## Research Themes

# Publications

## Books

[L6] Probabilistic models of population evolution. Scaling limits and interactions. Springer, 2016.

[L5] Stochastic Differential Equations, Backward SDEs, Partial Differential Equations (with A. Rascanu), Stochastic Modelling and Applied Probability 69, Springer 2014.

[L4] Markov processes and applications. Algorithms, networks, genome and finance. Translated and revised from the 2007 French edition. Wiley Series in Probability and Statistics. John Wiley & Sons, Ltd., Chichester; Dunod, Paris, 2008.

[L3] Processus de Markov et applications. Algorithmes, réseaux, génome et finance, Dunod, Paris, 2007.

[L2] Méthodes de Monte-Carlo pour les équations de transport et de diffusion (with Lapeyre, Bernard; Sentis, Rémi), Mathématiques & Applications (Berlin) [Mathematics & Applications] ,29. Springer-Verlag, Berlin, 1998. x+176 pp. ISBN: 3-540-63393-6 Engl. translation by Alan Craig and Fionn Craig: Introduction to Monte-Carlo methods for transport and diffusion equations. Oxford Texts in Applied and Engineering Mathematics, 6, Oxford University Press, Oxford, 2003. x+163 pp. ISBN: 0-19-852593-1

[L1] Méthodes Probabilistes pour les Equations de la Physique (with M. Cessenat, G. Ledanois, P.L. Lions and R. Sentis, R. Dautray ed.), Eyrolles, 1989.

## Articles

[151] Functional central limit theorems for epidemic models with varying infectivity, with G. Pang, submitted.

[150] A Spatial Stochastic Epidemic Model: Law of Large Numbers and Central Limit Theorem, with S. Bowong et A. Emakoua, submitted.

[149] Estimating the state of the Covid-19 epidemic in France using a non-Markovian model, with R. Forien and G. Pang, submitted.

[148] Epidemic models with varying infectivity, with R. Forien and G. Pang, submitted.

[147] Multi-patch epidemic models with general infectious periods, with G. Pang, submitted.

[146] Functional limit theorems for non-Markovian epidemic models, with G. Pang, submitted.

[145] Approximation of the height process of a continuous state branching process with interaction, with I. Dramé, *Theor. Probability and Math. Statist.*, to appear.

[144] Fluctuations around a homogenized semilinear random PDE, with M. Hairer, *Archive for Rational Mechanics and Analysis*, to appear.

[143] Household epidemic models and McKean-Vlasov Poisson driven SDEs, with R. Forien, submitted.

[142] Moderate Deviations and Extinction of an Epidemic, *Electron. J. Probab.* **25**, paper no. 25, 1-27, 2020.

[141] Deviations from the law of large numbers, and extinction of an endemic disease, in *Mathematical modeling of random and deterministic phenomena*, Manou–Abi, S.M., Dabo–Niang, S., Salone, J.J. eds., pp. 3–30, ISTE and Wiley, 2020.

[140] The height process of a continuous state branching process with interaction, with Z. Li and A. Wakolbinger, * J Theor Probab * (2020). https://doi.org/10.1007/s10959-020-01054-5

[139] Extinction time and the total mass of the continuous state branching processes with competition, with Vi Le, *Stochastics* **92**, pp. 852-875, 2020.

[138] A SIR model on a refining spatial grid I - Law of Large Numbers, with Modeste N'zi and Ténan Yeo, *Applied Math. & Optimization*, to appear.

[137] Stochastic epidemics in a homogeneous community, with Tom Britton, Part I of *Stochastic Epidemic Models with Inference*, T. Britton and E. Pardoux eds., Lecture Notes in Math. **2255**, pp. 1–120, Springer 2019.

[136] Large deviation of the exit measure through a characteristic boundary for a Poisson driven SDE, with Brice Samegni-Kepgnou, *ESAIM P & S* **24**, pp. 148-185, 2020.

