ISSN 1050-5164 (print) ISSN 2168-8737 (online) THE ANNALS of APPLIED PROBABILITY

AN OFFICIAL JOURNAL OF THE INSTITUTE OF MATHEMATICAL STATISTICS

Articles On the stochastic behaviour of optional processes up to random times CONSTANTINOS KARDARAS 429 A polynomial time approximation scheme for computing the supremum of Gaussian processes...... RAGHU MEKA 465 Viral processes by random walks on random regular graphs MOHAMMED ABDULLAH,COLIN COOPER AND MOEZ DRAIEF 477 Central limit theorems for an Indian buffet model with random weights PATRIZIA BERTI,IRENE CRIMALDI,LUCA PRATELLI AND PIETRO RIGO 523 LargedeviationsforMarkoviannonlinearHawkesprocesses...... LINGJIONG ZHU 548 A generalized backward scheme for solving nonmonotonic stochastic recursions...... P.MOYAL 582 Limit theorems for nearly unstable Hawkes processes THIBAULT JAISSON AND MATHIEU ROSENBAUM 600 Spatial preferential attachment networks: Power laws and clustering coefficients EMMANUEL JACOB AND PETER MÖRTERS 632 On the expected total number of infections for virus spread on a finite network ANTAR BANDYOPADHYAY AND FARKHONDEH SAJADI 663 Limit theorems for Smoluchowski dynamics associated with critical continuous-state branching processes . . . . GAUTAM IYER,NICHOLAS LEGER AND ROBERT L. PEGO 675 Exponential moments of affine processes MARTIN KELLER-RESSEL AND EBERHARD MAYERHOFER 714 Universality in polytope phase transitions and message passing algorithms MOHSEN BAYATI,MARC LELARGE AND ANDREA MONTANARI 753 Arbitrage and duality in nondominated discrete-time models BRUNO BOUCHARD AND MARCEL NUTZ 823 Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition ...... NICOLAS FOURNIER 860 Gibbs measures on permutations over one-dimensional discrete point sets MAREK BISKUP AND THOMAS RICHTHAMMER 898 Large deviations for cluster size distributions in a continuous classical many-body system SABINE JANSEN,WOLFGANG KÖNIG AND BERND METZGER 930 Approximation algorithms for the normalizing constant of Gibbs distributions MARK HUBER 974 A decreasing step method for strongly oscillating stochastic models CAMILO ANDRÉS GARCÍA TRILLOS 986 Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms CHRISTOPHE ANDRIEU AND MATTI VIHOLA 1030

Vol. 25, No. 2—April 2015 The Annals of Applied Probability 2015, Vol. 25, No. 2, 429–464 DOI: 10.1214/13-AAP976 © Institute of Mathematical Statistics, 2015

ON THE STOCHASTIC BEHAVIOUR OF OPTIONAL PROCESSES UP TO RANDOM TIMES

BY CONSTANTINOS KARDARAS1 London School of Economics and Political Science

In this paper, a study of random times on filtered probability spaces is undertaken. The main message is that, as long as distributional properties of optional processes up to the random time are involved, there is no loss of generality in assuming that the random time is actually a randomised stopping time. This perspective has advantages in both the theoretical and practical study of optional processes up to random times. Applications are given to financial mathematics, as well as to the study of the stochastic behaviour of Brownian motion with drift up to its time of overall maximum as well as up to last-passage times over finite intervals. Furthermore, a novel proof of the Jeulin–Yor decomposition formula via Girsanov’s theorem is provided.

REFERENCES

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MSC2010 subject classifications. 60G07, 60G44. Key words and phrases. Random times, randomised stopping times, times of maximum, last pas- sage times. [11] GUO,X.andZENG, Y. (2008). Intensity process and compensator: A new filtration expansion approach and the Jeulin–Yor theorem. Ann. Appl. Probab. 18 120–142. MR2380894 [12] HE,S.W.,WANG,J.G.andYAN, J. A. (1992). Semimartingale Theory and Stochastic Cal- culus. Kexue Chubanshe, Beijing. MR1219534 [13] JACOD,J.andSHIRYAEV, A. N. (1998). Local martingales and the fundamental asset pricing theorems in the discrete-time case. Finance Stoch. 2 259–273. MR1809522 [14] JACOD,J.andSHIRYAEV, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin. MR1943877 [15] JEANBLANC,M.andSONG, S. (2011). An explicit model of default time with given survival probability. Stochastic Process. Appl. 121 1678–1704. MR2811019 [16] JEULIN, T. (1980). Semi-martingales et Grossissement D’une Filtration. Lecture Notes in Math. 833. Springer, Berlin. MR0604176 [17] JEULIN,T.andYOR, M. (1978). Grossissement d’une filtration et semi-martingales: Formules explicites. In Séminaire de Probabilités, XII (Univ. Strasbourg, Strasbourg, 1976/1977). Lecture Notes in Math. 649 78–97. Springer, Berlin. MR0519998 [18] JEULIN,T.andYOR, M., eds. (1985). Grossissements de Filtrations: Exemples et Applications. Lecture Notes in Math. 1118. Springer, Berlin. MR0884713 [19] KALLSEN, J. (2003). σ -localization and σ -martingales. Teor. Veroyatn. Primen. 48 177–188. MR2013413 [20] KARATZAS,I.andKARDARAS, C. (2007). The numéraire portfolio in semimartingale finan- cial models. Finance Stoch. 11 447–493. MR2335830 [21] KARATZAS,I.andSHREVE, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York. MR1121940 [22] KARDARAS, C. (2010). Finitely additive probabilities and the fundamental theorem of asset pricing. In Contemporary Quantitative Finance 19–34. Springer, Berlin. MR2732838 [23] KARDARAS, C. (2010). Numéraire-invariant preferences in financial modeling. Ann. Appl. Probab. 20 1697–1728. MR2724418 [24] KARDARAS, C. (2014). On the characterisation of honest times that avoid all stopping times. Stochastic Process. Appl. 124 373–384. MR3131298 [25] KARDARAS, C. (2014). A time before which insiders would not undertake risk. In Inspired by Finance (the Musiela Festschrift) (Y. Kabanov, M. Rutkowski and T. Zariphopoulou, eds.) 349–362. Springer, Cham. [26] KRAMKOV,D.andSÎRBU, M. (2006). On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 16 1352– 1384. MR2260066 [27] KUSUOKA, S. (1999). A remark on default risk models. In Advances in Mathematical Eco- nomics, Vol.1(Tokyo, 1997). Adv. Math. Econ. 1 69–82. Springer, Tokyo. MR1722700 [28] LANDO, D. (1998). On Cox processes and credit risky securities. Review of Derivatives Re- search 2 610–612. [29] MEYER, P. A. (1972). La mesure de H. Föllmer en théorie des surmartingales. In Séminaire de Probabilités, VI (Univ. Strasbourg, Année Universitaire 1970–1971; Journées Prob- abilistes de Strasbourg, 1971). Lecture Notes in Math. 258 118–129. Springer, Berlin. MR0368131 [30] NIKEGHBALI,A.andYOR, M. (2006). Doob’s maximal identity, multiplicative decom- positions and enlargements of filtrations. Illinois J. Math. 50 791–814 (electronic). MR2247846 [31] PARTHASARATHY, K. R. (2005). Probability Measures on Metric Spaces. AMS Chelsea Pub- lishing, Providence, RI. MR2169627 [32] PROFETA,C.,ROYNETTE,B.andYOR, M. (2010). Option Prices as Probabilities: ANew Look at Generalized Black–Scholes Formulae. Springer, Berlin. MR2582990 [33] PROTTER, P. (1990). Stochastic Integration and Differential Equations: A New Approach. Ap- plications of Mathematics (New York) 21. Springer, Berlin. MR1037262 [34] ROGERS,L.C.G.andWILLIAMS, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 1. Cambridge Univ. Press, Cambridge. MR1796539 [35] SHIRYAEV,A.N.andCHERNY, A. S. (2000). Some distributional properties of a Brownian motion with a drift and an extension of P. Lévy’s theorem. Theory Probab. Appl. 44 412– 418. [36] TSIRELSON, B. (1998). Within and beyond the reach of Brownian innovation. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998) 311–320 (elec- tronic). Doc. Math., Extra Vol. III. MR1648165 [37] YOR, M. (1978). Grossissement d’une filtration et semi-martingales: Théorèmes généraux. In Séminaire de Probabilités, XII (Univ. Strasbourg, Strasbourg, 1976/1977). Lecture Notes in Math. 649 61–69. Springer, Berlin. MR0519996 The Annals of Applied Probability 2015, Vol. 25, No. 2, 465–476 DOI: 10.1214/13-AAP997 © Institute of Mathematical Statistics, 2015

