In Honor of Gopinath Kallianpur Takeyuki Hida Rajeeva L

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Takeyuki Hida Rajeeva L. Kararidikar Hiroshi Kunita Balram S. Rajput Shinzo Watanabe Jie Xiong Editors

Springer Science+Business Media, LLC Takeyuki Hida Rajeeva L. Karandikar Department of Mathematics Indian Statistical Institute Nagoya University 110016 New Dehli, India Nagoya, Japan Hiroshi Kunita Balram S. Rajput Department of Applied Sciences Department of Mathematics Kyushu University University of Tennessee Kyushu, Japan Knoxville, TN 47996

Shinzo Watanabe Jie Xiong Department of Mathematics Department of Mathematics Kyoto University University of Tennessee Kyoto, Japan Knoxville, TN 37996

Library of Congress Cataloging-in-Publication Data

Stochastics in finite and infinite dimensions: in honor of Gopinath Kallianpur / Takeyuki Hida ... [et al.], editors. p. cm - (Trends in mathematics) ISBN 978-1-4612-6643-3 ISBN 978-1-4612-0167-0 (eBook) DOI 10.1007/978-1-4612-0167-0

1. Stochastic processes. I. Kallianpur, G. II. Hida, Takeyuki, 1927- III. Series.

QA274.S825 2000 519.2-dc21 00-062123 CIP

AMS Subject Classifications: 60-06,60F25,60G35,60G60,60H05,60H10,60H15,60H30,93E11

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

ISBN 978-1-4612-6643-3 SPIN 10739229

Reformatted from authors' files by TgXniques, Inc., Cambridge, MA.

987654321 Trends in Mathematics

Trends in Mathematics is a book series devoted to focused collections of articles arising from conferences, workshops or series of lectures.

Volumes of TIMS must be of high scientific qUality. Articles without proofs, or which do not contain significantly new results, are not appropriate. High quality survey papers, however, are welcome. Contributions must be submitted to peer review in a process that emulates the best journal procedures, and must be edited for correct use of language. As a rule, the language will be English, but selective exceptions may be made. Articles should conform to the highest standards of bibliographic reference and attribution.

Proposals to the Publisher are welcomed at either: Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. [email protected] or Birkhauser Verlag AG, PO Box 133, CH-4010 Basel, Switzerland [email protected] Stochastics in Finite and Infinite Dimensions In Honor of Gopinath Kallianpur

Takeyuki Hida Rajeeva L. Karan'dikar Hiroshi Kunita Balram s. Rajput Shinzo Watanabe Jie Xiong Editors

Birkhauser Boston • Basel • Berlin Takeyuki Hida Rajeeva L. Karandikar Department of Mathematics Indian Statistical Institute Nagoya University 110016 New Dehli, India Nagoya, Japan Hiroshi Kunita Balram S. Rajput Department of Applied Sciences Department of Mathematics Kyushu University University of Tennessee Kyushu, Japan Knoxville, TN 47996 Shinzo Watanabe JieXiong Department of Mathematics Department of Mathematics Kyoto University University of Tennessee Kyoto, Japan Knoxville, TN 37996

Library of Congress Cataloging-in-Publication Data

Stochastics in finite and infinite dimensions: in honor of Gopinath Kallianpur I Takeyuki Hida ... let al.l, editors. p. em - (Trends in mathematics) ISBN 0-8176-4137-8 (alk. paper) - ISBN 3-7643-4137-8 (alk. paper) 1. Stochastic processes. I. Kallianpur, G. II. Hida, Takeyuki, 1927- III. Series.

QA274.S825 2000 519.2-dc21 00-062123 CIP

AMS Subject Classifications: 60-06, 60F25, 60035, 60G60, 60H05, 60HlO, 60H15, 60H30, 93Ell

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser Boston, clo Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use cf general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

ISBN 0-8176-4137-8 SPIN 10739229 ISBN 3-7643-4137-8

Reformatted from authors' files by TEXniques, Inc., Cambridge, MA.

9 8 7 6 5 4 3 2 1 Gopinath Kallianpur Contents

Preface ...... IX A Glimpse into the Life and Work of Gopinath Kallianpur B. V. Rao ...... xi

Publications of Gopinath Kallianpur ...... xxv

Precise Gaussian Lower Bounds on Heat Kernels S. Aida ...... 1

Feynman Integrals Associated with Albeverio-HflIegh-Krohn and Laplace Transform Potentials N. Asai, 1. Kubo, and H.-H. Kuo ...... 29

Random Iteration of I.I.D. Quadratic Maps K. B. Athreya and R. N. Bhattacharya ...... 49

Monte Carlo Algorithms and Asymptotic Problems in Nonlinear Filtering A. Budhiraja and H. J. Kushner ...... 59

A Covariant Quantum Stochastic Dilation Theory P. S. Chakraborty, D. Goswami, and K. B. Sinha ...... 89

Interacting Particle Filtering with Discrete-Time Observations: Asymptotic Behaviour in the Gaussian Case P. Del Moral and J. Jacod ...... 101

Hidden Markov Chain Filtering for Generalised Bessel Processes R. Elliott and E. Platen ...... 123

On the Zakai Equation of Filtering with Gaussian Noise L. Gawarecki and V. Mandrekar ...... 145

Prediction and Translation of Fractional Brownian Motions Y. Hu ...... 153

Time Maps in the Study of Feynman's Operational Calculus via Wiener and Feynman Path Integrals G. W. Johnson and L. Johnson ...... 173 Vlll Contents

Two Applications of Reproducing Kernel Hilbert Spaces in Stochastic Analysis T. Koski and P. Sundar ...... 195

Stochastic Linear Controlled Systems with Quadratic Cost Revisited N. V. K rylov ...... 207

Numerical Solutions for a Class of SPDEs with Application to Filtering T. G. Kurtz and J. Xiong ...... 233

Nonlinear Diffusion Approximations of Queuing Networks B. Margolius and W. A. Woyczynski ...... 259

On Equations of Stochastic Fluid Mechanics R. Mikulevicius and B. Rozovskii ...... 285

Infinite Level Asymptotics of a Perturbative Chern-Simons Integral 1. Mitoma ...... 303

Risk-Sensitive Dynamic Asset Management with Partial Information H. Nagai ...... 321

Existence of a Strong Solution for an Integro-Differential Equation and Superposition of Diffusion Processes y. Ogura, M. Tomisaki and M. Tsuchiya ...... 341

On the Consistency of the Maximum Likelihood Method in Testing Multiple Quantum Hypotheses K. R. Parthasarathy ...... 361 Large Deviations for Double Ito Equations V. Perez-Abreu and C. Tudor ...... 379

The Domain of a Generator and the Intertwining Property 1. Shigekawa ...... 401 Preface