[135] Approximation of a generalized CSBP with interaction, with Ibrahima Dramé, *Electron. Commun. Probab.*, **23** , 73, 1-14, 2018.

[134] Large Deviation Principle for Reflected Poisson driven SDEs in Epidemic Models, with Brice Samegni-Kepgnou, *Stochastic Analysis and Applications* **37**, pp. 836-864, 2019.

[133] Small jumps asymptotic of the moving optimum Poissonian SDE, with Elma Nassar, *Stochastic Processes and Applications* **129**, pp. 2320-2340, 2019.

[132] Phenotypic lag and population extinction in the moving-optimum model: insights from a small-jumps limit, with Michael Kopp and Elma Nassar, *J. Math. Biol.*, **77**, pp. 1431-1458, 2018.

[131] Large deviations for infectious diseases models, with Peter Kratz, *Séminaire de Probabilités* **XLIX**, C. Donati-Martin, A. Lejay, A. Rouault eds., *Lecture Notes in Math.* **2215**, pp. 221-327, 2018.

[130] Nonlinear filtering with degenerate noise, with David Jaurès Fotsa-Mbogne, *Electron. Commun. Probab.* **22** , Paper No. 44, 14 pp., 2017

[129] Large Deviation principle for Epidemic Models, with Brice Samegni-Kepgnou, *Journal Appli. Prob.***54**, 905-920, 2017.

[128] Continuity of the Feynman-Kac formula for a generalized parabolic equation, with A. Rascanu, *Stochastics*, **89** ,726-752, 2017.

[127] On the long time behavior of the solution of an SDE driven by a Poisson Point Process, with E. Nassar, *Adv. in Appl. Probab.* **49**, 344-367, 2017.

[126] TMEM187-IRAK1 Polymorphisms Associated with Rheumatoid Arthritis Susceptibility in Tunisian and French Female Populations: Influence of Geographic Origin, with O. Khalifa, N. Balandrauid, N. Lambert, I. Auger, J. Roudier, A. Sénéchal, D. Geneviêve, C. Picard, G. Lefranc, I. Touitou, B. M'Madi Mrenda, C. Benedito, A.-L. Gagez, Y.-M. Pers, C. Jorgensen, T. Mahjoub et F. Apparailly, *Journal of Immunology Research*, **2017**, Article ID 4915950, 12 pages, 2017.

[125] Non-binary branching process and non-Markovian exploration process, with I. Dramé and A.B. Sow, *ESAIM P & S* , **21**, 1-33, 2017

[124] Averaging for SDE-BSDE with null recurrent fast component Application to homogenization in a non periodic media, with Khaled Bahlali and A. Elouaflin, *Stochastic Processes and Applications*, **127**, 1321-1353, 2017.

[123] Stochastic variational inequalities on non-convex domains, with R. Buckdahn, L. Maticiuc and A. Rascanu, *J. Differential Equations*, **259**, 7332-7374, 2015.

[122] A mixing tree-valued process arising under neutral evolution with recombination, with A. Depperschmidt, P. Pfaffelhuber, *Electron. J. Probab.*, **20** no. 94, 1-22, 2015.

[121] A Wong-Zakai theorem for stochastic PDEs, with M. Hairer, *J. Math. Soc. Japan* **67**, 1-54, 2015.

[120] A path-valued Markov process indexed by the ancestral mass, with A. Wakolbinger, *ALEA Lat. Am. J. Probab. Math. Stat.*, **12**, 193-212, 2015.

[119] Height and the total mass of the forest of genealogical trees of a large population with general competition, with Vi Le, *ESAIM : Probability and Statistics*, **19**, 172-193, 2015.

[118] Branching processes with competition and generalized Ray Knight Theorem, with M. Ba, *Annales de l'Institut Henri Poincaré - Probabilités et Statistiques* **51**, 1290-1313, 2015.