A POLYNOMIAL TIME APPROXIMATION SCHEME FOR COMPUTING THE SUPREMUM OF GAUSSIAN PROCESSES1

BY RAGHU MEKA Microsoft Research

We give a polynomial time approximation scheme (PTAS) for computing the supremum of a Gaussian process. That is, given a finite set of vectors V ⊆ Rd + [ | |] , we compute a (1 ε)-factor approximation to EX←N d supv∈V v,X deterministically in time poly(d)·|V |Oε(1). Previously, only a constant factor deterministic polynomial time approximation algorithm was known due to the work of Ding, Lee and Peres [Ann. of Math. (2) 175 (2012) 1409–1471]. This answers an open question of Lee (2010) and Ding [Ann. Probab. 42 (2014) 464–496]. The study of supremum of Gaussian processes is of considerable impor- tance in probability with applications in functional analysis, convex geome- try, and in light of the recent breakthrough work of Ding, Lee and Peres [Ann. of Math. (2) 175 (2012) 1409–1471], to random walks on finite graphs. As such our result could be of use elsewhere. In particular, combining with the work of Ding [Ann. Probab. 42 (2014) 464–496], our result yields a PTAS for computing the cover time of bounded-degree graphs. Previously, such al- gorithms were known only for trees. Along the way, we also give an explicit oblivious estimator for semi- norms in Gaussian space with optimal query complexity. Our algorithm and its analysis are elementary in nature, using two classical comparison inequal- ities, Slepian’s lemma and Kanter’s lemma.

REFERENCES

ALDOUS,D.andFILL, J. (1994). Reversible Markov chains and random walks on graphs. Available at http://www.stat.berkeley.edu/~aldous/RWG/book.html. BALL, K. (2001). Convex Geometry and Functional Analysis. Handbook of the Geometry of Banach Spaces 1. North-Holland, Amsterdam. MR1863692 DADUSH,D.andVEMPALA, S. (2012). Deterministic construction of an approximate M-ellipsoid and its applications to derandomizing lattice algorithms. In Proceedings of the Twenty-Third Annual ACM–SIAM Symposium on Discrete Algorithms 1445–1456. SIAM, Philadelphia. MR3205304 DING, J. (2014). Asymptotics of cover times via Gaussian free fields: Bounded-degree graphs and general trees. Ann. Probab. 42 464–496. MR3178464 DING,J.,LEE,J.R.andPERES, Y. (2012). Cover times, blanket times, and majorizing measures. Ann. of Math.(2)175 1409–1471. MR2912708 ENGEBRETSEN,L.,INDYK,P.andO’DONNELL, R. (2002). Derandomized dimensionality reduc- tion with applications. In Proceedings of the Thirteenth Annual ACM–SIAM Symposium on Dis- crete Algorithms 705–712. SIAM, Philadelphia, PA.

MSC2010 subject classifications. Primary 60C05; secondary 68Q87. Key words and phrases. Gaussian processes, derandomization, cover time, random walks, ε-nets. FEIGE,U.andZEITOUNI, O. (2009). Deterministic approximation for the cover time of trees. Preprint. Available at arXiv:0909.2005. KAHN,J.,KIM,J.H.,LOVÁSZ,L.andVU, V. H. (2000). The cover time, the blanket time, and the Matthews bound. In 41st Annual Symposium on Foundations of Computer Science (Redondo Beach, CA, 2000) 467–475. IEEE Comput. Soc. Press, Los Alamitos, CA. MR1931843 KANTER, M. (1977). Unimodality and dominance for symmetric random vectors. Trans. Amer. Math. Soc. 229 65–85. MR0445580 LEDOUX,M.andTALAGRAND, M. (1991). Probability in Banach Spaces: Isoperimetry and Pro- cesses. Springer, Berlin. MR1102015 LEE, J. (2010). Open question: Cover times and the Gaussian free field. Available at https://tcsmath.wordpress.com/2010/12/09/open-question-cover-times-and-the-gaussian-free- field/. LOVÁSZ, L. (1993). Random walks on graphs: A survey. Combinatorics,PaulErdos˝ Is Eighty, Vol. 2 (Keszthely, 1993). Bolyai Soc. Math. Stud. 2353–2397. János Bolyai Math. Soc., Budapest. MR1395866 PISIER, G. (1999). The Volume of Convex Bodies and Banach Space Geometry. Cambridge Univ. Press, Cambridge. SIVAKUMAR, D. (2002). Algorithmic derandomization via complexity theory. In IEEE Conference on Computational Complexity 619–626. ACM, New York. SLEPIAN, D. (1962). The one-sided barrier problem for Gaussian noise. Bell Syst. Tech. J. 41 463– 501. MR0133183 TALAGRAND, M. (2005). The Generic Chaining: Upper and Lower Bounds of Stochastic Processes. Springer, Berlin. MR2133757 WINKLER,P.andZUCKERMAN, D. (1996). Multiple cover time. Random Structures Algorithms 9 403–411. MR1605407 The Annals of Applied Probability 2015, Vol. 25, No. 2, 477–522 DOI: 10.1214/13-AAP1000 © Institute of Mathematical Statistics, 2015