Over the last fifty years Gopinath Kallianpur has made significant con• tributions to several areas of Probability Theory and Stochastic Processes. For these and his other exceptional accomplishments as a scholar, educator and administrator, Kallianpur is internationally recognized. Moreover, he and his work have been a source of inspiration for generations of students and collaborators. Kallianpur celebrated his seventy-fifth birthday this year. We congrat• ulate him upon this occasion and to express our profound appreciation for his multifaceted achievements, we have dedicated this book to him on behalf of all of his friends, students and colleagues, many of whom have contributed articles here. Because of space limitations, however, it was not possible to invite everyone to contribute, and we regret this. Several topics that have been of interest to Kallianpur are reflected in this volume, including filtering theory, control theory, solutions to SDE and SPDE and approximations thereof, Feynman integrals, infinite dimen• sional diffusions, reproducing kernel Hilbert spaces, and fractional Brown• ian motion. Independent of our efforts, the Indian Statistical Institute planned in Kallianpur's honour a conference on Stochastic Processes at the campus of the Institute in Calcutta from December 18 to 23, 2000. We coordinated our efforts with those of the Institute. All contributors to this volume were invited to speak at this conference, at which time this volume was presented to Kallianpur. Many people helped bring about this tribute, and we take this oppor• tunity to thank them. Our sincere thanks to all the contributors, and, in particular, to Bhamidi V. Rao, for his very incisive and perceptive bio• graphical article "A Glimpse into the Life and Work of Gopinath Kallian• pur." Our thanks go also to all the referees, to Ann Kostant, executive editor, Birkhauser, and to the very cooperative Birkhauser staff. We are most appreciative of the excellent production work of Elizabeth Loew. Fi• nally, our very special thanks to Krishna, Kallianpur's charming wife, who at our request opened her family's photo archives and graciously provided us with several delightful photos of Kallianpur which are included in this volume.

Takeyuki Hida, Rajeeva Karandikar, Hiroshi Kunita Balram Rajput, Shinzo Watanabe, Jie Xiong March,2000 A Glimpse into the Life and Work of Gopinath Kallianpur

Gopinath Kallianpur is internationally recognized for his lifelong dedi• cation and important contributions to the fields of Statistics and Probabil• ity, for his deep insights and the inspiration he has so graciously offered to his students and colleagues over the years, and for his remarkable scholarly and administrative achievements. Kallianpur's life spans a period of three quarters of a century, and his contributions, particularly those related to his research, are wide in scope. It is therefore difficult to include in this rather short article all or even a substantial number of his achievements, as well as the events and persons that helped mould his personal and aca• demic life. What we have included here is a (nonrandom!) sampling of them.

Life and Professional Career

Gopinath Kallianpur was born on April 16, 1925 to parents Shankar and Ramabai in Mangalore, Karnataka, India. He received his Bachelors degree in 1945 and Masters degree in 1946 from the University of Madras. Immediately after graduation he moved to Bombay and worked as a lec• turer at Wilson College, where in 1947, he met a bright young student Krishna; they were later married. Krishna has two masters degrees, one in economics from Bombay University and the other in education from Michigan State University. Kallianpur's wide ranging achievements are due in no small measure to the support he received from his wife Krishna. Their two daughters Asha and Kalpana, under the inspiring guidance and influence of their parents, have also made notable achievements. Asha is a physician and has an M.D. from the University of North Carolina, Chapel Hill, and Kalpana is a physicist, with a Ph.D in physics from the University of Texas at Austin. While in Bombay Kallianpur came in contact with D.D. Kosambi, a well-known mathematician and versatile scholar. He attended a series of lectures on probability by Kosambi, which reinforced his interest in pursuing an academic career in mathematics and probability. He also spent a month learning about summability from S. Minakshisundaram, one of India's best mathematicians, at Andhra University which was then in Guntur. To pursue higher studies, he became a graduate student in the Statistics Department of the University of North Carolina, Chapel Hill, in the fall of 1949. Under the supervision of H. Robbins, he obtained his xii B.V. Rao doctoral degree in 1951 in the then developing field of stochastic processes. After graduation, he held the position of lecturer at the University of California, Berkeley, during 1951-52, and was a member of the Institute for Advanced Study, Princeton, from 1952 to 1953. At about this time P.C. Mahalanobis, founder and director of the In• dian Statistical Institute (1.S.1.), Calcutta, was searching for young prob• abilists and statisticians to teach and do research at his Institute. Maha• lanobis needed little time to recognize Kallianpur's talent and potential, and offered him a position at the Institute, which Kallianpur immediately accepted. Kallianpur joined the Institute in 1953 as a "worker" and stayed in Calcutta until the summer of 1956. During his stay he took an active part in both the theoretical and applied activities of the Research and Training School of the Institute. It was a period during which many dis• tinguished scientists, notable among them R.A. Fisher from Cambridge, N. Wiener from M.1.T, and Yu. V. Linnik from Leningrad, visited the Institute and delivered a series of lectures. Wiener, for example, lectured on prediction theory among other topics, and held extensive mathemati• cal discussions with Kallianpur and P.R. Masani, who was then working at the Institute of Sciences in Bombay as Professor of Mathematics and Head of the Department. He had a special invitation from Mahalanobis to attend Wiener's lectures. During the course of discussions, Wiener sug• gested collaborative work to Kallianpur. With this in mind Kallianpur in 1956 accepted the position of associate professor at the Department of Statistics at Michigan State University in East Lansing, and went on leave from the Institute. Unfortunately, upon his arrival in the U.S.A. he was faced with a serious health problem and was hospitalized for a considerable period of time. Partly because of this illness, the projected research collaboration with Wiener did not develop; nevertheless, Wiener's ideas had a strong influence on some of Kallianpur's future research. Aware of Kallianpur's ill health and in order to induce his return to 1.S.1., Mahalanobis granted Kallianpur a five-year leave of absence and offered him an added incentive. This provided Kallianpur with the option of living and holding his classes for 1.S.1. students at Giridih, a place known for its natural beauty and pleasant climate all year around. (Giridih is located about 250 miles away from Calcutta, and 1.S.1. maintains an experimental station there for applied research). Kallianpur was touched by the generous and thoughtful offer of Mahalanobis, but, partly for reasons of health, decided to stay in the U.S.A. Kallianpur fully recovered within a year and he remained at Michigan State University from 1956 until 1959. He spent the period of 1959- Gopinath Kallianpur - A Glimpse into his Life and Work Xlll