[117] Lambda-lookdown model with selection, with B. Bah, *Stochastic Processes and Applications*, **125**, 1089-1126, 2015.

[116] Numerical methods in the context of compartmental models in epidemiology, with P. Kratz and B. Samegni Kepnou, *ESAIM : Proceedings and Surveys* **48**, 169-189, 2014.

[115] Invariant measure selection by noise. An example, with J. Mattingly, *Discrete Contin. Dyn. Syst.* **34**, no. 10, 4223-4257, 2014.

[114] HLA-DRB1 genotypes and the risk of developing anti citrullinated protein antibody (ACPA) positive rheumatoid arthritis, with Nathalie Balandraud, Christophe Picard, Denis Reviron, Cyril Landais, Eric Toussirot, Nathalie Lambert, Emmanuel Telle, Caroline Charpin, Daniel Wendling, Isabelle Auger, Jean Roudier, *PLOS ONE* **8**, N 5, May 201

[113] Random homogenisation of a highly oscillatory singular potential, with M. Hairer and A. Piatnitski, *Stoch. PDE : Anal. Comp.* **1**, 571-605, 2013.

[112] Muller's ratchet clicks in finite time, with Julien Audiffren, *Stochastic Processes and Applications* **123**, 2370 - 2397, 2013. .

[111] Trees under attack: a Ray-Knight representation of Feller's branching diffusion with logistic growth, with V. Le, A. Wakolbinger, *Probab. Theory Relat. Fields* **155**, 583-619, 2013.

[110] The effect of competition on the height and length of the forest of genealogical trees of a large population, with Mamadou Ba, in *Malliavin Calculus and Related Topics, a Festschrift in Honor of David Nualart*, F. Viens, J. Feng, Y. Hu, E. Nualart Eds., Springer Proceedings in Mathematics and Statistics **34**, 445-467, 2012.

[109] A look-down model with selection, with B. Bah, A. B. Sow, in *Stochastic Analysis and Related Topics*, Laurent Decreusefond et Jamal Najim Ed, Springer Proceedings in Mathematics and Statistics Vol **22**, 1-28, 2012.

[108] Homogenization of a singular random on dimensional parabolic PDE with time varying coefficients, with Piatnitski, Andrey, *Annals of Probab.* **40**, pp. 1316-1356, 2012.

[107] Binary trees, exploration processes, and an extended Ray-Knight Theorem, with Ba, Mamadou and Sow, Ahmadou, Bamba, *Journal of Applied Probability* **49**, 210-225, 2012.

[106] From Brownian motion with a local time drift to Feller's branching diffusion with logistic growth, with A. Wakolbinger, *Electronic. Comm. Probab.* **16**, 720-731, 2011.

[105] Homogenization of a periodic degenerate semilinear elliptic PDE, with A. B. Sow, *Stochastics and Dynamics* **11**, 475-493, 2011.

[104] Compound Poisson approximation and testing for gene clusters with multigene families, with S. Grusea, O. Chabrol, P. Pontarotti, *J. Comput. Biol.* **18** , 579-594, 2011.

[103] From exploration paths to mass excursions - variations on a theme of Ray and Knight, with Wakolbinger, Anton, in *Surveys in Stochastic Processes*, Proc. 33rd SPA Conf. Berlin, 2009, J. Blath, P. Imkeller, S. Roelly eds. EMS Series of Congress reports, 87-106, 2011.

[102] Survival of a single mutant in one dimension, with Andjel, Enrique and Miller, Judith, *Electronic J. Probab* **15**, 2010.

[101] A weak convergence theorem for particle motion in a stochastic field, with Elskens, Yves, Annals of Applied Probab., 20, 2022-2039, 2010.

[100] Evolution of the ancestral recombination graph along the genome in case of a selective sweep, with Léocard, Stéphanie, *J. Math. Biology* **61**, 819-841, 2010.

