26 December 1926|
|Alma mater||University of Illinois at Urbana–Champaign|
- 1 Biography
- 2 Major awards
- 3 Research
- 4 Volume in honor of Zakai's 65 birthday
- 5 References
Moshe Zakai was born in Sokółka, Poland, to his parents Rachel and Eliezer Zakheim with whom he immigrated to Israel in 1936. He got the BSc degree in Electrical Engineering from the Technion – Israel Institute of Technology in 1951. He joined the Scientific Department of the Defense Minister of Israel where he was assigned to research and development of radar systems. From 1956–1958 he did graduate work at the University of Illinois on an Israeli Government Fellowship, and was awarded the PhD in Electrical Engineering. He then returned to the Scientific Department as head of the communication research group. In 1965 he joined the faculty of the Technion as an associate professor. In 1969 he was promoted to the rank of Professor and in 1970 he was appointed the holder of the Fondiller Chair in Telecommunication. He was appointed Distinguished Professor in 1985. From 1970 until 1973 he served as the Dean of the faculty of Electrical Engineering and from 1976 to 1978 he served as Vice President of Academic Affairs. He retired in 1998 as Distinguished Professor Emeritus.
Moshe Zakai is married to Shulamit (Mita) Briskman, they have 3 children and 12 grandchildren.
- 1973 Fellow of the Institute of Electrical and Electronic Engineers (IEEE)
- 1988 Fellow of the Institute of Mathematical Statistics
- 1989 Foreign member of the US National Academy of Engineering
- 1993 Member of the Israel Academy of Sciences and Humanities
- 1993 The IEEE Control Systems Award
- 1994 The Rothschild Prize in Engineering
Zakai's main research concentrated on the study of the theory of stochastic processes and its application to information and control problems; namely, problems of noise in communication radar and control systems. The basic class of random processes which represent the noise in such systems are known as "white noise" or the "Wiener process" where the white noise is "something like a derivative" of the Wiener process. Since these processes vary quickly with time, the classical differential and integral calculus is not applicable to such processes. In the 1940s Kiyoshi Itō developed a stochastic calculus (the Ito calculus) for such random processes.
The relation between classical and Ito calculi
From the results of Ito it became clear, back in the 1950s, that if a sequence of smooth functions which present the input to a physical system converge to something like a Brownian motion, then the sequence of outputs of the system do not converge in the classical sense. Several papers written by Eugene Wong and Zakai clarified the relation between the two approaches. This opened up the way to the application of the Ito calculus to problems in physics and engineering. These results are often referred to as Wong-Zakai corrections or theorems.
The problem of the optimal filtering (separating the signal from the noise) of a wide class of linear dynamical system is known as the Kalman filter. This led to the same problem for nonlinear dynamical systems. The results for this case were highly complicated. Around 1967, Zakai derived a considerably simpler solution for the optimal filter. It is known as the Zakai equation, and has been the starting point for further research work in this field.
Comparing practical solutions with the optimal solution
In many cases the optimal design of communication or radar operating under noise is too complicated to be practical, while practical solutions are known. In such cases it is extremely important to know how close the practical solution is to the theoretically optimal one. Several papers written by Zakai jointly with coauthors (Ben-Zion Bobrovsky, Ofer Zeitouni, Jacob Ziv and Eduardo Mayer-Wolf) introduced new approaches and results that gave a new impact to this topic.
Extension of the Ito calculus to the two-parameter processes
White noise and Brownian motion (the Wiener process) are functions of a single parameter, namely time. For problems such as rough surfaces it is necessary to extend the Ito calculus to two parameter "Brownian sheets". Several papers which he wrote jointly with Wong extend the Ito integral to a "two-parameter" time. They also showed that every functional of the Brownian sheet can be represented as an extended integral.
The Malliavin calculus and its application
In addition to the Ito calculus, Paul Malliavin developed in the 1970s a "stochastic calculus of variations" known as the "Malliavin calculus". It turned out that in this setup it is possible to define a stochastic integral which will include the Ito integral. The papers of Zakai with David Nualart, Ali Süleyman Üstünel, Zeitouni and Mayer-Wolf promoted the understanding and applicability of the Malliavin calculus.
The monograph of Üstünel and Zakai deals with the application of the Malliavin calculus to derive relations between the Wiener process and other processes which are in some sense "similar" to the probability law of the Wiener process.
In the last decade he extended to transformations which are in some sense a "rotation" of the Wiener process and with Ustunel and Mayer-Wolf extended to some general cases results of information theory which were known for simpler spaces.
Volume in honor of Zakai's 65 birthday
- Eddy Mayer-Wolf, Ely Merzbach, Adam Shwartz (Eds.), Stochastic Analysis: Liber Amicorum for Moshe Zakai, Academic Press, San Diego, California, 1991.
- On his life and research, see pages xi–xiv of the volume in honor of Zakai's 65 birthday.
- For the list of publications until 1990, see pages xv–xx. For publications between 1990 and 2000, see . For later publications search for M Zakai in arXiv.
- "IEEE Control Systems Award Recipients". IEEE. Retrieved March 30, 2011.
- "IEEE Control Systems Award". IEEE Control Systems Society. Retrieved March 30, 2011.