1961 at the Mathematics Department of Indiana University, Bloomington, where he was an associate professor. Subsequently, he went back to East Lansing, accepting a position of professor, and he stayed there during 1961-1963. In the fall of 1963, he joined the Mathematics Department of the University of Minnesota, Minneapolis, as professor, where he remained until 1976. He was already working intensively in the field of Gaussian and stationary processes, and he also began work in filtering theory, a field in which his interest was inspired by his discussions with Wiener at I.S.I. At about this time C. Striebel, a member of the Research Group of Lockheed Aircraft working on filtering and control problems, joined the Department at Minneapolis. Her arrival and their mutual interest in the general area of filtering problems led to an active and fruitful collaboration between the two. On July 24, 1976, Kallianpur became the first director of the In• dian Statistical Institute under its new Memorandum of Association. He worked relentlessly and vigourously, and made significant contributions to both the academic and administrative functioning of the Institute. He took immediate steps to improve the research atmosphere of the Institute at Calcutta and also at its recently established centre at New Delhi. He ar• ranged for the visits of many distinguished scientists to the Institute, and persuaded the Ford Foundation to provide funds to enable young prob• abilists and statisticians from the I.S.I. to visit the U.S.A. He also took

G. Kallianpur with Professor Kantorovich in his office at 1.8.1., 1978. xiv B.V. Rao

Krishna, Rani Mahalanobis, and G. Kallianpur in the "Amrapali", 1978. steps to improve the living conditions and other infrastructural facilities, and he was instrumental in setting up a Flume Laboratory at Calcutta. One of the most visible and lasting legacies of his tenure as the direc• tor is the founding of a new centre of I.S.I. at Bangalore, Karnataka. The creation of the Bangalore centre was a direct result of his initiative and persistent efforts. In this he had to act with great tact and administrative skill because there was a considerable amount of apprehension among the workers at Calcutta regarding the establishment of a new centre. This cen• tre, like the other two, continues to flourish academically and has become an important part of the mission of I.S.I. The steps initiated by Kallianpur as the director for the overall im• provement of the Institute continue to have a visible impact on the work• ings of the Institute. As constructive and fruitful as these steps were, some of them were also quite audacious. In fact, he himself was to remi• nisce later: "I was faced with many challenging problems whose solution brought a sense of fulfillment to me .... " I, like many others at I.S.I., want to reassure him that this period indeed brought a sense of achievement and satisfaction not only to him, but to all of us at the Institute. In 1978, Kallianpur returned to the University of Minnesota, and in 1979 he was appointed Alumni Distinguished Professor at the University of North Carolina at Chapel Hill. Assisted by M. R. Leadbetter and S. Cambanis, he set up a Center for Stochastic Processes at Chapel Hill. Gopinath Kallianpur - A Glimpse into his Life and Work xv

Left to right: Subimal Dutt, President of the Indian Statisti• cal Institute, B.D. Pande, Governor of West Bengal, C.R. Rao, G. Kallianpur, and Indira Gandhi (addressing), Prime Minister of India. [Photo taken on December 29, 1981 on the occasion of the Golden Jubilee celebrations of the I.S.I.]

This center provided an important avenue for significant interaction among faculty, students, and a large number of distinguished visitors from all over the world, and contributed towards substantial research in the general area of stochastic processes. Throughout his academic career, Kallianpur has been very active in serving the scientific profession in various capacities. He served on the Editorial Boards of several international journals and on committees of professional societies. At present, he is the Editor of the Journal of Ap• plied Mathematics and Optimization, and he is on the Editorial Board of Sankhya. The esteem in which Kallianpur and his contributions are held by the international scientific community is partly shown by the profes• sional honours he has been awarded. These include his election to the Fellowships of the Institute of Mathematical Statistics (IMS), the Inter• national Indian Statistical Association (lISA), and the Indian National Science Academy (INSA). He is also a member of the International Sta• tistical Institute. Recently Moscow State University honoured him by appointing him to the 1996 Kolomogorov Professorship. We close this section by alluding to some of Kallianpur's personal characteristics. He is a soft spoken and quiet person, an erudite scholar with a passion for research and learning. Because of his education in India xvi B.V. Rao during the twilight years of the British Raj, and later in the U.S., one can see many traits of western culture in his personal life. On the other hand, having been born and brought up in India, Indian culture and civilization have influenced his life much more: his first book begins with a quotation from a Buddhist Sutta of around 480 B.C., and if one happens to visit the Kallianpur home in Chapel Hill one would find there enchanting paintings and other works of art from India. It would be fair to say that he is a fine example of a rare blend of Eastern and Western values.

G. Kallianpur with Professor Harald Cramer, 1.8.1. convocation, 1977.

Contributions as a Teacher

Kallianpur loves teaching as well as working with and directing gradu• ate students. His stimulating lectures, his dedication, and his rare ability to explain the most difficult concepts in an elegantly simple manner have earned him the recognition by generations of students as a caring and in• spiring teacher. He has the rare ability to attract bright students and turn them into first rate researchers. So far 22 students have obtained doctoral degrees under his guidance. After graduation, a substantial number of his students remained in academic life and many of them, following their Gopinath Kallianpur - A Glimpse into his Life and Work XVll esteemed teacher, have established themselves as leading researchers and teachers in their own right. Kallianpur's Ph.D. Students: A. Amirdjanova, D. Baldwin, J. D. Borwankar, C. Bromley, A. Budhiraja, S.K. Christensen, A. Dasgupta, G. Hardy, H.P. Hucke, Y.T. Kim, R. LePage, P. MandaI, V. Mandrekar, H. Oodaira, W.J. Park, V. Perez-Abreu, U.V.R. Rao, D. Rhoades, R. Selukar, S. Sukhatme, J. Xiong, G.J. Zimmerman.

Research Contributions

Kallianpur's work in probability theory has had a considerable influ• ence on the development of the subject. As noted earlier, his research contributions are so prolific and wide ranging that it is not possible for me to describe them all; we briefly touch upon some of them. Kallianpur-Robbins law: The two-dimensional Brownian motion has a unique feature that is not shared by the Brownian motion of any other dimension. Specifically, it is recurrent with respect to nonempty open sets, in the sense that every nonempty open set is visited by the Brownian path. However, it is not point recurrent, in the sense that given any specific point, almost no path attains it for any positive time. This is why the asymptotic behaviour of the occupation times of two-dimensional Brownian paths is singularly different from that of the Brownian paths in dimensions other than two. Kallianpur and H. Robbins studied this asymptotic behaviour. To describe their main result, let (Bt)t"~o denote a two-dimensional Brownian motion with Bo = 0, and consider the additive functional A(t) = J; !(Bs) ds, where! is a bounded real integrable func• tion on R2 satisfying 2~ JR2 ! dx = 1. The Kallianpur-Robbins law states that as t ---+ 00, A(t) / logt converges in distribution to an exponential ran• dom variable with parameter one. It is very interesting to note that a few years ago J. W. Pitman and M. Yor made an extensive study of a similar Ito integral and obtained information about the winding property around points of Brownian paths. More recently, Y. Kasahara, M. Kono, and N. Kosugi studied the Kallianpur-Robbins law for a class of self-similar Gaussian processes. Kallianpur-Striebel formula: This is one of the most fundamental and important results of nonlinear filtering theory. It provides the solution to one of the filtering problems in the form of an integral on a suitable function space and plays a key role in the further development of the theory. The problem under consideration can be described as follows: There is a system process or signal process (Xt , 0 ::; t ::; T), defined on a probability space (0, B, P). This process cannot be observed directly; xviii B.V. Rao instead one observes the process

