Moshe Zakai

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Moshe Zakai
Moshe zakai.jpg
Born(1926-12-22)22 December 1926
Sokółka, Poland
Died27 November 2015(2015-11-27) (aged 88)
Haifa, Israel
NationalityIsraeli
Alma mater University of Illinois at Urbana–Champaign
SpouseShulamit (Mita) Briskman
Scientific career
Fields Electrical engineering
Institutions Technion

Moshe Zakai (December 22, 1926 – November 27, 2015) was a Distinguished Professor at the Technion, Israel in electrical engineering, member of the Israel Academy of Sciences and Humanities and Rothschild Prize winner. [1]

Contents

Biography

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 to 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 was married to Shulamit (Mita) Briskman, they have 3 children and 12 grandchildren.

Major awards

Research

Background

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. [4] These results are often referred to as Wong-Zakai corrections or theorems.

Nonlinear filtering

The solution to the problem of the optimal filtering 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 and were initially studied by Stratonovich in 1959 - 1960 and later by Kushner in 1964, leading to the Kushner-Stratonovich equation, a non-linear stochastic partial differential equation (SPDE) for the conditional probability density representing the optimal filter. Around 1967, Zakai derived a considerably simpler SPDE for an unnormalized version of the optimal filter density. It is known as the Zakai equation, [5] and it has the great advantage of being a linear SPDE. The Zakai equation 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.

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. [6] [7]

The Malliavin calculus and its application

In addition to the Ito calculus, Paul Malliavin developed in the 1970s a "stochastic calculus of variations", now 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 and Zeitouni promoted the understanding and applicability of the Malliavin calculus. [8] [9] [10] [11] [12]

The monograph of Üstünel and Zakai [13] 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 [14] [15] and with Ustunel extended to some general cases results of information theory which were known for simpler spaces. [16]

Further information

Related Research Articles

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created and started by the Japanese mathematician Kiyosi Itô during World War II.

In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows the computation of derivatives of random variables. Malliavin calculus is also called the stochastic calculus of variations. P. Malliavin first initiated the calculus on infinite dimensional space. Then, the significant contributors such as S. Kusuoka, D. Stroock, J-M. Bismut, S. Watanabe, I. Shigekawa, and so on finally completed the foundations.

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock prices, random growth models or physical systems that are subjected to thermal fluctuations.

<span class="mw-page-title-main">Itô calculus</span> Calculus of stochastic differential equations

Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion. It has important applications in mathematical finance and stochastic differential equations.

In signal processing, a nonlinearfilter is a filter whose output is not a linear function of its input. That is, if the filter outputs signals R and S for two input signals r and s separately, but does not always output αR + βS when the input is a linear combination αr + βs.

In stochastic processes, the Stratonovich integral or Fisk–Stratonovich integral is a stochastic integral, the most common alternative to the Itô integral. Although the Itô integral is the usual choice in applied mathematics, the Stratonovich integral is frequently used in physics.

In mathematics, the Clark–Ocone theorem is a theorem of stochastic analysis. It expresses the value of some function F defined on the classical Wiener space of continuous paths starting at the origin as the sum of its mean value and an Itô integral with respect to that path. It is named after the contributions of mathematicians J.M.C. Clark (1970), Daniel Ocone (1984) and U.G. Haussmann (1978).

In filtering theory the Zakai equation is a linear stochastic partial differential equation for the un-normalized density of a hidden state. In contrast, the Kushner equation gives a non-linear stochastic partial differential equation for the normalized density of the hidden state. In principle either approach allows one to estimate a quantity function from noisy measurements, even when the system is non-linear. The application of this approach to a specific engineering situation may be problematic however, as these equations are quite complex. The Zakai equation is a bilinear stochastic partial differential equation. It was named after Moshe Zakai.

<span class="mw-page-title-main">Paul Malliavin</span> French mathematician

Paul Malliavin was a French mathematician who made important contributions to harmonic analysis and stochastic analysis. He is known for the Malliavin calculus, an infinite dimensional calculus for functionals on the Wiener space and his probabilistic proof of Hörmander's theorem. He was Professor at the Pierre and Marie Curie University and a member of the French Academy of Sciences from 1979 to 2010.

In the theory of stochastic processes, filtering describes the problem of determining the state of a system from an incomplete and potentially noisy set of observations. While originally motivated by problems in engineering, filtering found applications in many fields from signal processing to finance.

Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations.

Václav Edvard "Vic" Beneš is a Czech-American, a mathematician known for his contributions to the theory of stochastic processes, queueing theory and control theory, as well as the design of telecommunications switches.

In mathematics, the Skorokhod integral, often denoted , is an operator of great importance in the theory of stochastic processes. It is named after the Ukrainian mathematician Anatoliy Skorokhod and japanese mathematician Masuyuki Hitsuda. Part of its importance is that it unifies several concepts:

<span class="mw-page-title-main">Jean-Michel Bismut</span> French mathematician

Jean-Michel Bismut is a French mathematician who has been a professor at the Université Paris-Sud since 1981. His mathematical career covers two apparently different branches of mathematics: probability theory and differential geometry. Ideas from probability play an important role in his works on geometry.

This page lists articles related to probability theory. In particular, it lists many articles corresponding to specific probability distributions. Such articles are marked here by a code of the form (X:Y), which refers to number of random variables involved and the type of the distribution. For example (2:DC) indicates a distribution with two random variables, discrete or continuous. Other codes are just abbreviations for topics. The list of codes can be found in the table of contents.

In filtering theory the Kushner equation is an equation for the conditional probability density of the state of a stochastic non-linear dynamical system, given noisy measurements of the state. It therefore provides the solution of the nonlinear filtering problem in estimation theory. The equation is sometimes referred to as the Stratonovich–Kushnerequation. However, the correct equation in terms of Itō calculus was first derived by Kushner although a more heuristic Stratonovich version of it appeared already in Stratonovich's works in late fifties. However, the derivation in terms of Itō calculus is due to Richard Bucy.

David Nualart is a Spanish mathematician working in the field of probability theory, in particular on aspects of stochastic processes and stochastic analysis.

In stochastic calculus, the Ogawa integral is a stochastic integral for non-adapted processes as integrands. The corresponding calculus is called non-causal calculus in order to distinguish it from the anticipating calculus of the Skorokhod integral. The term causality refers to the adaptation to the natural filtration of the integrator.

In probability theory, Lévy's stochastic area is a stochastic process that describes the enclosed area of a trajectory of a two-dimensional Brownian motion and its chord. The process was introduced by Paul Lévy in 1940, and in 1950 he computed the characteristic function and conditional characteristic function.

References

  1. ↑ "Obituary: Moshe Zakai, 1926–2015". IMS Bullentin. Retrieved 5 January 2016.
  2. ↑ "IEEE Control Systems Award Recipients" (PDF). IEEE . Retrieved March 30, 2011.
  3. ↑ "IEEE Control Systems Award". IEEE Control Systems Society. Archived from the original on December 29, 2010. Retrieved March 30, 2011.
  4. ↑ 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.
  5. ↑ Zakai, Moshe (1969). "On the optimal filtering of diffusion processes". Probability Theory and Related Fields. 11 (3): 230–243. doi: 10.1007/BF00536382 .
  6. ↑ 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 .
  7. ↑ 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 .
  8. ↑ Nualart, David; Zakai, Moshe (1988). "Generalized multiple stochastic integrals and the representation of Wiener functionals". Stochastics. 23 (3): 311–330. doi:10.1080/17442508808833496.
  9. ↑ Nualart, David; Zakai, Moshe (1989). "The partial Malliavin calculus". Séminaire de Probabilités XXIII. Lecture Notes in Mathematics. Vol. 1372. pp. 362–381. doi:10.1007/BFb0083986. ISBN   978-3-540-51191-5.
  10. ↑ 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 .
  11. ↑ 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 .
  12. ↑ 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 .
  13. ↑ Üstünel, Ali Süleyman (2000). Transformation of Measure on Wiener Space. Springer. p. 320. ISBN   978-3-540-66455-0.
  14. ↑ 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.
  15. ↑ Moshe, Zakai (2005). Émery, Michel (ed.). "Rotations and Tangent Processes on Wiener Space". Séminaire de Probabilités XXXVIII. Lecture Notes in Mathematics. Springer Berlin / Heidelberg. 1857: 165–186. arXiv: math/0301351 . doi:10.1007/978-3-540-31449-3_15. ISBN   978-3-540-23973-4.
  16. ↑ 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. arXiv: math/0409548 . doi:10.1109/TIT.2005.853297. ISSN   0018-9448.
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