- Wong, Eugene; Moshe Zakai (July 1965). "On the relation between ordinary and stochastic differential equations". International Journal of Engineering Science 3 (2): 213–229. doi:10.1016/0020-7225(65)90045-5.
- Zakai, Moshe (1969). "On the optimal filtering of diffusion processes". Probability Theory and Related Fields 11 (3): 230–243. doi:10.1007/BF00536382.
- Ziv, Jacob; Zakai, Moshe (May 1969). "Some lower bounds on signal parameter estimation". IEEE Transactions on Information Theory 15 (3): 386–391. doi:10.1109/TIT.1969.1054301.
- Zakai, Moshe; Ziv, Jacob (May 1972). "Lower and upper bounds on the optimal filtering error of certain diffusion processes". IEEE Transactions on Information Theory 18 (3): 325–331. doi:10.1109/TIT.1972.1054825.
- Chazan, D.; Zakai, Moshe; Ziv, Jacob (January 1975). "Improved Lower Bounds on Signal Parameter Estimation". IEEE Transactions on Information Theory 18 (3): 90–93. doi:10.1109/TIT.1975.1055325.
- Bobrovsky, Ben-Zion; Zakai, Moshe (December 1975). "A lower bound on the estimation error for Markov processes". IEEE Transactions on Automatic Control 20 (5): 785–788. doi:10.1109/TAC.1975.1101088.
- Bobrovsky, B. Z.; Mayer-Wolf, E; Zakai, M. (1987). "Some Classes of Global Cramer-Rao Bounds". Annals of Statistics 15 (4): 1421–1438. doi:10.1214/aos/1176350602.
- Bobrovsky, Ben-Zion; Zakai, Moshe M.; Zeitouni, Ofer (July 1988). "Error bounds for the nonlinear filtering of signals with small diffusion coefficients". IEEE Transactions on Information Theory 34 (4): 710–721. doi:10.1109/18.9770.
- Zeitouni, Ofer; Zakai, Moshe (1992). "On the Optimal Tracking Problem". SIAM Journal on Control and Optimization 30 (2): 429–439. doi:10.1137/0330026.
- Wong, Eugene; Zakai, Moshe (1976). "Weak Martingales and Stochastic Integrals in the Plane". Annals of Probability 4 (4): 570–586. doi:10.1214/aop/1176996028.
- Merzbach, Ely; Moshe Zakai (1980). "Predictable and dual predictable projections of two-parameter stochastic processes". Probability Theory and Related Fields 53 (3): 263–269. doi:10.1007/BF00531435.
- Nualart, David; Zakai, Moshe (1988). "Generalized multiple stochastic integrals and the representation of Wiener functionals". Stochastics 23 (3): 311–330. doi:10.1080/17442508808833496.
- Nualart, David; Zakai, Moshe (1989). The partial Malliavin calculus. Lecture Notes in Mathematics 1372. pp. 362–381. doi:10.1007/BFb0083986.
- Üstünel, Ali Süleyman; Zakai, Moshe (1989). "On Independence and Conditioning On Wiener Space". Annals of Probability 17 (4): 1441–1453. doi:10.1214/aop/1176991164.
- Üstünel, Ali Süleyman; Zakai, Moshe (1993). "Applications of the degree theorem to absolute continuity on Wiener space". Probability Theory and Related Fields 95 (4): 509–520. doi:10.1007/BF01196731.
- Üstünel, Ali Süleyman; Zakai, Moshe (1997). "The Construction of Filtrations on Abstract Wiener Space". Journal of Functional Analysis 143 (1): 10–32. doi:10.1006/jfan.1996.2973.
- Üstünel, Ali Süleyman (2000). Transformation of Measure on Wiener Space. Springer. p. 320. ISBN 978-3-540-66455-0.
- Üstünel, Ali Süleyman; Zakai, Moshe (1995). "Random rotations of the Wiener path". Probability Theory and Related Fields 103 (3): 409–429. doi:10.1007/BF01195481. ISSN 0178-8051.
- Moshe, Zakai (2005). "Rotations and Tangent Processes on Wiener Space". In Émery, Michel. Séminaire de Probabilités XXXVIII. Lecture Notes in Mathematics (Springer Berlin / Heidelberg) 1857: 165–186. doi:10.1007/978-3-540-31449-3_15. ISBN 978-3-540-23973-4.
- Zakai, Moshe (September 2005). "On mutual information, likelihood ratios, and estimation error for the additive Gaussian channel". IEEE Transactions on Information Theory 51 (9): 3017–3024. doi:10.1109/TIT.2005.853297. ISSN 0018-9448.
- Mayer-Wolf, Eddy; Zakai, Moshe (2007). "Some relations between mutual information and estimation error in Wiener space". Annals of Applied Probability 17 (3): 1102–1116. doi:10.1214/105051607000000131.
- Mayer-Wolf, Eduardo; Üstünel, Ali Süleyman; Zakai, Moshe (2008). "Some covariance inequalities in Wiener space". Journal of Functional Analysis 255 (9): 2563–2578. doi:10.1016/j.jfa.2008.06.023. ISSN 0022-1236.