[99] Viscosity solutions for systems of parabolic variational inequalities, with Maticiuc, Luciani; Rascanu, Aurel; Zalinescu, Adrian, *Bernoulli* **16**, 258-273, 2010.

[98] On the height and length of the ancestral recombination graph, with Salamat, Majid, *J. Appl. Prob.* **46**, 669-689, 2009.

[97] A probabilistic formula for a Poisson equation with Neumann boundary conditions, with Benchérif-Madani, Abdellatif, *Stoch. Anal. Appl.* **27**, 739-746, 2009.

[96] Homogenization of periodic semilinear parabolic degenerate PDES, with A. B. Sow, R. Rhodes, *Ann. I. H. Poincaré - AN* **26**,979-998, 2009.

[95] Homogenization of semilinear PDEs with discontinuous averaged coefficients, with K. Bahlali, A. Elouaflin, *Electronic J. of Probability* **14**, 477-499, 2009.

[94] Homogenization of periodic linear degenerate PDEs, with M. Hairer, *J. Funct. Anal.* **255**, No. 9, 2462-2487, 2008.

[93] Homogenization of a singular random one-dimensional PDE, with Iftimie, Bogdan; Piatnitski, Andrey, *Ann. Inst. Henri Poincaré Probab. Stat.* **44**, no. 3, 519-543, 2008.

[92] Homogenization of a semilinear parabolic PDE with locally periodic coefficients: a probabilistic approach, with Benchérif-Madani, Abdellatif, *ESAIM Probab. Stat.* **11**, 385-411, 2007.

[91] Homogenization of periodic semilinear hypoelliptic PDEs, with Diédhiou, Alassane, *Ann. Fac. Sci. Toulouse Math.* **16**, no. 2, 253-283, 2007.

[90] Malliavin calculus for the stochastic 2D Navier-Stokes equation, with Mattingly, Jonathan C., *Comm. Pure Appl. Math.* **59** (2006), 1742-1790, 2006.

[89] Homogenization of a singular random one dimensional PDE, with Piatnitski, Andrey, in *Multi scale problems and asymptotic analysis*, 291-303, GAKUTO, Internat. Ser. Math. Sci. Appl. **24**, Gakkotosho, Tokyo, 2006.

[88] Singular homogenization with stationary in time and periodic in space coefficients, with Diop, M. A.; Iftimie, B.; Piatnitski, A. L., *J. Funct. Anal.* **231**, no. 1, 1-46, 2006.

[87] On the Poisson equation and diffusion approximation III, with Veretennikov, A. Yu., *Ann. Probab.* **33**, no. 3, 1111-1133, 2005.

[86] Homogenization of a diffusion with locally periodic coefficients, with Benchérif-Madani, Abdellatif, in *Séminaire de Probabilités* XXXVIII, 363-392, Lecture Notes in Math. **1857**, Springer, Berlin, 2005.

[85] Backward stochastic differential equations associated to a symmetric Markov process, with Bally, V.; Stoica, L., Potential Anal., 22, no. 1, 17-60, 2005.

[84] Quenched large deviations for one dimensional nonlinear filtering, with Zeitouni, Ofer,
*SIAM J. Control Optim.* **43**, no. 4, 1272-1297, 2004/05.

[83] Malliavin calculus for highly degenerate 2D stochastic Navier-Stokes equations, with Hairer, Martin; Mattingly, Jonathan C., *C. R. Math. Acad. Sci. Paris* **339**, no. 11, 793-796, 2004.

[82] Probabilistic interpretation of a system of quasilinear parabolic PDEs, with Sow, A. B.,
*Stochastics & Stoch. Rep.* **7**

[81] $L\sp p$ solutions of backward stochastic differential equations, with Briand, Ph.; Delyon, B.; Hu, Y.; Stoica, L., *Stochastic Process. Appl.* **108**, no. 1, 109-129, 2003.

[80] On Poisson equation and diffusion approximation II, with Veretennikov, A. Yu., *Ann. Probab.* **31**, no. 3, 1166-1192, 2003.