Zt It XTdT + Wt, where Wt is a standard Wiener process independent of the system process. Let us fix a time point t. There is a functional of the system process, that is, a random variable 9 on the space n, which is measurable relative to the a-field generated by the X-process up to time t, and which is assumed integrable. The problem1 is to estimate 9 given the observations (Zn 0 ~ T ~ t). Assuming squared error loss, the estimate is of course, 6, the conditional expectation of 9 given a(Zn 0 ~ T ~ t). Thus, the task is to give a computable formula for this expression; this is exactly what the Kallianpur-Striebel formula achieves. To describe the formula, it is convenient to work within the product space setup in such a way that the process X (and hence g) depends on the first coordinate and W on the second. Denoting a generic point of the product space by w = (u, v), and assuming that the system process is jointly measurable and has square integrable paths, the Kallianpur-Striebel formula states that In g(u')q(w, u') P(du') 6(w) = fnl q(w, u') P(du') , where 0 1 denotes the first co-ordinate space of the product space 0, and q(w, u') is given by t q(w, u') = exp[l Xs(u') dZs(w) - ~ It X;(u') ds]. The above formula is useful for a fixed value of t. If the data is coming continuously - as is the case in most engineering and other technological applications - one requires a practical method so that the estimate at time t +.6., .6. > 0, can be obtained by updating the estimate at time t by some recursive procedure. Kallianpur and C. Striebel showed in a subse• quent paper that this can indeed be achieved. More precisely, if (Xt ) is a diffusion process, ! a function in the domain of its infinitesimal generator and g(u) = !(Xt(u)), then the process 6(t), the conditional expectation of 9 given the observation a-field up to time t, itself satisfies an appro• priate SDE. Taking the function space Bayes formula as a starting point, lTo quote the authors, the space n on which the system process is defined corre• sponds to the parameter space in the usual Bayes approach to the theory of estimation. Thus the probability P is the a priori distribution for the unknown parameter. The process (ZT) 0 ~ T ~ t) is the observed random vector. We wish to estimate the parametric function g. Gopinath Kallianpur - A Glimpse into his Life and Work XIX

M. H. A. Davis has developed a general methodology for obtaining sam• ple path solutions to a large class of problems. The Bayes formula plays a prominent role in the robust or pathwise theory of nonlinear filtering. FKK equations: The definitive formulation of the filtering problem from the innovation standpoint was given in a classic paper by M. Fu• jisaki, Kallianpur and H. Kunita. In the Kallianpur-Striebel setup it was assumed that the noise Wt is independent of the signal process X t . To enlarge the scope of applications, it is essential to drop this assump• tion. Of course, the dependence cannot be too arbitrary either. The basic assumption in the Fujisaki-Kallianpur-Kunita formulation is the most natural one, namely, any future noise is independent of the present. More precisely, for any time t, a(Xs, Ws : 0 ::; s ::; t) is independent of a(Wu - W t : t ::; u ::; T). In a fundamental paper, they showed that even in this case, under suitable conditions, the nonlinear filter 6(t) sat• isfies an appropriate SDE. It should be mentioned here that this problem was also independently studied by H. Kushner who derived the same equa• tions. These equations have come to be known as the Fujisaki-Kallianpur• Kunita (FKK) equations. An exact statement of these equations involves the innovation process. Since the appearance of their paper there has been an explosion of research in nonlinear filtering theory. These equations can be regarded as the equations governing the measure valued process, /It, the conditional distribution of X t given the observation a-field upto t. The SDE of the nonlinear filter is one of the first naturally arising examples of an SDE governing an infinite dimensional process. White Noise theory: The classical stochastic calculus theory of fil• tering is elegant, mathematically appealing, and has provided enormous stimulus to the theory of SDE's. However its practical validity was open to criticism. A. V. Balakrishnan argues that the results obtained by these methods cannot be implemented and are hence not suitable for applica• tions. One of the main objections is that, in practice, the natural space of the observations and the noise is a Hilbert space - the RKHS - which has zero Wiener measure. To use the count ably additive theory, one has to complete the space either by going to the path space consisting of con• tinuous functions or to the space of tempered distributions of Schwartz. The alternative possibility is to hang on to the finitely additive proba• bility on the Hilbert space and try to develop the statistics of the white noise. This is not to diminish the importance of the Wiener process. The Wiener process undoubtedly plays a central role with its versatile appli• cations and well-developed theory. The only suggestion is to separate its role as a model for white noise. The signal, it should be noted, is still based on conventional count ably additive probability space. xx B.V. Rao