VIRAL PROCESSES BY RANDOM WALKS ON RANDOM REGULAR GRAPHS

BY MOHAMMED ABDULLAH1,COLIN COOPER AND MOEZ DRAIEF1 University of Birmingham, King’s College London and Imperial College London

We study the SIR epidemic model with infections carried by k particles making independent random walks on a random regular graph. Here we as- sume k ≤ n,wheren is the number of vertices in the random graph, and  is some sufficiently small constant. We give an edge-weighted graph re- duction of the dynamics of the process that allows us to apply standard re- sults of Erdos–Rényi˝ random graphs on the particle set. In particular, we show how the parameters of the model give two thresholds: In the subcrit- ical regime, O(ln k) particles are infected. In the supercritical regime, for a constant β ∈ (0, 1) determined by the parameters of the model, βk get in- fected with probability β,andO(ln k) get infected with probability (1 − β). Finally, there is a regime in which all k particles are infected. Furthermore, the edge weights give information about when a particle becomes infected. We exploit this to give a completion time of the process for the SI case.

REFERENCES

[1] ABDULLAH,M.,COOPER,C.andFRIEZE, A. (2012). Cover time of a random graph with given degree sequence. Discrete Math. 312 3146–3163. MR2957935 [2] ALDOUS,D.andFILL, J. Reversible Markov chains and random walks on graphs. Un- published manuscript. Available at http://stat-www.berkeley.edu/pub/users/aldous/RWG/ book.html. [3] AMINI,H.,DRAIEF,M.andLELARGE, M. (2009). Marketing in a random network. In Net- work Control and Optimization: Second Euro-NF Workshop, NET-COOP 2008 Paris, France, September 8–10, 2008. Revised Selected Papers. Lecture Notes in Computer Sci- ence 5425 17–25. Springer, Berlin. [4] AMINI,H.,DRAIEF,M.andLELARGE, M. (2013). Flooding in weighted sparse random graphs. SIAM J. Discrete Math. 27 1–26. MR3032902 [5] BAUMANN,H.,CRESCENZI,P.andFRAIGNIAUD, P. (2009). Parsimonious flooding in dy- namic graphs. In Proceedings of the 28th ACM Symposium on Principles of Distributed Computing 260–269. ACM, New York. [6] CHAINTREAU,A.,HUI,P.,SCOTT,J.,GASS,R.,CROWCROFT,J.andDIOT, C. (2007). Impact of human mobility on opportunistic forwarding algorithms. IEEE Transactions Mobile Computing 6 606–620. [7] COOPER,C.andFRIEZE, A. (2005). The cover time of random regular graphs. SIAM J. Dis- crete Math. 18 728–740. MR2157821 [8] COOPER,C.andFRIEZE, A. (2008). The cover time of the giant component of a random graph. Random Structures Algorithms 32 401–439. MR2422388

MSC2010 subject classifications. Primary 05C81, 60J20; secondary 05C80. Key words and phrases. Random walks, epidemics, random graphs. [9] COOPER,C.,FRIEZE,A.andRADZIK, T. (2009). Multiple random walks in random regular graphs. SIAM J. Discrete Math. 23 1738–1761. MR2570201 [10] DALEY,D.J.andGANI, J. (1999). Epidemic Modelling: An Introduction. Cambridge Studies in Mathematical Biology 15. Cambridge Univ. Press, Cambridge. MR1688203 [11] DICKMAN,R.,ROLLA,L.T.andSIDORAVICIUS, V. (2010). Activated random walkers: Facts, conjectures and challenges. J. Stat. Phys. 138 126–142. MR2594894 [12] DIMITRIOU,T.,NIKOLETSEAS,S.andSPIRAKIS, P. (2006). The infection time of graphs. Discrete Appl. Math. 154 2577–2589. MR2277961 [13] DRAIEF,M.andGANESH, A. (2011). A random walk model for infection on graphs: Spread of epidemics & rumours with mobile agents. Discrete Event Dyn. Syst. 21 41–61. MR2764438 [14] DRAIEF,M.andMASSOULIÉ, L. (2010). Epidemics and Rumours in Complex Networks. Lon- don Mathematical Society Lecture Note Series 369. Cambridge Univ. Press, Cambridge. MR2582458 [15] JANSON, S. (1999). One, two and three times log n/n for paths in a complete graph with random weights. Combin. Probab. Comput. 8 347–361. MR1723648 [16] JANSON,S.,ŁUCZAK,T.andRUCINSKI, A. (2000). Random Graphs. Wiley, New York. MR1782847 [17] LAM,H.,LIU,Z.,MITZENMACHER,M.,SUN,X.andWANG, Y. (2012). Information dis- semination via random walks in d-dimensional space. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms 1612–1622. ACM, New York. MR3205318 [18] LOVÁSZ, L. (1993). Random walks on graphs: A survey. In Combinatorics, Paul Erd˝os Is Eighty, Vol.2(Keszthely, 1993). Bolyai Soc. Math. Stud. 2 353–397. János Bolyai Math. Soc., Budapest. MR1395866 [19] PERES,Y.,SINCLAIR,A.,SOUSI,P.andSTAUFFER, A. (2011). Mobile geometric graphs: Detection, coverage and percolation. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms 412–428. SIAM, Philadelphia, PA. MR2857136 [20] PETTARIN,A.,PIETRACAPRINA,A.,PUCCI,G.andUPFAL, E. (2011). Tight bounds on in- formation dissemination in sparse mobile networks. In Proceedings of the 30th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing 355–362. ACM, New York. [21] PITTEL, B. (1987). On spreading a rumor. SIAM J. Appl. Math. 47 213–223. MR0873245 [22] RHODES,C.andNEKOVEE, M. (2008). The opportunistic transmission of wireless worms between mobile devices. Physica A: Statistical Mechanics and Its Applications 387 6837– 6844. [23] VA N D E R HOFSTAD, R. (2013). Random graphs and complex networks. Available at http://www.win.tue.nl/~rhofstad/NotesRGCN.pdf. The Annals of Applied Probability 2015, Vol. 25, No. 2, 523–547 DOI: 10.1214/14-AAP1002 © Institute of Mathematical Statistics, 2015