[79] Homogenization of a nonlinear random parabolic partial differential equation, with Piatnitski, A. L., *Stochastic Process. Appl.* **104**, no. 1, 1-27, 2003.

[78] On the Poisson equation and diffusion approximation I, with Veretennikov, A. Yu., *Ann. Probab.* **29**, no. 3, 1061-1085, 2001.

[77] BSDEs, convergence in law and homogenization of semilinear parabolic PDEs, with Gaudron, Guillaume, *Ann. Inst. H. Poincaré Probab. Statist.* **37**, no. 1, 1-42, 2001.

[76] On the smoothness of an invariant measure of a Markov chain with respect to a parameter, with Veretennikov, A. Yu., (in Russian) *Dokl. Akad. Nauk* **370**, no. 2, 158-160, 2000.

[75] Backward stochastic variational inequalities, with Rascanu, Aurel, *Stochastics & Stoch. Rep.* **67**, no. 3-4, 159-167, 1999.

[74] Homogenization of linear and semilinear second order parabolic PDEs with periodic coefficients: a probabilistic approach, *J. Funct. Anal.* **167**, no. 2, 498-520, 1999.

[73] The critical exponent for a stochastic PDE to hit zero, with Mueller, Carl, in *Stochastic analysis, control, optimization and applications*, 325-338, Systems Control Found. Appl., Birkhäuser Boston, Boston, MA, 1999.

[72] Forward-backward stochastic differential equations and quasilinear parabolic PDEs, with Tang, Shanjian, *Probab. Theory Related Fields* **114** , no. 2, 123-150, 1999.

[71] BSDEs, weak convergence and homogenization of semilinear PDEs, in *Nonlinear analysis, differential equations and control* (Montreal, QC, 1998), 503-549, NATO Sci. Ser. C Math. Phys. Sci., **528**, Kluwer Acad. Publ., Dordrecht, 1999.

[70] Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order, in *Stochastic analysis and related topics VI* (Geilo, 1996), 79-127, Progr. Probab., **42**, Birkhäuser Boston, Boston, MA, 1998.

[69] Malliavin calculus for white noise driven parabolic SPDEs, with Bally, Vlad, *Potential Anal.* **9**, no. 1, 27-64, 1998.

[68] Backward stochastic differential equations with subdifferential operator and related variational inequalities, with Rascanu, Aurel, *Stochastic Process. Appl.* **76**, no. 2, 191-215, 1998.

[67] Generalized BSDEs and nonlinear Neumann boundary value problems, with Zhang, Shuguang, *Probab. Theory Related Fields*, **110** no. 4, 535-558, 1998.

[66] Generalized discontinuous backward stochastic differential equations, in *Backward stochastic differential equations*, (Paris, 1995-1996), 207-219, Pitman Res. Notes Math. Ser. **364**, Longman, Harlow, 1997.

[65] Averaging of backward stochastic differential equations, with application to semi-linear PDE's, with Veretennikov, A. Yu., *Stochastics & Stoch. Rep.* **60**, no. 3-4, 255-270, 1997.

[64] SPDEs with reflection and Malliavin calculus, with Donati-Martin, C., *Bull. Sci. Math.* **121**, no. 5, 405-422,1997.

[63] Probabilistic interpretation of a system of semi-linear parabolic partial differential equations, with Pradeilles, Frédéric; Rao, Zusheng, *Ann. Inst. H.Poincaré Probab. Statist.* **33**, no. 4, 467-490, 1997.

[62] Backwards SDE with random terminal time and applications to semilinear elliptic PDE, with Darling, R. W. R., *Ann. Probab.* **25**, no. 3, 1135-1159, 1997.

[61] Backward stochastic differential equations and integral-partial differential equations, with Barles, Guy; Buckdahn, Rainer, *Stochastics & Stoch. Rep.* **60**, no. 1-2, 57-83, 1997.

