| Russian Mathematical Surveys |
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Mathematical Life Albert Nikolaevich Shiryaev (on his 90th birthday) A. V. Bulinski, A. A. Gushchin, M. V. Zhitlukhin, V. V. Kozlov, A. D. Manita, A. A. Muravlev, A. A. Novikov, I. V. Pavlov, D. V. Treschev, A. S. Holevo, E. B. Yarovaya, P. A. Yaskov
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    On 12 October 2024 the outstanding scientist, academician of the Russian Academy of Sciences Albert Nikolaevich Shiryaev observed his 90th birthday. Albert Shiryaev was born in Shchelkovo, Moscow Oblast, and soon his parents moved to Podlipki (now located within the city of Korolev). Many of his relatives were connected with the factories operating in Korolev at different times. From the 7th to 10th grade he studied in Moscow school no. 338 and Boarding School no. 1 under the auspices of the Ministry of Foreign Affairs (when his parents, who worked in the Ministry of Foreign Affairs, were on a foreign mission). During his school years his interests were quite diverse: he attended seminars on the history of diplomacy at the Institute of International Relations, went to clubs at the Bauman Technical School, was engaged in sports (Russian hockey, football, figure skating) and ballroom dancing. In the 9th grade he began to master the two-volume book Mathematical Analysis by F. Franklin, recently translated into Russian, and solved successfully a large number of problems for each chapter. To enroll in the Faculty of Mechanics and Mathematics at Moscow State University, as a school graduate with a medal of excellence, he needed only to pass an interview. When it turned out that one of the interviewers was S. A. Galpern, who had been the editor of the translation of Franklin’s book, the fate of the applicant was decided. At the university Shiryaev did not abandon his passion for sports and was engaged in the newly created alpine skiing section, becoming a Candidate for Master of Sport in this field. As a student at the Faculty of Mechanics and Mathematics, he attended various seminars and eventually went to the Department of Probability Theory, where Roland L’vovich Dobrushin supervised his third-year paper. His first scientific research on “The central limit theorem for complex Markov chains” was performed during his 4th and 5th years at the university. Immediately after graduating from Moscow State University (1952–1957), on a recommendation of A. N. Kolmogorov he was hired to the Department of Probability Theory of the Steklov Mathematical Institute of the USSR Academy of Sciences, and he has been working in this institute ever since, rising from a senior laboratory assistant to a chief researcher. Soon after starting work in the Steklov Institute, Shiryaev was invited to work part-time at the Department of Probability Theory of Moscow University, of which he is currently (since 1996) the head. At the Steklov Institute his scientific activity began with a joint research with V. P. Leonov in the nonlinear theory of random processes, based on the study of higher moments and semi-invariants. In the paper “On the technique of calculating semi-invariants” (Teor. Ver. Prilozhen., 1959), co-authored with Leonov, the higher moments
$$
\begin{equation*}
m_X^{(\nu)}=\mathsf{E} X_{t_1}^{\nu_1}\cdots X_{t_n}^{\nu_n}
\end{equation*}
\notag
$$
of random processes $X=(X_t)_{t\geqslant0}$ were examined and their connection with the corresponding semi-invariants, which are simpler in structure, was established.
The formulae so obtained were applied to find the moment and semi-invariant characteristics of nonlinear transformations $Y_t=Q(X_t)$, $t\geqslant0$. Shiryaev also obtained relevant results for spectral moments and semi-invariants and stated ergodicity conditions in these terms (previously, similar conditions were known only for Gaussian processes, for which all semi-invariants, starting from the third, are equal to zero). After results on semi-invariants, Kolmogorov involved Shiryaev in the planned work at the Steklov Institute on a topic related to the quickest detection of changes (‘disorders’) in the statistical structure of observed processes. The simplest ‘disorder problem’ can be formulated as follows. Let $X=(X_t)_{t\geqslant0}$ be an observable random process such that where $I(\,\cdot\,)$ is the indicator function, $B=(B_t)_{t\geqslant0}$ is a Brownian motion, and $\theta$ is the random time of an appearance of a ‘disorder’, which needs to be detected as soon as possible with a low probability of ‘false alarm’.The solution of a number of ‘disorder’ problems led Shiryaev to a series of works on statistical sequential analysis; a summary of the results obtained in these works and the general theory was presented in his two books: Statistical sequential analysis (Nauka, Moscow, 1969, 2nd ed.: 1976) and Stochastic problems of disorder (Moscow Center of Continuous Mathematical Education, Moscow, 2016). Both monographs were translated into English by Springer. In the Markov diffusion case these problems lead to the solution of Dirichlet–Stephan problems for differential equations. Unlike the Dirichlet problem, where the domain in which the equation operates is fixed and the required solution must satisfy a certain condition on its boundary, in Dirichlet–Stephan problems the domain in which the equation operates can change over time, and it is required to find not only the solution but also the ‘optimal’ domain. At the same time, Shiryaev found that the condition of ‘smooth pasting’ is often fulfilled at the boundaries of the optimal domain. The mathematical formulation of the disorder problem under consideration was as follows. Let
$$
\begin{equation*}
\mathsf{P}_\pi(\theta=0)=\pi, \qquad \mathsf{P}_\pi(\theta\geqslant t\,|\,\theta >0)=(1-\pi)e^{-t},
\end{equation*}
\notag
$$
and let the task under consideration be to find the Bayesian function
$$
\begin{equation}
V^*(\pi)=\inf_{\tau}[\mathsf{P}_\pi(\tau<\theta)+c\mathsf{E}_\pi(\tau-\theta)^+],
\end{equation}
\tag{1}
$$
where $\tau$ is the time of issuing a ‘warning’ signal, that is, a non-negative random variable measurable with respect to the process observed (the event $\{\tau\leqslant t\}$ belongs to the $\sigma$-algebra $\mathcal{F}_t^X=\sigma(X_s, s\leqslant t)$ for any $t\geqslant0$), $\mathsf{P}_\pi(\tau<\theta)$ is the false alarm probability, $\mathsf{E}_\pi(\tau-\theta)^+$ is the average delay time, and $c>0$ acts as a Lagrange multiplier.
Let $\pi_t=\mathsf{P}_\pi(\theta\leqslant t\,|\,\mathcal{F}_t^X)$. Shiryaev showed that
$$
\begin{equation}
V^*(\pi)=\inf_\tau \mathsf{E}_\pi\biggl[(1-\pi_\tau)+ c\int_0^\tau \pi_s\,ds\biggr],
\end{equation}
\tag{2}
$$
while the process $(\pi_t)_{t\geqslant0}$ satisfies the stochastic differential equation
$$
\begin{equation}
d\pi_t=\biggl(\lambda-\frac{\mu^2}{\sigma^2}\,\pi_t^2\biggr)(1-\pi_t)\,dt+\frac{\mu}{\sigma^2} \pi_t(1-\pi_t)\,dX_t.
\end{equation}
\tag{3}
$$
He also showed that the process $X=(X_t)_{t\geqslant0}$ admits the so-called innovation representation:
$$
\begin{equation*}
X_t=\mu\int_0^t \pi_s\,ds+\sigma \overline B_t,\qquad t\geqslant0,
\end{equation*}
\notag
$$
where the process $\overline B=(\overline B_t)_{t\geqslant0}$ is a Brownian motion relative to the flow of $\sigma$-algebras $\mathbb{F}^X=(\mathcal{F}_t^X)_{t\geqslant0}$. Thus, equation (3) can be written as
$$
\begin{equation*}
d\pi_t=\lambda(1-\pi_t)\,dt+\frac{\mu}{\sigma^2}\pi_t(1-\pi_t)\,d\overline B_t,
\end{equation*}
\notag
$$
from which it follows that the process $(\pi_t)_{t\geqslant0}$ has the infinitesimal operator
$$
\begin{equation*}
\mathcal{A}=\lambda(1-\pi)\,\frac{d}{d\pi}+ \frac{\mu^2}{\sigma^2}\pi(1-\pi)\,\frac{d^2}{d \pi^2}\,.
\end{equation*}
\notag
$$
Shiryaev established that $V^*(\pi)$ is the solution to the following Dirichlet–Stephan problem (with respect to the function $V^*(\pi)$ and boundary $A^*$):
$$
\begin{equation*}
\begin{alignedat}{2} \mathcal{A} V(\pi)&=-c\pi, &&\qquad 0<\pi<A, \\ V(\pi)&=1-\pi,&&\qquad \pi\geqslant A, \\ \frac{d V(\pi)}{d\pi}&=-1,&& \qquad \pi=A, \\ \frac{dV(\pi)}{d\pi}&\to0,&& \qquad \pi\downarrow 0. \end{alignedat}
\end{equation*}
\notag
$$
He showed that this problem admits a solution $(V^*(\pi),A^*)$, and the optimal stopping time $\tau^*$ has the form $\tau^*=\inf\{t\colon \pi_t\geqslant A^*\}$. Thus, for the class $\mathfrak{M}_\alpha=\{\tau\colon\mathsf{P}_\pi(\tau<\theta) \leqslant \alpha\}$ (the class of stopping times $\tau$ for which the false alarm probability does not exceed the specified value $\alpha$) the delay time is found: $R(\alpha,\lambda)= \mathsf{E}_0(\tau^*-\theta\,|\,\tau^*\geqslant\theta)$.
