On the Kolmogoroff equation (Q2725510)

This review is on one of the most famous mathematical manuscripts of theNEWLINElast century. It was written by the French soldier Wolfgang Doeblin [in his nativeNEWLINEGerman spelling Döblin], son of the German writer Alfred Döblin, in theNEWLINEwinter of 1939/40 at the front in eastern France [cf.\ \textit{B.~Bru}, ``UnNEWLINEhiver en campagne.'' C.~R. Acad. Sci., Paris, Sér.~I, Math.\ 331, Spec.\NEWLINEIss., 1037--1058 (2000; Zbl 1035.01519)]. Its publication is linked to aNEWLINEtruly tragic story. Wolfgang Doeblin, based in the Ardenne mountains withNEWLINEhis military unit, worked on this manuscript during the war phaseNEWLINEafter the German attack on Poland which is known in France as \textit{la drôle de guerre}.NEWLINEHe decided to send it as a sealed envelope (\textit{pli cacheté}) to theNEWLINEarchive of the Académie des Sciences in Paris. It was not the first oneNEWLINEhe deposited this way. Obviously he wanted to continue working on it and toNEWLINEpublish it after returning from a war he was expecting to be short. WhetherNEWLINEhe was aware of the significance of his findings, or only jealously triedNEWLINEto withhold them from possible competitors, has to be left to speculation.NEWLINEDoeblin never returned from this war. His unit in disorganization, he tookNEWLINEhis life on June~21, 1940, in a barn in Housseras in the Vosges mountains.NEWLINEHis manuscript was shut away in the archive for 60 years, and madeNEWLINEaccessible only in 2000 after his brother Stephan had agreed.NEWLINENEWLINENEWLINENEWLINEIts first sentence paraphrases its program: \textit{``We consider a movableNEWLINEparticle, travelling randomly on the line (or some part of it).''} DoeblinNEWLINEundertakes to lay the foundations of an area that we nowadays callNEWLINEstochastic analysis. He intends to construct a solution of theNEWLINEone-dimensional Kolmogorov equationNEWLINENEWLINE\[NEWLINE-v_t = a v_x + \frac{1}{2} \sigma^2 v_{xx},NEWLINE\]NEWLINENEWLINE(\(v_t, v_x\) denote the temporal or spatial derivative of theNEWLINEfunction \(v\) etc.) by detaching the description of its dynamics fromNEWLINEthe notions of continuum mechanics, and translating it into theNEWLINEterms of probability theory. As opposed to continuum mechanics, theyNEWLINEdeal with trajectories of idealized diffusing particles. DoeblinNEWLINEstarts the implementation of this program with the second sentence:NEWLINE\textit{``Imagine a well defined probability \(F(x,y;s,t)\) for theNEWLINEparticle, which at time \(s\) is in state \(x\), to be at time \(t\) \({(>s)}\)NEWLINEto the left of a state \(y\), a probability independent of theNEWLINEprevious history of the motion.''}NEWLINENEWLINENEWLINENEWLINENEWLINEDoeblin alludes to a notion whichNEWLINEfrom today's perspective could be interpreted as distributionNEWLINEfunction of a \textit{transition probability density} \(p(x,y; s,t)\),NEWLINEassociated by the relationship \(F(x,y; s,t) = \int_{-\infty}^yNEWLINEp(x,z; s,t) dz.\) The \textit{independence of the previous history}NEWLINEfinds an expression in the Chapman-Kolmogorov equation, aNEWLINErudimentary form of the Markov property. It states that for theNEWLINEfuture behaviour of the particle, viewed from the present, only itsNEWLINEcurrent state matters, and not the previous history of its motion.NEWLINETranslated into the terms of continuum mechanics, it provides anNEWLINEinterpretation of the coefficients \(a\) and \(\sigma\) of theNEWLINEKolmogorov equation.NEWLINENEWLINEAccording to the definition of \(F\) for states \(x\) and times \(s\) the functionNEWLINENEWLINE\[NEWLINEa(x,s) = \lim_{t\downarrow s} \frac{1}{t-s} \int_{|y-x|<1} (y-x) dNEWLINEF(x,y; s,t)NEWLINE\]NEWLINE depicts the infinitesimal average trend of the motion,NEWLINEthe \textit{drift of the diffusing particle}, and NEWLINE\[NEWLINE\sigma^2(x,s) = \lim_{t\downarrow s} \frac{1}{t-s} \int_{|y-x|<1} (y-x)^2 d F(x,y; s,t)NEWLINE\]NEWLINE its \textit{diffusion intensity}, or in contemporaryNEWLINEterms its \textit{volatility}.NEWLINENEWLINENEWLINENEWLINEBased on this interpretation Doeblin is the first one to