[60] Reflected solutions of backward SDE's, and related obstacle problems for PDE's, with El Karoui, N.; Kapoudjian, C.; Peng, S.; Quenez, M. C., *Ann. Probab.* **25**, no. 2, 702-737, 1997.

[59] Backward stochastic differential equations reflected in a convex domain, with Gégout-Petit, A., *Stochastics & Stoch. Rep.* **57**, no. 1-2, 111-128, 1996.

[58] Asymptotic stability of the optimal filter with respect to its initial condition, with Ocone, Daniel, *SIAM J. Control Optim.* **34**, no. 1, 226-243, 1996.

[57] Backward stochastic differential equations and applications, *Proceedings of the International Congress of Mathematicians*, Vol. 1, 2 (Zürich, 1994), 1502-1510, Birkhäuser, Basel, 1995.

[56] Stratonovich stochastic differential equations driven by general semimartingales, with Kurtz, Thomas G.; Protter, Philip, *Ann. Inst. H. Poincaré Probab. Statist.* **31**, no. 2, 351-377, 1995.

[55] Markov field properties of solutions of white noise driven quasi-linear parabolic PDEs, with Nualart, D., *Stochastics & Stoch. Rep.* **48**, no. 1-2, 17-44, 1994.

[54] White noise driven parabolic SPDEs with measurable drift, with Bally, V.; Gyöngy, I., *J. Funct. Anal.* **120**, no. 2, 484-510, 1994.

[53] Symmetric reflected diffusions, with Williams, R. J., *Ann. Inst. H. Poincaré Probab. Statist.* **30**, no. 1, 13-62, 1994.

[52] Backward doubly stochastic differential equations and systems of quasilinear SPDEs, with Peng, Shi Ge, *Probab. Theory Related Fields* **98**, no. 2, 209-227, 1994.

[51] A stochastic Feynman-Kac formula for anticipating SPDEs, and application to nonlinear smoothing, with Ocone, Daniel, *Stochastics & Stoch. Rep.* **45**, no. 1-2, 79-126, 1993.

[50] On the regularization effect of space-time white noise on quasi-linear parabolic partial differential equations, avec Gyöngy, Istvan, *Probab. Theory Related Fields* **97**, no. 1-2, 211-229, 1993.

[49]] Absolute continuity of the law of the solution of a parabolic SPDE, with Zhang, Tu Sheng, *J. Funct. Anal.* **112**, no. 2, 447-458, 1993.

[48] White noise driven SPDEs with reflection, with Donati-Martin, C., *Probab. Theory Related Fields* **95**, no. 1, 1-24, 1993.

[47] Stochastic partial differential equations, a review, *Bull. Sci. Math.* **117**, no. 1, 29-47, 1993.

[46] On quasi-linear stochastic partial differential equations, with Gyöngy, I., *Probab. Theory Related Fields* **94**, no. 4, 413-425, 1993.

[45] Backward stochastic differential equations and quasilinear parabolic partial differential equations, with Peng, S., in *Stochastic partial differential equations and their applications* (Charlotte, NC, 1991), 200-217, Lecture Notes in Control and Inform. Sci., **176**, Springer, Berlin, 1992.

[44] White noise driven quasilinear SPDEs with reflection, with Nualart, D., *Probab. Theory Related Fields* **93**, no. 1, 77-89, 1992.

[43] Lyapounov exponent of linear stochastic systems with large diffusion term, with Wihstutz, V., *Stochastic Process. Appl.* **40**, no. 2, 289-308, 1992.

[42] Second order stochastic differential equations with Dirichlet boundary conditions, with Nualart, David, *Stochastic Process. Appl.* **39**, no. 1, 1-24, 1991.

[41] Finite-dimensional approximate filters in the case of high signal-to-noise ratio, with Roubaud, M.-C., in *Stochastic analysis*, 433-448, Academic Press, Boston, MA, 1991.