Although the concept of a Gaussian cylindrical measure on Hilbert spaces, a prototype of a finitely additive measure, existed in the well• known works of 1. Gross, 1. E. Segal, and others, but in order to develop the statistical model one needs to devise a common space where the signal and noise both live. Since there are inherent problems in going to prod• uct spaces in the finitely additive context, a delicate, "correct and suit• able" approach has to be found. This is precisely what Kallianpur and R. L. Karandikar did in a series of papers; these results are expounded in their joint book White Noise Theory of Prediction, Filtering and Smooth• ing. This beautiful theory has several directions for future investigations. For instance, it is not clear how to incorporate dependence of the signal and noise in this setup. The present writer is well aware of the criticism to this approach by conventional theorists. Some feel that the finitely additive theory is eso• teric, while others even feel that it is unwarranted. But there is no denying the fact that this theory is mathematically challenging, and in the present case, as hinted above, it is a necessity. Advocates of finitely additive prob• ability include: L.J. Savage and L.E. Dubins who, following a suggestion of de Finetti, initiated the study of gambling systems in the finitely ad• ditive setup; and D. Heath and W. Sudderth who, in a different context, made the following interesting comment, "A Bayesian who seeks to avoid incoherent inferences might be advised to abandon improper count ably additive priors and use only finitely additive priors." Fisher consistency: Inspired by the lectures of R.A. Fisher at the 1.S.1, Kallianpur and C.R. Rao introduced the concept of Fisher consis• tency in the theory of statistical estimation. To describe their idea and result, let {F(., On denote a family of distributions, {p(x, On the corre• sponding densities on the real line, g(O) a function of the parameter, and XI, ... ,Xn LLd. observations with the underlying distribution F(.,O). Let T be an estimator of the parametric function g( 0) based on these ob• servations. Fisher obtained an information theoretic lower bound for the variance of an unbiased estimator of g. Unfortunately, as demonstrated by J. L. Hodges, the asymptotic variance of a consistent estimator can be arbitrary small. Thus, any definition of efficiency based on the concept of "least asymptotic variance" is void. According to Kallianpur and Rao, a Fisher consistent estimator T is a weakly continuous function whose domain includes the underlying set of distributions as well as all empirical distributions, and further satis• fies T(F(., 0)) = g(O). The idea is the following: For a given sample the estimator is the value of T at the empirical distribution of the sample. Under the assumption of Frechet differentiability of the estimator T and Gopinath Kallianpur - A Glimpse into his Life and Work xxi the usual regularity conditions on the family {p{x,O)}, they showed that the information bound given by Fisher is indeed a lower bound for the variance of an estimator. A reviewer questioned their interpretation and this paper failed to produce a ripple in the statistical literature of the time. It is heartening to note that this kind of approach has gained ground in recent years. Kallianpur considered several other aspects of statistical esti• mation, including optimal properties of regular Bayes estimators, limiting distributions of Von Mises statistical functions, asymptotic distributions of V-statistics, and estimating a two-dimensional area using the line grid method. This last mentioned work actually goes back to his stay at the Institute in the early fifties and has its origins in the crop cutting experi• ments conducted by the LS.L Environmental Pollution models: In recent years Kallianpur has been interested in applications of stochastic calculus methods in more concrete areas. One of them is building models of environmental pollution and the behaviour of voltage potentials of spatially extended neurons. The water pollution model, initially proposed by H. Kwakernaak, can roughly be described as follows: Think of a river, represented by an interval, say, [0, l], in which pollutants are being deposited. It is assumed that the deposit times form a Poisson process. At each such time the place of deposit and the quantity of deposit have a given joint distribution on [0, l] x (0,00). Moreover these are independent at different deposit times. The quantity to be described is the time evolution of u(t, x), the concentration of the chemical at location x at time t. The equation satisfied by u can be written as an appropriate SDE in an infinite-dimensional space. Of course, one can replace the interval [0, l] by any reasonable space. For instance, if it is replaced by a subset of R3, then the resulting setup can be regarded as modeling atmospheric pollution. Kallianpur and J. Xiong have studied this problem extensively in a series of papers; most of their results are included in their monograph Stochastic Differential Equations in Infinite-dimensional Spaces. Kallianpur not only brings his expertise in stochastic calculus to develop all the details concerning the existence of solutions of the SDE and the properties of the resulting process, he also brings his expertise in filtering theory to implement the model when the observations are corrupted, as is usually the case in practice. Mathematical Finance: The second application area of Kallianpur's recent interest is Mathematical Finance. A flavour of it is provided in his recent monograph with R.L. Karandikar entitled Introduction to Option Pricing Theory. A welcome feature of this book is that the two important concepts "Absence of Arbitrage" and "Completeness of Markets" are dealt with in depth in the context of a general model. A refreshingly new view- XXlI B.V. &0

G. Kallianpur in front of the Kallianpur home at Chapel Hill, N.C., April 2000. point put forward is that the underlying filtration should not be treated as a mere technicality but, while considering the trading strategies, one must consider strategies that are predictable relative to the filtration generated by the observation process. As mentioned earlier, the works of Kallianpur referred to above consti• tute only a sample from among his many contributions. We now mention very briefly some of the others: He, along with coauthors, contributed ex• tensively to our understanding of the Feynman integral by using an astute generalization of analytic continuation methods, or by viewing it as a dis• tribution on an abstract Wiener space, by using the second quantization of a basic self-adjoint operator on the Cameron-Martin-Maruyama space, or by using Hilbert space valued traces. He has also made deep studies of the interrelations among multiple Stratonovich integrals, multiple Wiener integrals, multiple as well as iterated Ogawa integrals. His zero-one law for Gaussian processes has inspired several authors who extended this zero• one law for stable and other more general infinitely divisible processes and their non-linear functionals. This also spurred research in the related Gopinath Kallianpur - A Glimpse into his Life and Work XXlll area that pertains to studying the geometric and algebraic structure of the topological support of Gaussian, stable and other infinitely divisible probability measures.

Acknowledgments. In preparing this article I have benefitted from three books [1-3], each of which touches upon some aspect of the life and work of Kallianpur. I thank Moshe Zakai, Shinzo Watanabe, Murray Rosenblatt, S. Bhaskara Rao, Balram S. Raj put , John Long, Rajeeva L. Karandikar, Jagdish C. Gupta and Indra M. Chakravarty for their help. Any inaccu• racies that remain are of my own making.

References

[1] S. Cambanis, J.K Ghosh, R.L. Karandikar, and P.K Sen (editors), Stochastic Processes: A festschrift in honour of Gopinath Kallianpur, Springer-Verlag, New York, 1991.

[2] J.K Ghosh, S.K Mitra, and KR. Parthasarathy (editors), Glimpses of India's Statistical Heritage, Wiley Eastern Ltd, Bombay, 1992.

[3] Ashok Rudra, Prasanta Chandra Mahalanobis - A biography, Oxford University Press, Calcutta, 1996.

B.V. Rao Indian Statistical Institute Calcutta Publications of Gopinath Kallianpur

Books

Stochastic Filtering Theory. Applications of Mathematics 13, Springer• Verlag, New York, Berlin, 1980.

Stokhasticheskaya teoriya filtratsii. (Russian) [Stochastic Filtering Theory] translated from the English by V.M. Shurenkov, "Nauka", Moscow, 1987.

White Noise Theory of Prediction, Filtering and Smoothing (with RL. Karandikar). Stochastic Monographs 3, Gordon & Breach Science Pub• lishers, New York, 1988.

Stochastic Differential Equations in Infinite-dimensional Spaces (with J. Xiong), Institute of Mathematical Statistics Lecture Notes-Monograph Series 26, 1993. Expanded version of lectures delivered as part of the Barrett Lectures at the University of Tennessee, Knoxville, TN, March 25-27, 1993.

Introduction to Option Pricing Theory (with RL. Karandikar), Birkhauser, Boston, 1999.

Books Edited

Measure Theory Applications to Stochastic Analysis (with D. Kolzow), Lecture Notes in Mathematics 695, Springer-Verlag, Berlin, 1977.

Statistics and Probability, Essays in Honor of C.R. Rao (with P.R Krishnaiah and J.K. Ghosh), North-Holland Publishing Co., Amsterdam• New York, 1982.

Theory and Application of Random Fields, Lecture Notes in Control and Information Sciences, 49, Springer-Verlag, Berlin, New York, 1983.

Stochastic Methods in Biology (with M. Kimura and T. Hida), Lecture Notes in Biomathematics 70, Springer-Verlag, Berlin, New York, 1987. xxvi Publications of Gopinath Kallianpur

Papers

Integrale de Stieltjes stochastique et un theoreme sur les fonctions aleatoires d'ensembles, G.R. Acad. Sci. Paris, 232 (1951), 922-923.

Ergodic property of the Brownian motion process (with H. Robbins), Proc. Nat. Acad. Sci. USA, 39 (1953), 525-533.

The sequence of sums of independent random variables (with H. Rob• bins), Duke Math. J., 21 (1954), 285-307.