CENTRAL LIMIT THEOREMS FOR AN INDIAN BUFFET MODEL WITH RANDOM WEIGHTS

BY PATRIZIA BERTI,IRENE CRIMALDI1,LUCA PRATELLI AND PIETRO RIGO Università di Modena e Reggio-Emilia, IMT Institute for Advanced Studies, Accademia Navale and Università di Pavia The three-parameter Indian buffet process is generalized. The possibly different role played by customers is taken into account by suitable (random) weights. Various limit theorems are also proved for such generalized Indian buffet process. Let Ln be the number of dishes experimented by the first n = n customers, and let Kn (1/n) i=1 Ki where Ki is the number of dishes tried by customer i. The asymptotic distributions of Ln and Kn, suitably centered and scaled, are obtained. The convergence turns out to be stable (and not only in distribution). As a particular case, the results apply to the standard (i.e., nongeneralized) Indian buffet process.

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LARGE DEVIATIONS FOR MARKOVIAN NONLINEAR HAWKES PROCESSES

BY LINGJIONG ZHU1 New York University

Hawkes process is a class of simple point processes that is self-exciting and has clustering effect. The intensity of this point process depends on its entire past history. It has wide applications in finance, neuroscience and many other fields. In this paper, we study the large deviations for nonlinear Hawkes processes. The large deviations for linear Hawkes processes has been studied by Bordenave and Torrisi. In this paper, we prove first a large deviation prin- ciple for a special class of nonlinear Hawkes processes, that is, a Markovian Hawkes process with nonlinear rate and exponential exciting function, and then generalize it to get the result for sum of exponentials exciting functions. We then provide an alternative proof for the large deviation principle for a linear Hawkes process. Finally, we use an approximation approach to prove the large deviation principle for a special class of nonlinear Hawkes processes with general exciting functions.

REFERENCES

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MSC2010 subject classifications. 60G55, 60F10. Key words and phrases. Large deviations, rare events, point processes, Hawkes processes, self- exciting processes. [12] HAWKES, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58 83–90. MR0278410 [13] HAWKES,A.G.andOAKES, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Probab. 11 493–503. MR0378093 [14] JAGERS, P. (1975). Branching Processes with Biological Applications. Wiley, London. MR0488341 [15] JOÓ, I. (1984). Note on my paper: “A simple proof for von Neumann’s minimax theorem” [Acta Sci. Math. (Szeged) 42 (1980), no. 1-2, 91–94; MR0576940 (81i:49008)]. Acta Math. Hungar. 44 363–365. MR0764631 [16] KARABASH,D.andZHU, L. (2012). Limit theorems for marked Hawkes processes with ap- plication to a risk model. Preprint. Available at arXiv:1211.4039. [17] KORALOV,L.B.andSINAI, Y. G. (2007). Theory of Probability and Random Processes, 2nd ed. Springer, Berlin. MR2343262 [18] LINIGER, T. (2009). Multivariate Hawkes processes. Ph.D. thesis, ETH, Zürich. [19] LIPTSER,R.S.andSHIRYAEV, A. N. (2001). Statistics of Random Processes. II, 2nd ed. Springer, Berlin. MR1800858 [20] OAKES, D. (1975). The Markovian self-exciting process. J. Appl. Probab. 12 69–77. MR0362522 [21] STABILE,G.andTORRISI, G. L. (2010). Risk processes with non-stationary Hawkes claims arrivals. Methodol. Comput. Appl. Probab. 12 415–429. MR2665268 [22] VARADHAN, S. R. S. (2001). Probability Theory. Amer. Math. Soc., Providence, RI. MR1852999 [23] VARADHAN, S. R. S. (2008). Large deviations. Ann. Probab. 36 397–419. MR2393987 [24] ZHU, L. (2014). Limit theorems for a Cox–Ingersoll–Ross process with Hawkes jumps. J. Appl. Probab. To appear. [25] ZHU, L. (2014). Process-level large deviations for nonlinear Hawkes point processes. Ann. Inst. Henri Poincaré Probab. Stat. 50 845–871. MR3224291 [26] ZHU, L. (2013). Nonlinear Hawkes processes. Ph.D. thesis, New York Univ. [27] ZHU, L. (2013). Ruin probabilities for risk processes with non-stationary arrivals and subex- ponential claims. Insurance Math. Econom. 53 544–550. MR3130449 [28] ZHU, L. (2013). Central limit theorem for nonlinear Hawkes processes. J. Appl. Probab. 50 760–771. MR3102513 [29] ZHU, L. (2013). Moderate deviations for Hawkes processes. Statist. Probab. Lett. 83 885–890. MR3040318 The Annals of Applied Probability 2015, Vol. 25, No. 2, 582–599 DOI: 10.1214/14-AAP1004 © Institute of Mathematical Statistics, 2015

A GENERALIZED BACKWARD SCHEME FOR SOLVING NONMONOTONIC STOCHASTIC RECURSIONS

BY P. M OYAL Université de Technologie de Compiègne

We propose an explicit construction of a stationary solution for a stochas- tic recursion of the form X ◦ θ = ϕ(X) on a partially-ordered Polish space, when the monotonicity of ϕ is not assumed. Under certain conditions, we show that an extension of the original probability space exists, on which a solution is well defined, and construct explicitly this extension using a ran- domized contraction technique. We then provide conditions for the existence of a solution on the original space. We finally apply these results to the sta- bility study of two nonmonotonic queuing systems.

REFERENCES

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LIMIT THEOREMS FOR NEARLY UNSTABLE HAWKES PROCESSES

BY THIBAULT JAISSON AND MATHIEU ROSENBAUM École Polytechnique Paris and Université Pierre et Marie Curie (Paris 6)

Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high- frequency finance. However, in practice, the statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properly. By nearly unstable, we mean that the L1 norm of their kernel is close to unity. We study in this work such processes for which the stability condition is almost violated. Our main result states that after suitable rescaling, they asymptotically behave like integrated Cox–Ingersoll– Ross models. Thus, modeling financial order flows as nearly unstable Hawkes processes may be a good way to reproduce both their high and low frequency stylized facts. We then extend this result to the Hawkes-based price model in- troduced by Bacry et al. [Quant. Finance 13 (2013) 65–77]. We show that un- der a similar criticality condition, this process converges to a Heston model. Again, we recover well-known stylized facts of prices, both at the microstruc- ture level and at the macroscopic scale.