Equations like (3) prompted Shiryaev to derive filtering equations (for $\mathsf{E}(\theta_t\,|\,\mathcal{F}_t^X)$), interpolation equations (for $\mathsf{E}(\theta_s \,|\, \mathcal{F}_t^X)$ when $s\leqslant t$), and extrapolation equations (for $\mathsf{E}(\theta_s \,|\, \mathcal{F}_t^X)$ when $s\geqslant t$) in the case of processes $\theta=(\theta_t)_{t\geqslant0}$ generated by martingales and diffusion processes $X=(X_t)_{t\geqslant0}$. These equations formed the basis for Shiryaev’s investigations of issues related to linear and nonlinear filtering, parameter estimation, optimal control, and so on. A summary of this work was presented in his joint book with R. Sh. Liptser Statistics of random processes (Nauka, Moscow, 1974), which has had two editions in English (Springer, 1978, 2001). In that book the concepts of ‘weak’ solutions of stochastic differential equations and the concept of an ‘innovation’ process (for diffusion processes) were introduced, a multidimensional version of the Cameron–Martin theorem was derived, and conditions for the absolute continuity ($\widetilde{\mathsf{P}} \ll \mathsf{P}$) and singularity ($\widetilde{\mathsf{P}}\perp \mathsf{P}$) of two probability measures $\mathsf{P}$ and $\widetilde{\mathsf{P}}$ for diffusion processes were investigated. A significant part of Shiryaev’s work (with co-authors) was devoted to the issues of contiguity ($\widetilde{\mathsf{P}}^n\!\!\vartriangleleft\!\mathsf{P}^n$) and asymptotic separation ($\widetilde{\mathsf{P}}^n \!\!\vartriangle\! \mathsf{P}^n$) for two sequences of measures $(\mathsf{P}^n)_{n\geqslant1}$ and $(\widetilde{\mathsf{P}}^n)_{n\geqslant1}$. The main interest was in finding conditions for these properties which can be formulated in so-called ‘predictable’ terms. If the measures $\mathsf{P}^n$ and $\widetilde{\mathsf{P}}^n$ are given on filtered spaces $(\Omega^n,\mathcal{F}^n,(\mathcal{F}_t^n){t\geqslant0})$, then necessary and sufficient conditions for contiguity and asymptotic separation can be given in terms of the properties of the Hellinger processes $h(\alpha,\mathsf{P}^n,\widetilde{\mathsf{P}}^n)= (h_t(\alpha,\mathsf{P}^n,\widetilde{\mathsf{P}}^n))_{t\geqslant0}$. The criteria obtained in the cases of discrete and continuous time were included in two books: P. E. Greenwood and A. N. Shiryaev, Contiguity and the statistical invariance principle (Gordon & Breach, 1985); J. Jacod and A. N. Shiryaev, Limit theorems for stochastic processes (Springer, 1987 and the 2nd edition, 2003); the second monograph was later translated into Russian (Nauka, Moscow, 1994). These books also explored the question of the applicability of the statistical principle of invariance, which is well known in statistics but was considered in a more general setting. In particular, this principle states that for semimartingales $X^n=(X_t^n)_{t\geqslant0}$, $n\geqslant1$, the weak convergence of their laws $\operatorname{Law}(X^n\,|\,\mathsf{P}^n)$ to some probability measure $\mathsf{Q}$ under the assumption of contiguity $(\widetilde{\mathsf{P}}^n_t) \vartriangleleft (\mathsf{P}_t^n)$ for all $t\geqslant0$ also ensures the convergence $\operatorname{Law}(X^n\,|\,\widetilde{\mathsf{P}}^n)$ to a certain probability measure $\mathrm{\widetilde Q}$ such that $\mathrm{\widetilde Q} \ll \mathsf{Q}$. A large cycle of works by Shiryaev, in collaboration with Jacod, Lipster, L. I. Galtchouk, Yu. M. Kabanov, and others, was related to the development of the theory of weak convergence of semimartingales $X^n=(X_t^n)_{t\geqslant0}$ as $n\to\infty$. The consideration of the class of semimartingales is justified by the fact that this class is sufficiently ‘rich’: it includes important processes such as discrete-time ones, most processes with independent increments, martingales, many Markov processes, diffusion processes, solutions of stochastic differential equations, and so on. For semimartingales the concept of the triplet $(B,C,\nu)$ of predictable characteristics is defined, generalizing the characteristics of processes with independent increments (drift, Gaussian component variance, Lévy measure) and playing a fundamental role in the theory of semimartingales, particularly in questions of their weak convergence in the space $D$ (of right-continuous functions with left limits) endowed with a complete separable metric topology. Typically, the proof of functional limit theorems $X^n\xrightarrow{\operatorname{Law}} X$ follows the scheme below (Yu. V. Prohorov):
$$
\begin{equation*}