A note on the Robbins-Monro stochastic approximation method, Ann. Math. Statist., 25 (1954), 386-388.

On a limit theorem for dependent random variables (in Russian), Dokl. Akad. Nauk. SSSR (NS), 101 (1955), 13-16

On an ergodic property of a certain class of Markov processes, Proc. Amer. Math. Soc., 6 (1955), 159-169.

On Fisher's lower bound for the asymptotic variance of a consistent estimate (with C.R. Rao), Sankhya 15 (1955), 331-342; corrigenda 16, 206.

A note on perfect probability, Ann. Math. Statist. 30 (1959), 169-172.

A problem in optimum filtering with finite data, Ann. Math. Statist. 30 (1959), 659-669.

On the amount of information contained in a O"-field, in Contributions to Probability and Statistics, 1. Olkin et al. eds., Stanford Univ. Press (1960), 265-273.

The topology of weak convergence of probability measures, J. Math. Mech. 10 (1961), 947-969.

The equivalence and singularity of Gaussian measures, with H. Oodaira, in Proc. Sympos. Time Series Analysis, M. Rosenblatt ed., Wiley (1962), 279-291.

Von Mises functionals and maximum likelihood estimation, Sankhya 25 (1963), 149-158. Publications of Gopinath Kallianpur xxvii

On the connection between multiplicity theory and O. Hanner's time domain analysis of weakly stationary processes (with V. Mandrekar), in Essays in Probability and Statistics, R.C. Bose et al. eds., Univ. of North Carolina Press (1964), 1-13.

Von Mises functionals and maximum likelihood estimation, in Con• tributions to Statistics, Statistical Publishing Society, Calcutta (1965), 137-146.

Multiplicity and representation theory of weakly stationary processes (with V. Mandrekar), Theory Probab. Appl. 10 (1965), 553-581.

Semi-groups of isometries and the representation and multiplicity of weakly stationary stochastic processes (with V. Mandrekar), Ark. Mat. 6 (1966), 319-335.

Estimation of stochastic processes: Arbitrary system process with additive white noise observation errors (with C. Striebel), Ann. Math. Statist. 39 (1968), 785-801.

Stochastic differential equations occurring in the estimation of contin• uous parameter stochastic processes (with C. Striebel), Theory Probab. Appl. 14 (1969), 567-594.

Stochastic differential equations in statistical estimation problems (with C. Striebel), in Multivariate Analysis II, P.R. Krishnaiah ed., Aca• demic Press (1969), 367-388.

Zero-one laws for Gaussian processes, Trans. Amer. Math. Soc. 149 (1970), 199-211.

A note on uniform convergence of stochastic processes (with N. C. Jain), Ann. Math. Statist. 41(1970), 1360-1362.

The role of reproducing kernel Hilbert spaces in the study of Gaussian stochastic processes, in Advances in Probability II, P. Ney ed., Dekker 1970),49-83.

Norm convergent expansions for Gaussian processes (with N.C. Jain), Proc. Amer. Math. Soc. 25 (1970), 890-895.

Supports of Gaussian measures (with M.G. Nadkarni), Proc. Sixth Berkeley Symp. Probab. Math. Statist. 2, Univ. of California, Berkeley (1970/71), 375-387. xxviii Publications of Gopinath Kallianpur

The Bernstein-von Mises theorem and Bayes estimation in Markov processes, with J. Borwanker and B.L.S. Prakasa Rao, Ann. Math. Statist. 42 (1971), 1241-1253.

A stochastic differential equation of Fisk type for estimation and non• linear filtering problems (with C. Striebel), SIAM J. Appl. Math. 21 (1971),61-72.

Abstract Wiener processes and their reproducing kernel Hilbert spaces, Z. Wahr. verw. Geb. 17 (1971), 113-123.

Nonlinear filtering, in Optimizing Methods in Statistics, J.S. Rustagi ed., Academic Press (1971), 211-232.

Spectral theory for H-valued stationary processes (with V. Man• drekar), J. Multivariate Anal. 1 (1971), 1-16.

Stochastic differential equations for the nonlinear filtering problem (with M. Fujisaki and H. Kunita), Osaka J. Math. 9 (1972), 19-40.

Oscillation function of a multiparameter Gaussian process (with N.C. Jain), Nagoya Math. J. 47 (1972), 15-28.

Homogeneous chaos expansions, in Statistical Models and Turbulence, Lecture Notes in Physics 12, M. Rosenblatt et al. eds., Springer-Verlag (1972), 230-254.

Non-anticipative representations of equivalent Gaussian processes (with H. Oodaira), Ann. Probab. 1 (1973), 104-122.

Non-anticipative canonical representations of equivalent Gaussian pro• cesses, in Multivariate Analysis III, P.R. Krishnaiah ed., Academic Press (1973), 31-44.

Canonical representations of equivalent Gaussian processes, Sankhya A 35 (1973), 405-416

The Markov property for generalized Gaussian random fields, Ann. Inst. Fourier 24 no. 2 (1974), 143-167.

Canonical representations of equivalent Gaussian processes, in Stochas• tic Processes and Related Topics, M.L. Puri ed., Academic Press (1975), 195-221. Publications of Gopinath Kallianpur XXIX

The square of a Gaussian Markov process and non-linear prediction (with T. Hida), J. Multivariate Anal. 5 (1975), 451-461.

A general stochastic equation for the non-linear filtering problem, in Optimization Techniques IFIP Technical Conference, Lecture Notes in Computer Science 27, G.!. Marchuk ed., Springer-Verlag (1975),198-204.

A stochastic equation for the optimal non-linear filter, in Multivariate Analysis IV, P.R. Krishnaiah ed., North Holland (1977), 267-281.

Non-anticipative transformations of the two-parameter Wiener pro• cess and a Girsanov theorem (with N. Etemadi), J. Multivariate Anal. 7 (1977), 28-49.

A linear stochastic system with discontinuous control, in Proc. of the Intern'tl Symp. on Stochastic Differential Equations, K. Ito ed., Wiley (1978), 127-140.

Freidlin-Wentzell estimates for abstract Wiener processes (with H. Oodaira), Sankhya A 40 (1978), 116-137.

Representation of Gaussian random fields (with C. Bromley), in Stochastic Differential Systems, Lecture Notes in Control and Informa• tion Sciences 25, B. Grigelionis ed., Springer-Verlag (1980), 129-142.

Gaussian random fields (with C. Bromley), Appl. Math. Optimization 6 (1980), 361-376.

A stochastic equation for the conditional density in a filtering problem, in Multivariate Analysis V, P.R. Krishnaiah ed., North Holland (1980), 137-150.

Some ramifications of Wiener's ideas on nonlinear prediction, in Nor• bert Wiener Collected Works III, P. Masani ed., MIT Press (1981), 402- 424.