[40] Boundary value problems for stochastic differential equations, with Nualart, D., *Ann. Probab.* **19**, no. 3, 1118-1144, 1991.

[39] Filtrage non linéaire et équations aux dérivées partielles stochastiques associées. (French) [Nonlinear filtering and associated stochastic partial differential equations], in *Ecole d'été de Probabilités de Saint-Flour* XIX 1989, 67-163, Lecture Notes in Math. **1464**, Springer, Berlin, 1991.

[38] An introduction to Malliavin calculus and some of its applications, with Michel, Dominique, in Recent advances in stochastic calculus (College Park, MD, 1987), 65-104, Progr. Automat. Info. Systems, Springer, New York, 1990.

[37] Monotonicity methods for white noise driven quasi-linear SPDEs, with Buckdahn, R., in *Diffusion processes and related problems in analysis*, Vol. I (Evanston, IL, 1989), 219-233, Progr. Probab. **22**, Birkhäuser Boston, Boston, MA, 1990.

[36] Applications of anticipating stochastic calculus to stochastic differential equations, in *Stochastic analysis and related topics* II (Silivri, 1988), 63-105, Lecture Notes in Math. **1444**, Springer, Berlin, 1990.

[35] Stochastic Volterra equations with anticipating coefficients, with Protter, Philip, *Ann. Probab.* **18**, no. 4, 1635-1655, 1990.

[34] Differential calculus and integration by parts on Poisson space (avec Carlen, Eric A.), in *Stochastics, algebra and analysis in classical and quantum dynamics* (Marseille, 1988), 63-73, Math. Appl., **59**, Kluwer Acad. Publ., Dordrecht, 1990.

[33] Adapted solution of a backward stochastic differential equation, with Peng, S. G., *Systems Control Lett.* **14**, no. 1, 55-61, 1990.

[32] A Lie algebraic criterion for nonexistence of finite-dimensionally computable filters, with Ocone, Daniel, in *Stochastic partial differential equations and applications* II (Trento, 1988), 197-204, Lecture Notes in Math. **1390**, Springer, Berlin, 1989.

[31] Piecewise monotone filtering with small obervation noise, with Fleming, W. H., in *SIAM J. Control Optim.* **27**, no. 5, 1156-1181, 1989.

[30] Linear stochastic differential equations with boundary conditions, with Ocone, Daniel, *Probab. Theory Related Fields* **82**, no. 4, 489-526, 1989.

[29] Stochastic variational inequalities of parabolic type, with Haussmann, U. G., *Appl. Math. Optim.* **20**, no. 2, 163-192, 1989.

[28] A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations, with Ocone, Daniel, *Ann. Inst. H. Poincaré Probab. Statist* **25**, no. 1, 39-71, 1989.

[27] Stochastic calculus with anticipating integrands, with Nualart, D., *Probab. Theory Related Fields* **78**, no. 4, 535-581, 1988.

[26] Lyapunov exponent and rotation number of two-dimensional linear stochastic systems with small diffusion, with Wihstutz, V., *SIAM J. Appl. Math.* **48**, no. 2, 442-457, 1988.

[25] A conditionally almost linear filtering problem with non-Gaussian initial condition, with Haussmann, U. G., *Stochastics* **23**, no. 2, 241-275, 1988.

[24] Uniqueness for diffusions with piecewise constant coefficients, with Bass, R. F., *Probab. Theory Related Fields* **76**, no. 4, 557-572, 1987.

[23] A two-sided stochastic integral and its calculus, with Protter, P., *Probab. Theory Related Fields* **76**, no. 1, 15-49, 1987.

[22] Two-sided stochastic calculus for SPDEs, in *Stochastic partial differential equations and applications* (Trento, 1985), 200-207, Lecture Notes in Math. **1236**, Springer, Berlin, 1987.