propose aNEWLINE\textit{pathwise solution of Kolmogorov's equation}, an equation of stateNEWLINEthat one can translate into a stochastic integral equation in the senseNEWLINEof Itô's calculus. If \(X_t\) is the random position of the idealized particleNEWLINEat time \(t\), then according to Doeblin NEWLINE\[NEWLINEX_t = x + \beta_{H_t} + \int_0^tNEWLINEa(s, X_s) ds,NEWLINE\eqno(1)NEWLINE\]NEWLINE where \(\beta_{H_t}\) is a Brownian motionNEWLINE\(\beta\) running with the clock \(H_t = \int_0^t \sigma^2(s, X_s) ds\).NEWLINEOnly after the construction of the stochastic integral of theNEWLINEBrownian motion, Itô, about 5 years later, finds his version of thisNEWLINEfirst sample path solution of Kolmogorov's equation in the formNEWLINENEWLINE\[NEWLINEX_t = x + \int_0^t \sigma(s, X_s) dB_s + \int_0^t a(s, X_s)NEWLINEds.NEWLINE\eqno(2)NEWLINE\]NEWLINENEWLINENEWLINENEWLINENEWLINEThe translation between (1) andNEWLINE(2) is provided by a theorem on the time change ofNEWLINEBrownian motion proved in 1969 by Dubins and Schwarz. It states in aNEWLINEmore general form that a continuous local martingale \(M\) withNEWLINEquadratic variation \(\langle M\rangle\) can be considered asNEWLINE\(\beta_{\langle M\rangle}\), i.e., a Brownian motion \(\beta\), runningNEWLINEwith the clock \(\langle M\rangle\) that is intrinsic to \(M\). ApplyingNEWLINEthis theorem to the continuous local martingale \(\int_0^\cdotNEWLINE\sigma(s, X_s) dB_s\) with quadratic variation \(\int_0^\cdotNEWLINE\sigma^2(s, X_s) ds\), one just obtains Doeblin's version of theNEWLINEsolution of Kolmogorov's equation from Itô's. To find hisNEWLINEdescription of a pathwise solution of Kolmogorov's equation DoeblinNEWLINEtherefore circumvented the use of Itô's integral by anticipating aNEWLINElink between martingales and time changed Brownian motions that wasNEWLINEclarified only much later.NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINEViewed from a modern perspective, Doeblin obtains the most interesting andNEWLINEfar-reaching insight in section~XV of his manuscript. Here he discovers aNEWLINEversion of the fundamental theorem of stochastic analysis, today known asNEWLINEItô's formula, again by replacing the stochastic integral of Itô'sNEWLINEversion by a time changed Brownian motion running with the appropriateNEWLINEclock. Itô's version states for a function \(f\) which is continuouslyNEWLINEdifferentiable in the time variable and twice continuously differentiableNEWLINEin the space variable, and a pathwise solution \(X\) of Kolmogorov's equationNEWLINEgiven by (2)NEWLINENEWLINE\[NEWLINEf(t,X_t) = f(0,x) + \int_0^t [f_x \sigma](s, X_s) d B_s + \int_0^t \Bigl[f_x a + f_t + \frac{1}{2}NEWLINEf_{xx}\sigma^2\Bigr](s,X_s) ds. NEWLINE\]NEWLINENEWLINEIn his manuscript written about 5~years before Itô, Doeblin comes up withNEWLINEthe alternative description avoiding stochastic integralsNEWLINENEWLINE\[NEWLINEf(t,X_t) = f(0,x) + \beta_{H^f_t} + \int_0^t \Bigl[f_x a + f_t + \frac{1}{2} f_{xx}\sigma^2 \Bigr](s,X_s) ds,NEWLINE\]NEWLINENEWLINEwhere the Brownian motion \(\beta\) runs with the clockNEWLINENEWLINE\[NEWLINEH^f_t= \int_0^t [f_x \sigma]^2(s,X_s) ds.NEWLINE\]NEWLINENEWLINENEWLINENEWLINENEWLINEHow far ahead of his time Doeblin's \textit{pli cacheté}, as the paperNEWLINEhas come to be known, was can also beNEWLINEjudged from other observations. In one of its last sections he discusses aNEWLINEversion of a \textit{comparison theorem}, stating that trajectories of twoNEWLINEsolutions of Kolmogorov's equation with drift coefficients \(a_1 \leq a_2\)NEWLINEkeep the order prescribed by the drift coefficients. In the literature ofNEWLINEstochastic analysis this theorem is attributed to papers by \textit{T. Yamada} [J. Math. Kyoto Univ. 13, 497--512 (1973; Zbl 0277.60047)] and \textit{T. Yamada} and \textit{Y. Ogura} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 56, 3--19 (1981; Zbl 0468.60056)].NEWLINENEWLINENEWLINENEWLINEThe \textit{pli cacheté}NEWLINEconstitutes a milestone in stochastic analysisNEWLINEthat strangely remained insignificant for the development of the area, sinceNEWLINEit was hidden for 60~years.