Some remarks on the purely nondeterministic property of second order random fields, in Stochastic Differential Systems, Lecture Notes in Control and Information Sciences 36, M. Arato et al. eds., Springer-Verlag (1981), 98-109.

A generalized Cameron-Feynman integral, in Statistics and Probability: Essays in Honor of C.R. Rao, G. Kallianpur, P.R. Krishnaiah and J.K. Ghosh eds., North Holland (1982), 369-374. xxx Publications of Gopinath Kallianpur

On the diffusion approximation to a discontinuous model for a sin• gle neuron, in Contributions to Statistics, P.K. Sen eds., North Holland (1983), 247-258.

Nondeterministic random fields and Wold and Halmos decompositions for commuting isometries (with V. Mandrekar), in Prediction Theory and Harmonic Analysis, V. Mandrekar and H. Salehi eds., North Holland (1983), 165-190.

Commuting semigroups of isometries and Karhunen representation of stationary random fields (with V. Mandrekar), in Theory and Application of Random Fields, Lecture Notes in Control and Information Sciences 49, G. Kallianpur ed., Springer-Verlag (1983), 126-145.

A finitely additive white noise approach to nonlinear filtering (with RL. Karandikar), Appl. Math. Optimization 10 (1983), 159-185.

On the splicing of measures (with D. Ramachandran), Ann. Probab. 11 (1983), 819-822.

Some recent developments in nonlinear filtering theory (with RL. Karandikar), Acta Appl. Math. 1 (1983), 399-434.

Generalized Feynman integrals using analytic continuation in several complex variables (with C. Bromley), in Stochastic Analysis, M. Pinsky ed., Dekker (1984), 217-267.

Regularity property of Donsker's delta function (with H.H. Kuo), Appl. Math. Optimization 12 (1984), 89-95.

Measure valued equations for the optimum filter in the finitely additive nonlinear filtering theory (with RL. Karandikar), Z. Wahr. verw. Geb. 66 (1984), 1-17.

Infinite dimensional stochastic differential equation models for spa• tially distributed neurons (with R Wolpert), Appl. Math. Optimization 12 (1984), 125-172.

The nonlinear filtering problem for the unbounded case (with RL. Karandikar), Stochastic Proc. Appl. 18 (1984), 57-66.

The Markov property of the filter in the finitely additive white noise approach to nonlinear filtering (with RL. Karandikar), Stochastics 13 (1984), 177-198. Publications of Gopinath Kallianpur xxxi

A finitely additive approach to nonlinear filtering: A brief survey (with RL. Karandikar), in Multivariate Analysis VI, P.R Krishnaiah ed., North Holland (1985), 335-344.

White noise theory of filtering: Some robustness and consistency re• sults, in Stochastic Differential Systems, Lecture Notes in Control and In• formation Sciences 69, M. Metivier and E. Pardoux eds., Springer-Verlag (1985), 217-223.

Analytic and sequential Feynman integrals on abstract Wiener and Hilbert spaces and a Cameron-Martin formula (with D. Kannan and RL. Karandikar), Ann. Inst. H. Poincare Probab. Statist. 21 (1985), 323-361.

White noise calculus and nonlinear filtering (with RL. Karandikar), Ann. Probab. 13 (1985), 1033-1107.

White noise calculus for two-parameter filtering (with A.H. Korezli• oglu) , in Stochastic Differential Systems, Lecture Notes in Control and Information Sciences 96, H. Engelberg and W. Schmidt eds., Springer• Verlag (1986), 61-69.

Stochastic differential equations in duals of nuclear spaces with some applications, IMA Technical Report No. 244, Univ. of Minnesota (1986).

Weak convergence of stochastic neuronal models (with R Wolpert), Stochastic Methods in Biology, Lecture Notes in Biomathematics 70, Kimura et al. eds., Springer-Verlag (1987), 116-145.

The filtering problem for infinite dimensional stochastic processes, with RL. Karandikar, in Stochastic Differential Systems, Stochastic Control, Theory and Applications, W. Fleming and P.L. Lions eds., Springer-Verlag (1987), 215-223.

Stochastic differential equations for neuronal behavior (with S.K. Christensen), in Adaptive Statistical Procedures and Related Topics, J. Van Ryzin ed., IMS Lecture Notes Monograph Series 8 (1987), 237-272.

Stochastic evolution equations driven by nuclear space-valued mar• tingales (with V. Perez-Abreu), Appl. Math. Optimization 17 (1988), 237-272.

Smoothness properties of the conditional expectation in finitely addi• tive white noise filtering (with H. Hucke and RL. Karandikar), J. Multi• variate Anal. 27 (1988), 261-269. xxxii Publications of Gopinath Kallianpur

Weak convergence of solutions of stochastic evolution equations in nuclear spaces (with V. Perez-Abreu), in Stochastic Partial Differential Equations and Applications, Lecture Notes in Mathematics 1390, Da Prato and L. Thbaro eds., Springer-Verlag (1989), 133-139.

Some remarks on Hu and Meyer's paper and infinite dimensional cal• culus on finitely additive canonical Hilbert space (with C.W. Johnson), Theory Probab. Appl. 34 (1989), 742-752.

Diffusion equations in duals of nuclear spaces (with 1. Mitoma and R. Wolpert), Stochastics 20 (1990), 285-329.

Infinite dimensional stochastic differential equations with applications, in Stochastic Methods in Experimental Sciences, W. Kasprzak and A. Weron eds., World Scientific (1990), 208-219.

On the prediction theory of two parameter stationary random fields (with A.C. Miamee and H. Niemi), J. Multivariate Anal. 32 (1990), 120- 149.

Multiple Wiener integrals on abstract Wiener spaces and liftings of p-linear forms (with C.W. Johnson), in White Noise Analysis, T. Hida et al. eds., World Scientific (1990), 208-219.

Propagation of chaos and the McKean-Vlasov equation in duals of nu• clear spaces (with T.S. Chiang and P. Sundar), Appl. Math. Optimization 24 (1991), 55-83.

A skeletal theory of filtering, in Stochastic Analysis, E. Mayer-Wolf et al. eds., Academic Press (1991), 213-234.

Traces, natural extensions and Feynman distributions, in Gaussian Random Fields, K. Ito and T. Hida eds., World Scientific (1991), 14-27.

A line grid method in areal sampling and its connection with some early work of H. Robbins, Amer. J. Math. Manag. Sci. 11 (1991),40-53.

Parameter estimation in linear filtering (with R. Selukar), J. Multi• variate Anal. 39 (1991), 284-304.

Stochastic differential equation models for spatially distributed neu• rons and propagation of chaos for interacting systems, Math. Biosci. 112 (1992), 207-224. Publications of Gopinath Kallianpur xxxiii

Nuclear space-valued stochastic differential equations with applica• tions, in Probabilistic and Stochastic Methods in Analysis, with Applica• tions, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 372, Kluwer Acad. Pub!. (1992), 631-647.