[21] Grossissement d'une filtration et retournement du temps d'une diffusion. (French) [Enlargement of a filtration and time reversal of a diffusion], in *Séminaire de Probabilités XX*, 1984/85, 48-55, Lecture Notes in Math. **1204**, Springer, Berlin, 1986.

[20] Wide band limit of Lyapounov exponents, in *Stochastic differential systems* (Bad Honnef, 1985), 305-315, Lecture Notes in Control and Inform. Sci. **78**, Springer, Berlin, 1986.

[19] Time reversal of diffusions, with Haussmann, U. G., *Ann. Probab.* **14**, no. 4, 1188-1205, 1986.

[18] Almost sure and moment stability for linear Itô equations, with Arnold, L.; Oeljeklaus, E., in *Lyapunov exponents* (Bremen, 1984), 129-159, Lecture Notes in Math. **1186**, Springer, Berlin, 1986.

[17] Time-reversal of diffusion processes and nonlinear smoothing, in *Systems and optimization* (Enschede, 1984), 171-181, Lecture Notes in Control and Inform. Sci. **66**, Springer, Berlin, 1985.

[16] Sur les équations aux dérivées partielles stochastiques, de type parabolique.(French) [On stochastic partial differential equations of parabolic type], in *Colloquium in honor of Laurent Schwartz*, Vol. 2 (Palaiseau, 1983). Astérisque **132**, 71-87, 1985.

[15] Time reversal of diffusion processes, with Haussmann, U. G., in *Stochastic differential systems* (Marseille-Luminy, 1984), 176-182, Lecture Notes in Control and Inform. Sci. **69**, Springer, Berlin, 1985.

[14] Discretization and simulation of stochastic differential equations, with Talay, D., *Appl. Math.* **3**, no. 1, 23-47, 1985.

[13] Etude de la stabilité de la solution d'une EDS bilinéaire � coefficients périodiques. Application au mouvement des pales d'hélicoptêre. (French) [Study of the stability of the solution of a bilinear SDE with periodic coefficients. Application to the motion of helicopter blades], with Pignol, M., in *Analysis and optimization of systems*, Part 2 (Nice, 1984), 92-103, Lecture Notes in Control and Inform. Sci. **63**, Springer, Berlin, 1984.

[12] Asymptotic analysis of PDEs with wide-band noise disturbances, and expansion of the moments, with Bouc, R., *Stochastic Anal. Appl.* **2**, no. 4, 369-422, 1984.

[11] Analyse asymptotique du problême de filtrage non linéaire avec bruit d'observation à large bande. (French) [Asymptotic analysis of the nonlinear filtration problem with wide-band observation noise], in *Analysis and optimization of systems* (Versailles, 1982), 729-743, Lecture Notes in Control and Inform. Sci. **44**, Springer, Berlin, 1982.

[10] Smoothing of a diffusion process conditioned at final time, in *Stochastic differential systems* (Bad Honnef, 1982), 187-196, Lecture Notes in Control and Inform. Sci. **43**, Springer, Berlin, 1982.

[9] Equations of nonlinear filtering and application to stochastic control with partial observation, in *Nonlinear filtering and stochastic control* (Cortona, 1981), 208-248, Lecture Notes in Math. **972**, Springer, Berlin, 1982.

[8] Optimal control for partially observed diffusions, with Fleming, Wendell H., *SIAM J. Control Optim.* **20**, no. 2, 261-285, 1982.

[7] Equations du filtrage non linéaire, de la prédiction et du lissage. (French) [Nonlinear filtering, prediction and smoothing equations] *Stochastics* **6**, no. 3-4, 193-231, 1981/82.

[6] Nonlinear filtering, prediction and smoothing, in *Stochastic systems: the mathematics of filtering and identification and applications* (Les Arcs, 1980), pp. 529-557, NATO Adv. Study Inst. Ser. C: Math. Phys. Sci. **78**, Reidel, Dordrecht-Boston, Mass., 1981.

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