REFERENCES

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SPATIAL PREFERENTIAL ATTACHMENT NETWORKS: POWER LAWS AND CLUSTERING COEFFICIENTS1

BY EMMANUEL JACOB AND PETER MÖRTERS ENS Lyon and University of Bath

We define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mech- anism favoring short distances and high degrees. The competition of prefer- ential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limit law, which can be a power law with any exponent τ>2. The average clustering coefficient of the networks converges to a positive limit. Finally, a phase transition oc- curs in the global clustering coefficients and empirical distribution of edge lengths when the power-law exponent crosses the critical value τ = 3. Our main tool in the proof of these results is a general weak law of large numbers in the spirit of Penrose and Yukich.

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MSC2010 subject classifications. Primary 05C80; secondary 60C05, 90B15. Key words and phrases. Scale-free network, Barabási–Albert model, preferential attachment, dy- namical random graph, geometric random graph, power law, degree distribution, edge length distri- bution, clustering coefficient. [11] FLAXMAN,A.D.,FRIEZE,A.M.andVERA, J. (2006). A geometric preferential attachment model of networks. Internet Math. 3 187–205. MR2321829 [12] FLAXMAN,A.D.,FRIEZE,A.M.andVERA, J. (2007). A geometric preferential attachment model of networks. II. In Algorithms and Models for the Web-graph. Lecture Notes in Computer Science 4863 41–55. Springer, Berlin. MR2504906 [13] JANSSEN,J.,PRAŁAT,P.andWILSON, R. (2013). Geometric graph properties of the spatial preferred attachment model. Adv. in Appl. Math. 50 243–267. MR3003346 [14] JORDAN, J. (2010). Degree sequences of geometric preferential attachment graphs. Adv. in Appl. Probab. 42 319–330. MR2675104 [15] JORDAN, J. (2013). Geometric preferential attachment in nonuniform metric spaces. Electron. J. Probab. 18 1–15. MR3024102 [16] JORDAN,J.andWADE, A. (2013). Phase transitions for random geometric preferential attach- ment graphs. Preprint. Available at arXiv:1311.3776. [17] PENROSE, M. (2003). Random Geometric Graphs. Oxford Studies in Probability 5. Oxford Univ. Press, Oxford. MR1986198 [18] PENROSE,M.D.andYUKICH, J. E. (2003). Weak laws of large numbers in geometric prob- ability. Ann. Appl. Probab. 13 277–303. MR1952000 [19] RUDAS,A.,TÓTH,B.andVALKÓ, B. (2007). Random trees and general branching processes. Random Structures Algorithms 31 186–202. MR2343718 The Annals of Applied Probability 2015, Vol. 25, No. 2, 663–674 DOI: 10.1214/14-AAP1007 © Institute of Mathematical Statistics, 2015

ON THE EXPECTED TOTAL NUMBER OF INFECTIONS FOR VIRUS SPREAD ON A FINITE NETWORK

BY ANTAR BANDYOPADHYAY AND FARKHONDEH SAJADI Indian Statistical Institute, Delhi Centre In this work we consider a simple SIR infection spread model on a finite population of n agents represented by a finite graph G. Starting with a fixed set of initial infected vertices the infection spreads in discrete time steps, where each infected vertex tries to infect its neighbors with a fixed proba- bility β ∈ (0, 1), independently of others. It is assumed that each infected vertex dies out after an unit time and the process continues till all infected ver- tices die out. This model was first studied by [Ann. Appl. Probab. 18 (2008) 359–378]. In this work we find a simple lower bound on the expected number of ever infected vertices using breath-first search algorithm and show that it asymptotically performs better for a fairly large class of graphs than the upper bounds obtained in [Ann. Appl. Probab. 18 (2008) 359–378]. As a by prod- uct we also derive the asymptotic value of the expected number of the ever infected vertices when the underlying graph is the random r-regular graph 1 and β

REFERENCES

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MSC2010 subject classifications. Primary 60K35, 05C80; secondary 60J85, 90B15. Key words and phrases. Breadth-first search, local weak convergence, percolation on finite graphs, random r-regular graphs, susceptible infected removed model, virus spread. [10] KOZMA,G.andNACHMIAS, A. (2011). A note about critical percolation on finite graphs. J. Theoret. Probab. 24 1087–1096. MR2851246 [11] WÄSTLUND, J. (2012). Replica symmetry of the minimum matching. Ann. of Math.(2)175 1061–1091. MR2912702 The Annals of Applied Probability 2015, Vol. 25, No. 2, 675–713 DOI: 10.1214/14-AAP1008 © Institute of Mathematical Statistics, 2015

LIMIT THEOREMS FOR SMOLUCHOWSKI DYNAMICS ASSOCIATED WITH CRITICAL CONTINUOUS-STATE BRANCHING PROCESSES

BY GAUTAM IYER1,3,NICHOLAS LEGER3 AND ROBERT L. PEGO2,3 Carnegie Mellon University

We investigate the well-posedness and asymptotic self-similarity of so- lutions to a generalized Smoluchowski coagulation equation recently intro- duced by Bertoin and Le Gall in the context of continuous-state branching theory. In particular, this equation governs the evolution of the Lévy mea- sure of a critical continuous-state branching process which becomes extinct (i.e., is absorbed at zero) almost surely. We show that a nondegenerate scaling limit of the Lévy measure (and the process) exists if and only if the branch- ing mechanism is regularly varying at 0. When the branching mechanism is regularly varying, we characterize nondegenerate scaling limits of arbitrary finite-measure solutions in terms of generalized Mittag–Leffler series.