\begin{aligned} \, &\biggl[\begin{aligned} \, &\text{tightness of the sequence }\\ &\text{of distributions of $(X^n)_{n\geqslant1}$} \end{aligned}\biggr] \oplus \biggl[\begin{aligned} \, &\text{convergence of finite-dimensional}\\ &\text{distributions} \end{aligned}\biggr] \\ &\qquad\oplus\biggl[\begin{aligned} \, &\text{characterization of the process $X$ in}\\ &\text{terms of finite-dimensional distributions}\end{aligned}\biggr] \implies X^n \xrightarrow{\operatorname{Law}} X. \end{aligned}
\end{equation*}
\notag
$$
In the theory of weak convergence of semimartingales (developed by Jacod and Shiryaev) another scheme is used:
$$
\begin{equation*}
\begin{aligned} \, &\biggl[\begin{aligned} \, &\text{tightness of the sequence }\\ &\text{of distributions of $(X^n)_{n\geqslant1}$} \end{aligned}\biggr] \oplus\biggl[\begin{aligned} \, &\text{convergence of triplets}\\ &\text{$(B^n,C^n,\nu^n)$ to $(B,C,\nu)$} \end{aligned}\biggr] \\ &\qquad\oplus\biggl[\begin{aligned} \, &\text{characterization of the process $X$}\\ &\text{in terms of the triplet $(B,C,\nu)$} \end{aligned}\biggr] \implies X^n \xrightarrow{\operatorname{Law}} X. \end{aligned}
\end{equation*}
\notag
$$
In the book by Shiryaev and Jacod a systematic exposition related to the components of this second scheme was presented.
In 2006, the book by G. Peskir and A. N. Shiryaev Optimal stopping and free-boundary problems (Birkhäuser, Basel) was published, which, in addition to general results in the theory of optimal stopping rules, contains extensive material relating to solving specific optimal stopping problems of the type
$$
\begin{equation*}
V(x)=\sup_\tau \mathsf{E}_x \biggl[ M(X_\tau)+ \int_0^\tau L(X_t)\,dt+\sup_{t\leqslant\tau} K(X_t)\biggr],
\end{equation*}
\notag
$$
where $X=(X_t)_{t\geqslant0}$ is a Markov process and $\tau$ is a stopping time. The excessive characterization of the value function $V(x)$ is examined in detail, criteria for the validity of the conditions of ‘smooth pasting’ are stated, and a number of specific problems are solved, including situations where the stopping times are bounded ($\tau\leqslant T<\infty$).
A large cycle of Shiryaev’s works is related to financial mathematics. Many of his students have chosen this discipline as their main specialization. Shiryaev taught the very first course on financial mathematics at Moscow State University and involved many of his students in his seminars, especially the ones who were well-versed in the theory of martingales and stochastic analysis of random processes. In 1998, the two-volume monograph Essentials of stochastic finance (Fazis, Moscow; subsequent editions: Moscow Center of Continuous Mathematical Education, Moscow) was published, which serves as a desk book for students and all those involved in the application of mathematical methods to problems in financial mathematics and financial engineering. This monograph was been published three times in Russia and several times in English by World Scientific (1st ed., 1999). The first part of that monograph presents ‘facts and models’ of financial mathematics. The second part outlines the theory of arbitrage and calculations. The so-called extended version of the first fundamental theorem (Jacod and Shiryaev) is provided, various characterizations of the absence of arbitrage are given, and the calculations of the price of various options are performed (in particular, of the ‘Russian option’, introduced by L. Shepp and Shiryaev). The book Change of time and change of measure (World Scientific, 2010, and 2015), written by Shiryaev in collaboration with O. E. Barndorff-Nielsen and based on lectures given by the authors at Moscow University and at the universities of Aarhus (Denmark), Halmstad (Sweden), and Barcelona (Spain), had two editions. It is dedicated to the applications of time and measure change to problems of arbitrage, hedging, and calculation of rational prices in financial mathematics and financial engineering. Shiryaev pays great attention to pedagogical and organizational work. He has been the head of the Department of Probability Theory of the Faculty of Mechanics and Mathematics at Moscow State University since 1996. Under his supervision 70 students defended their Ph.D. theses, and more than 15 of them became Doctors of Sciences. His students worked and continue to work in leading universities in Russia and also Australia, Bulgaria, the United Kingdom, Germany, Georgia, Israel, Canada, the USA, Uzbekistan, Uruguay, Finland, and Sweden. Shiryaev has delivered many general and specialized lecture courses. His two-volume university textbook on probability