simulated basic examples. An application to a PK example is also presented.

References

[1] Mortensen, S., Klim, S., Dammann, B., Kristensen, N., Madsen, H., Overgaard, R. (2007) A Matlab framework for estimation of NLME models using stochastic differential equations., Journal of Pharmacokinetics and Pharmacodynamics 34, 623–642.

[2] Donnet, S., Samson, A. (2008) Parametric inference for mixed models defined by stochastic differential equations., ESAIM P& S 12, 196–218.

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Participants

Adam Lund Adeline Samson Alejandro Cencerrado Alexander Schnurr Alexandre Iolov

Ana Filipa Prior

Anders Jensen Anders Rønn-Nielsen Anna Barz Badreddine Azzouzi Benedikt Funke Bezirgen Veliyev Bo Markussen Carsten Wiuf Catherine Larédo Christian Schmidt Chunhao Cai Daniele Cappelletti David Skovmand Dominique Dehay Dragi Anevski Eleni Vradi Emeline Schmisser Ernst Eberlein

Ernst August v. Hammerstein

Gang Huang

Gayrat Urazboev

Helle Sørensen

Hilmar Mai

Hiroki Masuda

Imma Valentina Curato

Jakob Söhl

Jan Gairing

John Schoenmakers

Kamille Tågholt Gad

Laura Sacerdote

Lilia Hadvig

Mareile Große Ruse

Marietta Kokla Marina Kleptsyna Mark Podolskij Markus Reiß Markus Bibinger Martin Jacobsen Martin Jönsson Masayuki Uchida Massimiliano Tamborrino Mathias Trabs Maud Delattre Michael Sørensen Mikko Pakkanen Moritz Schauer Nakahiro Yoshida

Niels Richard Hansen

Niels Keiding Nina Munkholt Olivier Bouaziz Paula Milheiro-Oliveira Pavel Chigansky Peter Spreij Petra Posedel Romain Guy Sam Finch Søren Johansen

Seyed Nourollah Mousavi

Susanne Ditlevsen

Tilahun Ferede Asena

Umberto Picchini Valentine Genon-Catalot Verónica Rosas Viviane Baladi Yury Kutoyants Zhou Li

Equivalence for nonparametric drift estimation of a diffusion process and its Euler scheme

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Equivalence for nonparametric drift estimation of a

diffusion process and its Euler scheme

Catherine Laredo, Valentine Genon-Catalot

To cite this version:

Catherine Laredo, Valentine Genon-Catalot. Equivalence for nonparametric drift estimation of a

diffusion process and its Euler scheme. DYNSTOCH 2013, Apr 2013, Copenhagen, Denmark. pp.43.

�hal-02748689�

DYNSTOCH 2013

University of Copenhagen

April 17-19

Organizers:

Web:

Sponsors:

Contents

General Information

Venue

Wireless network

Maps

DYNSTOCH WORKSHOP 2013

PROGRAMME

Venue: Auditorium 4

The H.C. Ørsted Institute, Universitetsparken 5

Directions

Wednesday, April 17

12:30 - 14:00 LUNCH

14:00 - 14:30

Ernst Eberlein (University of Freiburg): TWO PRICE ECONOMIES

IN CONTINUOUS TIME

14:30 - 15:00

Markus Bibinger (Humboldt-Universität zu Berlin): VOLATITLITY

MATRIX ESTIMATION FROM NOISY OBSERVATIONS

15:00 - 15:30

Ernst August v. Hammerstein (University of Freiburg): OPTIMALITY

OF PAYOFFS IN LEVY MODELS.

15:30 - 16:00 BREAK

16:00 - 16:30

Alexander Schnurr (TU Dortmund): AN ORDINAL PATTERN

APPROACH TO DETECT AND TO MODEL DEPENDENCE

STRUCTURES BETWEEN FINANCIAL TIME SERIES

16:30 - 17:00

Hilmar Mai (Weierstrass Institute Berlin): MINIMAX ESTIMATION OF A

SUBORDINATOR'S DENSITY: UNIFORM RATES OF CONVERGENCE

17:00 - 17:30

Olivier Bouaziz (University of Paris 5): A LASSO ESTIMATOR FOR

EVENT-SPECIFIC RATE MODELS FOR RECURRENT EVENTS

17:30 - 18:00 Li Zhou (Université du Maine): ON APPROXIMATION OF BSDE

18:00 - WELCOME RECEPTION

Thursday, April 18

9:00 - 9:30

Nakahiro Yoshida (University of Tokyo): MARTINGALE EXPANSION

AND STATISTICS OF VOLATILITY

9:30 - 10:00

Kleptsyna Marina (Université du Maine): INFERENCE IN SYSTEMS

WITH MIXED FRACTIONAL BROWNIAN MOTION NOISES

10:00 - 10:30

Catherine Larédo (INRA & Université Paris Diderot): EQUIVALENCE

FOR NON PARAMETRIC DRIFT ESTIMATION OF A DIFFUSION

PROCESS AND ITS EULER SCHEME

10:30 - 11:00 BREAK

11:00 - 11:30

Adeline Samson (University Paris Descartes): PARAMETER ESTIMATION

IN THE STOCHASTIC MORRIS-LECAR NEURONAL MODEL

11:30 - 12:00

Masayuki Uchida (Osaka University): DISCRIMINANT ANALYSIS FOR