REFERENCES

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MSC2010 subject classifications. Primary 60J80; secondary 60G18, 35Q70, 82C28. Key words and phrases. Continuous-state branching process, critical branching, limit theorem, scaling limit, Smoluchowski equation, coagulation, self-similar solution, Mittag–Leffler series, reg- ular variation, Bernstein function. [10] JACOD,J.andSHIRYAEV, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin. MR1943877 [11] KYPRIANOU, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Ap- plications. Springer, Berlin. MR2250061 [12] KYPRIANOU,A.E.andPARDO, J. C. (2008). Continuous-state branching processes and self- similarity. J. Appl. Probab. 45 1140–1160. MR2484167 [13] LAMBERT, A. (2007). Quasi-stationary distributions and the continuous-state branching pro- cess conditioned to be never extinct. Electron. J. Probab. 12 420–446. MR2299923 [14] LEYVRAZ, F. (2003). Scaling theory and exactly solved models in the kinetics of irreversible aggregation. Phys. Reports 383 95–212. [15] LI, Z.-H. (2000). Asymptotic behaviour of continuous time and state branching processes. J. Austral. Math. Soc. Ser. A 68 68–84. MR1727226 [16] MENON,G.andPEGO, R. L. (2004). Approach to self-similarity in Smoluchowski’s coagula- tion equations. Comm. Pure Appl. Math. 57 1197–1232. MR2059679 [17] MENON,G.andPEGO, R. L. (2007). Universality classes in Burgers turbulence. Comm. Math. Phys. 273 177–202. MR2308754 [18] MENON,G.andPEGO, R. L. (2008). The scaling attractor and ultimate dynamics for Smolu- chowski’s coagulation equations. J. Nonlinear Sci. 18 143–190. MR2386718 [19] NORRIS, J. R. (1999). Smoluchowski’s coagulation equation: Uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent. Ann. Appl. Probab. 9 78–109. MR1682596 [20] PAKES, A. G. (2010). Critical Markov branching process limit theorems allowing infinite vari- ance. Adv. in Appl. Probab. 42 460–488. MR2675112 [21] PRABHAKAR, T. R. (1971). A singular integral equation with a generalized Mittag–Leffler function in the kernel. Yokohama Math. J. 19 7–15. MR0293349 [22] SCHILLING,R.L.,SONG,R.andVONDRACEKˇ , Z. (2010). Bernstein Functions: Theory and Applications. de Gruyter Studies in Mathematics 37. de Gruyter, Berlin. MR2598208 [23] SLACK, R. S. (1972). Further notes on branching processes with mean 1. Z. Wahrsch. Verw. Gebiete 25 31–38. MR0331539 The Annals of Applied Probability 2015, Vol. 25, No. 2, 714–752 DOI: 10.1214/14-AAP1009 © Institute of Mathematical Statistics, 2015

EXPONENTIAL MOMENTS OF AFFINE PROCESSES

BY MARTIN KELLER-RESSEL AND EBERHARD MAYERHOFER1 TU Berlin and Dublin City University

We investigate the maximal domain of the moment generating function of affine processes in the sense of Duffie, Filipovic´ and Schachermayer [Ann. Appl. Probab. 13 (2003) 984–1053], and we show the validity of the affine transform formula that connects exponential moments with the solution of a generalized Riccati differential equation. Our result extends and unifies those preceding it (e.g., Glasserman and Kim [Math. Finance 20 (2010) 1–33], Fil- ipovic´ and Mayerhofer [Radon Ser. Comput. Appl. Math. 8 (2009) 1–40] and Kallsen and Muhle-Karbe [Stochastic Process Appl. 120 (2010) 163–181]) in that it allows processes with very general jump behavior, applies to any convex state space and provides both sufficient and necessary conditions for finiteness of exponential moments.

REFERENCES

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MSC2010 subject classifications. Primary 60J25; secondary 91B28. Key words and phrases. Affine process, exponential moment, Riccati equation, financial model- ing. DUFFIE,D.,FILIPOVIC´ ,D.andSCHACHERMAYER, W. (2003). Affine processes and applications in finance. Ann. Appl. Probab. 13 984–1053. MR1994043 DUFFIE,D.andKAN, R. (1996). A yield-factor model of interest rates. Math. Finance 6 379–406. FILIPOVIC´ , D. (2009). Term-Structure Models: A Graduate Course. Springer, Berlin. MR2553163 FILIPOVIC´ ,D.andMAYERHOFER, E. (2009). Affine diffusion processes: Theory and applications. In Advanced Financial Modelling (H. Albrecher, W. J. Runggaldier and W. Schachermayer, eds.). Radon Ser. Comput. Appl. Math. 8 125–164. Berlin: de Gruyter. MR2648460 GLASSERMAN,P.andKIM, K.-K. (2010). Moment explosions and stationary distributions in affine diffusion models. Math. Finance 20 1–33. MR2599675 HESTON, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6 327–343. KALLSEN,J.andMUHLE-KARBE, J. (2010). Exponentially affine martingales, affine measure changes and exponential moments of affine processes. Stochastic Process. Appl. 120 163–181. MR2576885 KELLER-RESSEL, M. (2009). Affine processes—Theory and applications in mathematical finance. Ph.D. thesis, Vienna Univ. Technology. KELLER-RESSEL, M. (2011). Moment explosions and long-term behavior of affine stochastic volatility models. Math. Finance 21 73–98. MR2779872 KELLER-RESSEL,M.,SCHACHERMAYER,W.andTEICHMANN, J. (2011). Affine processes are regular. Probab. Theory Related Fields 151 591–611. MR2851694 KELLER-RESSEL,M.,SCHACHERMAYER,W.andTEICHMANN, J. (2013). Regularity of affine processes on general state spaces. Electron. J. Probab. 18 1–17. MR3040553 LANDO, D. (1998). On Cox processes and credit risky securities. Review of Derivatives Research 2 99–120. LEE, R. W. (2004). The moment formula for implied volatility at extreme strikes. Math. Finance 14 469–480. MR2070174 LEIPPOLD,M.andTROJANI, F. (2010). Asset pricing with matrix jump diffusions. FINRISK Work- ing paper series, January 2010. MAYERHOFER, E. (2012). Affine processes on positive semidefinite d × d matrices have jumps of finite variation in dimension d>1. Stochastic Process. Appl. 122 3445–3459. MR2956112 MAYERHOFER,E.,MUHLE-KARBE,J.andSMIRNOV, A. G. (2011). A characterization of the martingale property of exponentially affine processes. Stochastic Process. Appl. 121 568–582. MR2763096 MUHLE-KARBE, J. (2009). On utility-based investment, pricing and hedging in incomplete markets. Ph.D. thesis, TU München. NICOLATO,E.andVENARDOS, E. (2003). Option pricing in stochastic volatility models of the Ornstein–Uhlenbeck type. Math. Finance 13 445–466. MR2003131 SATO,K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge. MR1739520 SCHNEIDER,P.,SÖGNER,L.andVEZA, T. (2010). The economic role of jumps and recovery rates in the market for corporate default risk. Journal of Financial and Quantitative Analysis 45 1517– 1547. SPREIJ,P.andVEERMAN, E. (2010). The affine transform formula for affine jump-diffusions with general closed convex state spaces. Available at arXiv:1005.1099. VOLKMANN, P. (1973). Über die Invarianz konvexer Mengen und Differentialungleichungen in einem normierten Raume. Math. Ann. 203 201–210. MR0322305 WU, L. (2011). Variance dynamics: Joint evidence from options and high-frequency returns. J. Econometrics 160 280–287. MR2745884 The Annals of Applied Probability 2015, Vol. 25, No. 2, 753–822 DOI: 10.1214/14-AAP1010 © Institute of Mathematical Statistics, 2015

UNIVERSALITY IN POLYTOPE PHASE TRANSITIONS AND MESSAGE PASSING ALGORITHMS

BY MOHSEN BAYATI,MARC LELARGE1 AND ANDREA MONTANARI2 Stanford University, INRIA and ENS, and Stanford University

N We consider a class of nonlinear mappings FA,N in R indexed by sym- × metric random matrices A ∈ RN N with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Bolthausen [Comm. Math. Phys. 325 (2014) 333–366]. Within information theory, they are known as “approximate mes- sage passing” algorithms. We study the high-dimensional (large N) behavior of the iterates of F for polynomial functions F, and prove that it is universal; that is, it depends only on the first two moments of the entries of A, under a sub-Gaussian tail condition. As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves, for a broad class of random projections, a conjecture by David Donoho and Jared Tanner.