The Skorohod integral and the derivative operator of functionals of a cylindrical Brownian motion (with V. Perez-Abreu), Appl. Math. Optim. 25 (1992), 11-29.

Periodically correlated processes and their relationship to L2 [0, T]• valued stationary sequences (with H.L. Hurd), in Nonstationary Stochastic Processes and Their Applications, A.G. Miamee ed., World Sci. Publish• ing (1992), 256-284.

A Segal-Langevin type stochastic differential equation on a space of generalized functionals, (with r. Mitoma), Can ad. J.Math. 44 (1992), 524-552.

Distributions, Feynman integrals and measures on abstract Wiener spaces (with A.S. Ustunel), in Stochastic Analysis and Related Topics, H. Korezlioglu and A.S. Ustunel eds., Birkhauser (1992), 237-284.

Propagation of chaos for systems of interacting neurons (with T.S. Chiang and P. Sundar), in Stochastic Partial Differential Equations and Applications, G. Da Prato and r. Thbaro eds., Res. Notes Math. Ser.268, Pitman (1992), 98-110.

The analytic Feynman integral of the natural extension of pth homoge• neous chaos (with G.W. Johnson), in Measure Theory, Rend. Circ. Mat. Palermo Ser. 1128, 181-199.

A nuclear space-valued stochastic differential equation driven by Pois• son random measures (with J. Xiong), in Stochastic Partial Differential Equations and Their Applications, B.L. Rozovskii and R.B. Sower eds., Lecture Notes in Control and Inform. Sci. 176, Springer (1992),135-143.

Stochastic differential equations in infinite dimensions: A brief survey and some new directions of research (with J. Xiong), in Multivariate Anal• ysis: Future Directions, C.R. Rao ed., North-Holland (1993), 267-277. xxxiv Publications of Gopinath Kallianpur

Homogeneous chaos, p-forms, scaling and the Feynman integral (with G.W. Johnson), Trans. Amer. Math. Soc. 340 (1993), 503-548.

An introduction to white-noise analysis and nonlinear filtering, with R.L. Karandikar, in Mathematical Theory of Control, Lecture Notes in Pure and Appl. Math. 142, Dekker (1993),173-183.

Asymptotic behavior of a system of interacting nuclear-space-valued stochastic differential equations driven by Poisson random measures (with J. Xiong), Appl. Math. Optim. 30 (1994), 175-201.

Nonlinear transformations of the canonical Gauss measure on Hilbert space and absolute continuity (with R.L. Karandikar), Acta Appl. Math. 35 (1994), 63-102.

Remarks on the existence of k-traces (with G.W. Johnson), in Chaos Expansions, Multiple Wiener-Ito Integrals and Their Applications, C. Houdre and V. Perez-Abreu eds., Probab. Stochastics Ser., CRC Press (1994), 47-71.

Stochastic models of environmental pollution (with J. Xiong), Adv. in Appl. Probab. 26 (1994), 377-403.

The existence and uniqueness of solutions of nuclear space-valued equa• tions driven by Poisson random measures (with J. Xiong, G. Hardy and S. Ramasubramanian), Stoch. & Stoch. Reps. 50 (1994), 85-122.

Hilbert space valued traces and multiple Stratonovich integrals with statistical applications (with A. Budhiraja) , in Stochastic Analysis on Infinite-dimensional Spaces, Pitman Res. Notes Math. Ser., 310, Long• man Sci. Tech. (1994), 26-32.

Uniqueness and robustness of solution of measure-valued equations of nonlinear filtering (with A.G. Bhatt and R.L. Karandikar), Ann. Probab. 23 (1995), 1895-1938.

Estimation of Hilbert space valued parameters by the method of sieves (with R. Selukar), in Statistics and Probability, A Raghu Raj Bahadur Festschrift, J.K. Ghosh et al. eds., Wiley Eastern (1993), 325-347.

Hilbert space valued traces and multiple Stratonovich integrals with statistical applications (with A. Budhiraja), Probab. Math. Statist. 15 (1995)-Special Issue in Honor of the Birth Centenary of J. Neyman (1995), 127-163. Publications of Gopinath Kallianpur xxxv

Diffusion approximation of nuclear space-valued stochastic-differential equations driven by Poisson random measures (with J. Xiong), Ann. Appl. Probab. 5 (1995), 493-517.

Approximations to the solution of the Zakai equation using multi• ple Wiener and Stratonovich integral expansions (with A. Budhiraja), Stochastics & Stochas. Rep. 56 (1996), 271-315.

Large deviations for a class of stochastic partial differential equations (with J. Xiong), Ann. Probab. 24 (1996), 320-345.

Some recent developments in nonlinear filtering theory, in Ito's Stochastic Calculus and Probability Theory, N. Ikeda et al. eds., Springer (1996), 157-170.

On problems with a free boundary that arise in probability theory (uniqueness theorems) (with O.A. Oleinik), Russian Math. Surveys 51 (1996), 1203-1205.

The Feynman-Stratonovich semigroup and Stratonovich integral ex• pansions in nonlinear filtering (with A. Budhiraja), Appl. Math. Optim. 35 (1997), 91-116.

Stochastic filtering: A part of stochastic nonlinear analysis, in Proc. Sympos. Appl. Math. 52 (1997), 371-385.

The generalized Hu-Meyer formula for random kernels (with A. Bud• hiraja), Appl. Math. Optim. 35 (1997), 177-202.

Two results on multiple Stratonovich integrals (with A. Budhiraja), Statist. Sinica 7 (1997), 907-922.

(1997) On problems with a free boundary that arise in probability theory (existence theorems) (with O.A. Oleinik), Uspekhi Mat. Nauk 52 (1997); translation in Russian Math. Surveys 52, 222-223.

On interacting systems of Hilbert-space-valued diffusions (with A.G. Bhatt, R.L. Karandikar and J. Xiong), Appl. Math. Optim. 37 (1998), 151-188.

A curious example from statistical differential geometry (with Y.-T. Kim), Th. of Probab. & its Appl. 43 (1998), 116-140. xxxvi Publications of Gopinath Kallianpur

The Russian options, in Stochastic Processes and Related Topics, Trends in Math., Birkhauser (1998) 231-252.

Exponential integrability and application to stochastic quantization (with Y.Z. Hu), Appl. Math. Optim. 37 (1998), 295-353.

Stochastic Dyson series and the solution to associated stochastic evo• lution equations (with G.W. Johnson), in Stochastic Analysis and Math• ematical Physics, World Sci. Publishing (1998), 82-108.

Robustness of the nonlinear filter (with A.G. Bhatt and R.L. Karandikar), Stochastic Process. Appl. 81 (1999), 247-254.

Multiple fractional integrals (with A. Dasgupta), Probab. Theory Re• lat. Fields 115 (1999), 505-525.

Chaos decomposition of multiple fractional integrals and applications (with A. Dasgupta), Probab. Theory Relat. Fields 115 (1999), 527-548.