theory (Probability-1 and Probability-2) was reprinted seven times in Russia (the first single-volume edition was published in 1980), three editions of the textbook were published in English by Springer, and the book was also been translated into German and Chinese. This textbook has an associated problem book (Problems in probability theory, Moscow Center of Continuous Mathematical Education, Moscow, 2006) and a solution book (Probability in theorems and problems (with proofs and solutions), Moscow Center of Continuous Mathematical Education, Moscow, 2013), co-authored by Shiryaev with I. G. Ehrlich and P. A. Yaskov. This problem book was translated into English (Springer, 2012). Shiryaev actively participated in organizing Soviet-Japanese symposia on probability theory, and his energy and mathematical versatility contributed greatly to the success of the First World Congress of the Bernoulli Society (Tashkent, 1986). He dedicates much effort to organizing and conducting annual international conferences on stochastic methods in Divnomorskoe (preparations are under way for the 10th anniversary conference, dedicated to the 90th anniversary of the organization, by Kolmogorov, of the Department of Probability Theory at Moscow State University). Shiryaev took the initiative and inspired the organization of various scientific schools (many people remember the international symposium “Leaders and their students”, held at the Steklov Institute in 2010). He gave talks at many conferences both in Russia and abroad. In 1978 Shiryaev was a plenary speaker at the International Congress of Mathematicians in Helsinki. He was a forum speaker at international conferences in Brighton (UK), Sydney, Beijing, and Ulm. Shiryaev is a member of the European Academy (since 1990), an honorary member of the Royal Statistical Society of the United Kingdom (since 1985), and served as the president of the Russian Society of Actuaries in 1994–1995. He is a honourary doctor of Albert Ludwig University of Freiburg (Germany) and the University of Angers (France), as well as a honourary professor at the University of Amsterdam (Netherlands). He was awarded the titles of Honoured Researcher of the Russian Federation and Honoured Professor of Moscow State University. His international activities have been recognized by his colleagues — he was the president of the Bernoulli Society for Mathematical Statistics and Probability (1989–1991) and the president of the Bachelier Finance Society (1998–1999). Shiryaev received the A. A. Markov Prize (1974) for a cycle of works on stochastic equations of Markov processes, the A. N. Kolmogorov Prize (1994) for the cycle of works “The Kolmogorov ’disorder’ problem, methods of its solution, and their development”, and the P. L. Chebyshev Gold Medal for his outstanding achievements in the field of mathematics (2017). He is also a laureate of the A. Humboldt Prize (1996) and A. Wald International Prize (2011). Shiryaev is a member of the dissertation councils at the Steklov Institute and Moscow State University, as well as a member of the academic council of the Faculty of Mechanics and Mathematics at Moscow University. He is the editor-in-chief of the journal Teoriya Veroyatnostei i ee Primeneniya1 and a member of the editorial boards of several international journals, including Finance and Stochastics, Quantitative Finance, Markov Processes and Related Fields and the journal Vestnik Moskovskogo Universiteta. Ser. 1. Matematkia, Mekhanika.2 One of Shiryaev’s undeniable achievements is his editorial and publishing activity in the preservation of the memory of Andrey Nikolaevich Kolmogorov. Selected works of Kolmogorov and books of memoirs about him were published; in 2003, on the centenary of Kolmogorov’s birth, the three-volume set Kolmogorov: Truth is good (biobibliography), These running lines… (from the correspondence between Kolmogorov and P. S. Alexandroff), and Quiet echo of the heart (from diaries) was released (all in Russian). And this work continues. On 5 February, 2024, A. N. Shiryaev was awarded a Certificate of Merit from the President of the Russian Federation for his merits in the development of national science, many years of fruitful activity, and in connection with the 300th anniversary of the founding of the Russian Academy of Sciences. Colleagues, friends, and students wish Albert Nikolaevich health, long creative life, and inexhaustible energy in all of his endeavors. 
Citation in format AMSBIB
\Bibitem{BulGusZhi25} |
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