REFERENCES

[1] ADAMCZAK,R.,LITVAK,A.E.,PAJOR,A.andTOMCZAK-JAEGERMANN, N. (2011). Re- stricted isometry property of matrices with independent columns and neighborly poly- topes by random sampling. Constr. Approx. 34 61–88. MR2796091 [2] AFFENTRANGER,F.andSCHNEIDER, R. (1992). Random projections of regular simplices. Discrete Comput. Geom. 7 219–226. MR1149653 [3] ANDERSON,G.W.,GUIONNET,A.andZEITOUNI, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge. MR2760897 [4] BAI,Z.andSILVERSTEIN, J. W. (2009). Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. Springer, New York. MR2567175 [5] BAI,Z.D.andSILVERSTEIN, J. W. (1998). No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Probab. 26 316–345. MR1617051 [6] BAYATI,M.andMONTANARI, A. (2012). The LASSO risk for Gaussian matrices. IEEE Trans. Inform. Theory 58 1997–2017. MR2951312 [7] BOLTHAUSEN, E. (2014). An iterative construction of solutions of the TAP equations for the Sherrington–Kirkpatrick model. Comm. Math. Phys. 325 333–366. MR3147441 [8] BÜRGISSER,P.andCUCKER, F. (2010). Smoothed analysis of Moore–Penrose inversion. SIAM J. Matrix Anal. Appl. 31 2769–2783. MR2740632

MSC2010 subject classifications. Primary 60F05; secondary 68W40. Key words and phrases. Universality, random matrices, message passing, compressed sensing, polytope neighborliness. [9] DONOHO,D.andTANNER, J. (2009). Observed universality of phase transitions in high- dimensional geometry, with implications for modern data analysis and signal processing. Philos. Trans. R. Soc. Lond. Ser. AMath. Phys. Eng. Sci. 367 4273–4293. MR2546388 [10] DONOHO, D. L. (2005). Neighborly polytopes and sparse solution of underdetermined linear equations. Technical report, Statistics Dept., Stanford Univ., Stanford, CA. [11] DONOHO, D. L. (2006). High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension. Discrete Comput. Geom. 35 617–652. MR2225676 [12] DONOHO,D.L.,JAVANMARD,A.andMONTANARI, A. (2013). Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing. IEEE Trans. Inform. Theory 59 7434–7464. MR3124654 [13] DONOHO,D.L.,JOHNSTONE,I.andMONTANARI, A. (2013). Accurate prediction of phase transitions in compressed sensing via a connection to minimax denoising. IEEE Trans. Inform. Theory 59 3396–3433. MR3061255 [14] DONOHO,D.L.,MALEKI,A.andMONTANARI, A. (2009). Message passing algorithms for compressed sensing. Proc. Natl. Acad. Sci. USA 106 18914–18919. [15] DONOHO,D.L.,MALEKI,A.andMONTANARI, A. (2011). The noise-sensitivity phase tran- sition in compressed sensing. IEEE Trans. Inform. Theory 57 6920–6941. MR2882271 [16] DONOHO,D.L.andTANNER, J. (2005). Neighborliness of randomly projected simplices in high dimensions. Proc. Natl. Acad. Sci. USA 102 9452–9457 (electronic). MR2168716 [17] DONOHO,D.L.andTANNER, J. (2005). Sparse nonnegative solution of underdetermined lin- ear equations by linear programming. Proc. Natl. Acad. Sci. USA 102 9446–9451 (elec- tronic). MR2168715 [18] DONOHO,D.L.andTANNER, J. (2009). Counting faces of randomly projected poly- topes when the projection radically lowers dimension. J. Amer. Math. Soc. 22 1–53. MR2449053 [19] JAVANMARD,A.andMONTANARI, A. (2013). State evolution for general approximate mes- sage passing algorithms, with applications to spatial coupling. Information and Inference 2 115–144. [20] KABASHIMA,Y.,WADAYAMA,T.andTANAKA, T. (2009). A typical reconstruction limit for compressed sensing based on lp-norm minimization. J. Stat. Mech. L09003. [21] KRZAKALA,F.,MÉZARD,M.,SAUSSET,F.,SUN,Y.F.andZDEBOROVÁ, L. (2012). Statistical-physics-based reconstruction in compressed sensing. Physical Review X 2 021005. [22] LUBINSKY, D. S. (2007). A survey of weighted polynomial approximation with exponential weights. Surv. Approx. Theory 3 1–105. MR2276420 [23] MALEKI,A.,ANITORI,L.,YANG,A.andBARANIUK, R. (2011). Asymptotic analysis of complex LASSO via complex approximate message passing (CAMP). Available at arXiv:1108.0477. [24] MÉZARD,M.,PARISI,G.andVIRASORO, M. A. (1987). Spin Glass Theory and Beyond. World Scientific Lecture Notes in Physics 9. World Scientific, Teaneck, NJ. MR1026102 [25] RANGAN, S. (2011). Generalized approximate message passing for estimation with random linear mixing. In IEEE Intl. Symp. on Inform. Theory (St. Perersbourg), August 2011. IEEE, Piscataway, NJ. [26] RANGAN,S.,FLETCHER,A.K.andGOYAL, V. K. (2009). Asymptotic analysis of map es- timation via the replica method and applications to compressed sensing. In Neural Infor- mation Processing Systems (NIPS). Vancouver. [27] SCHNITER, P. (2010). Turbo reconstruction of structured sparse signals. In Proceedings of the Conference on Information Sciences and Systems. Princeton, NJ. [28] SCHNITER,P.andRANGAN, S. (2012). Compressive phase retrieval via generalized approxi- mate message passing. In Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on 815–822. IEEE, Piscataway, NJ. [29] TAO,T.andVU, V. (2012). Random matrices: The Universality phenomenon for Wigner en- sembles. Available at arXiv:1202.0068. [30] THOULESS,D.J.,ANDERSON,P.W.andPALMER, R. G. (1977). Solution of “solvable model of a spin glass.” Philosophical Magazine 35 593–601. [31] VERSHIK,A.M.andSPORYSHEV, P. V. (1992). Asymptotic behavior of the number of faces of random polyhedra and the neighborliness problem. Selecta Math. Soviet. 11 181–201. Selected translations. MR1166627 The Annals of Applied Probability 2015, Vol. 25, No. 2, 823–859 DOI: 10.1214/14-AAP1011 © Institute of Mathematical Statistics, 2015

ARBITRAGE AND DUALITY IN NONDOMINATED DISCRETE-TIME MODELS

BY BRUNO BOUCHARD1 AND MARCEL NUTZ2 CEREMADE, Université Paris Dauphine and CREST-ENSAE and Columbia University

We consider a nondominated model of a discrete-time financial market where stocks are traded dynamically, and options are available for static hedg- ing. In a general measure-theoretic setting, we show that absence of arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of martingale measures. In the arbitrage-free case, we show that optimal super- hedging strategies exist for general contingent claims, and that the minimal superhedging price is given by the supremum over the martingale measures. Moreover, we obtain a nondominated version of the Optional Decomposition Theorem.

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FINITENESS OF ENTROPY FOR THE HOMOGENEOUS BOLTZMANN EQUATION WITH MEASURE INITIAL CONDITION

BY NICOLAS FOURNIER Université Pierre et Marie Curie We consider the 3D spatially homogeneous Boltzmann equation for (true) hard and moderately soft potentials. We assume that the initial con- dition is a probability measure with finite energy and is not a Dirac mass. For hard potentials, we prove that any reasonable weak solution immediately belongs to some Besov space. For moderately soft potentials, we assume ad- ditionally that the initial condition has a moment of sufficiently high order (8 is enough) and prove the existence of a solution that immediately belongs to some Besov space. The considered solutions thus instantaneously become functions with a finite entropy. We also prove that in any case, any weak so- lution is immediately supported by R3.

REFERENCES

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GIBBS MEASURES ON PERMUTATIONS OVER ONE-DIMENSIONAL DISCRETE POINT SETS1

BY MAREK BISKUP AND THOMAS RICHTHAMMER UCLA and Universität Hildesheim We consider Gibbs distributions on permutations of a locally finite in- ⊂ R finite set X , where a permutation σ of X is assigned (formal) energy x∈X V(σ(x)− x). This is motivated by Feynman’s path representation of the quantum Bose gas; the choice X := Z and V(x):= αx2 is of principal in- terest. Under suitable regularity conditions on the set X and the potential V , we establish existence and a full classification of the infinite-volume Gibbs measures for this problem, including a result on the number of infinite cycles of typical permutations. Unlike earlier results, our conclusions are not limited to small densities and/or high temperatures.

REFERENCES

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LARGE DEVIATIONS FOR CLUSTER SIZE DISTRIBUTIONS IN A CONTINUOUS CLASSICAL MANY-BODY SYSTEM1

BY SABINE JANSEN,WOLFGANG KÖNIG AND BERND METZGER Ruhr-Universität Bochum, WIAS Berlin and TU Berlin, and WIAS Berlin An interesting problem in statistical physics is the condensation of clas- sical particles in droplets or clusters when the pair-interaction is given by a stable Lennard–Jones-type potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribu- tion for any inverse temperature β ∈ (0, ∞) and particle density ρ ∈ (0,ρcp) in the thermodynamic limit. Here ρcp > 0 is the close packing density. While in general the rate function is an abstract object, our second main result is the -convergence of the rate function toward an explicit limiting rate function − in the low-temperature dilute limit β →∞, ρ ↓ 0 such that −β 1 log ρ → ν for some ν ∈ (0, ∞). The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the decoupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter ν. Under additional assumptions on the potential, the -convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle.

REFERENCES

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APPROXIMATION ALGORITHMS FOR THE NORMALIZING CONSTANT OF GIBBS DISTRIBUTIONS

BY MARK HUBER Claremont McKenna College

Consider a family of distributions {πβ } where X ∼ πβ means that P(X = = − x) exp( βH (x))/Z(β).HereZ(β) is the proper normalizing constant, equal to x exp(−βH (x)).Then{πβ } is known as a Gibbs distribution, and Z(β) is the partition function. This work presents a new method for approx- imating the partition function to a specified level of relative accuracy using only a number of samples, that is, O(ln(Z(β)) ln(ln(Z(β)))) when Z(0) ≥ 1. This is a sharp improvement over previous, similar approaches that used a much more complicated algorithm, requiring O(ln(Z(β)) ln(ln(Z(β)))5) samples.

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A DECREASING STEP METHOD FOR STRONGLY OSCILLATING STOCHASTIC MODELS

BY CAMILO ANDRÉS GARCÍA TRILLOS Université Nice Sophia Antipolis

We propose an algorithm for approximating the solution of a strongly os- cillating SDE, that is, a system in which some ergodic state variables evolve quickly with respect to the other variables. The algorithm profits from ho- mogenization results and consists of an Euler scheme for the slow scale vari- ables coupled with a decreasing step estimator for the ergodic averages of the quick variables. We prove the strong convergence of the algorithm as well as a C.L.T. like limit result for the normalized error distribution. In addition, we propose an extrapolated version that has an asymptotically lower complexity and satisfies the same properties as the original version.

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CONVERGENCE PROPERTIES OF PSEUDO-MARGINAL MARKOV CHAIN MONTE CARLO ALGORITHMS

BY CHRISTOPHE ANDRIEU1 AND MATTI VIHOLA2 University of Bristol and University of Jyväskylä

We study convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms (Andrieu and Roberts [Ann. Statist. 37 (2009) 697– 725]). We find that the asymptotic variance of the pseudo-marginal algorithm is always at least as large as that of the marginal algorithm. We show that if the marginal chain admits a (right) spectral gap and the weights (normalised esti- mates of the target density) are uniformly bounded, then the pseudo-marginal chain has a spectral gap. In many cases, a similar result holds for the absolute spectral gap, which is equivalent to geometric ergodicity. We consider also unbounded weight distributions and recover polynomial convergence rates in more specific cases, when the marginal algorithm is uniformly ergodic or an independent Metropolis–Hastings or a random-walk Metropolis targeting a super-exponential density with regular contours. Our results on geomet- ric and polynomial convergence rates imply central limit theorems. We also prove that under general conditions, the asymptotic variance of the pseudo- marginal algorithm converges to the asymptotic variance of the marginal al- gorithm if the accuracy of the estimators is increased.

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