FUKUSHIMA, MASATOSHI

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TitleA BRIEF SURVEY ON STOCHASTIC CALCULUS INMARKOV PROCESSES (Stochastic Analysis of JumpProcesses and Related Topics)Author(s)FUKUSHIMA, MASATOSHICitation数理解析研究所講究録 (2010), 1672: 191-197Issue Date2010-01URL http://hdl.handle.net/2433/141173RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityA BRIEF SURVEY ON STOCHASTIC CALCULUSIN MARKOV PROCESSESMASATOSHI FUKUSHIMA (Osaka)1 From A.N. Kolmogorov to P. L\’evy and K. It\^o[K] A. Kolmogorov, \"Uber die analytischen Methoden in der Wahrscheilichkeitsrechnung, Math.Ann. 104 (1931), 415-458For a Markov process $(X_{t}, P_{x})$ , define $P_{t}f(x)=E_{x}[f(X_{t})]$ .Then$u(t,)=P_{t}f(x) \llcorner sati^{q}\llcorner fies\frac{\partial x}{\partial t}u(t,x)=\mathcal{G}_{x}u(t,x)$,the Kolmogorov equationwhere $\mathcal{G}$ is the it generator of the transition semigroup $\{P_{t}\cdot, t\geq 0\}$ :$\mathcal{G}v(x)=\lim_{t\downarrow 0}\frac{P_{t}v(x)-v(x)}{t}$ .In a special case, $\mathcal{G}$ is a second order differential operator:$\mathcal{G}n(x)=\frac{1}{2}a(x)v’’(\prime \mathfrak{r}\cdot)+b(x)v’(x)$ .[L] P. L\’evy, Theorie de l’Addition des $\dagger/^{r_{CJ?\eta.(xble.\backslash }}-Al\acute{e}at_{0};?^{\backslash }es_{:}$ Gauthier-Villars, Paris, 1937[I.1] K. It\^o, On stochastic processes (infinitely divisible laws of probability), Japan. Journ.Math. XVIII (1942), 261-301L\’evy-It\^o decomposition of the sample path $X_{t}$ of a L\’evy processas a sum of a Gaussian process $X_{t}^{(1)}$ and an independent process $X_{t}^{(2)}$expressed using a Poisson random measure $J$ with intensity $ds|(dx)$ by$X_{t}^{(2)}=1 i_{l}nnarrow\infty(\int_{(0,t)\cross(1/n,1)}xJ((l\xi)-f\int_{1/n,1)}.l:\iota/(dx))+\int_{(0_{:}t)\cross(1,\infty)}xJ(d\xi)$$\Rightarrow$ L\’evy-Khinchin formula of $\varphi(z)=\log E[e^{izX_{1}}]$ :$-\{z, Az\}+i\langle\gamma,$ $z \rangle+\int_{|x|<1}(e^{i\langle z,x)}-1-\uparrow\{z.x))_{1}/(dx)+\int_{|x|\geq 1}(e^{i(z,x\rangle}-1)\nu(dx)$ .SDE and It\^o’s formula[I.2] K. It\’o, Differential equations deterinining a Ma.rkoff process (in Japanese), Joum. Pan-Japan Math. Coll. No. 1077(1942), 1352-1440;(in English) in Kiyosi It\^o Selected Papers, 42-75, Springer-Verlag, 1986[I.3] K. It\^o, On stochastic differential equations, $Mem$ . Amer. Math. Soc. 4(1951), 1-51[I.4] K. It\^o, On a formula $co$ncerning stochast $i($ . differeiitials, Nagoya Math. J. 3 (1951), 55-65Solution $X_{t}$ of SDE $dX_{t}=\sqrt{a(X_{t})}(dB_{f}+b(X_{f})dt$ and It\^o’s formula$v(X_{t})-v(X_{0})= \int_{0}^{t}v’(X_{s})\sqrt{a(X_{6})}dB_{s}+\int_{0}^{t}(\frac{1}{2}av’’+bv’)(X_{s})ds$yields $\mathcal{G}v=\frac{1}{2}av’’+bv’$ .1672 2010 191-197 1912 Books on SDE[I.5] K. It\^o, Probability Theory, Iwanami (in Japanese), 1952[I.7] K. It\^o, Lectures on Stochastic Processes, Tata Institute of Fundamental Research, Bombay1960[C] Ju-Gyoung Cho, Stochastic Integral Equations, Pyongyang, 1963[M] H.P. McKean, Jr., Stochastic Integrals, Academic Press, New York and London, 1969[IW] N. Ikeda and S. Watanabe, Stochasttc Differential Equations and Diffusion Processes,North-Holland/Kodansha, 1980During the period 1955-1965, the Japanese probability school led by It\^o was mostlyconcerned with the study of the one dimensional diffusion processes and its possibleextensions to more general Markov processes, while the Russian probability school ledby Dynkin was equally concerned with the theory of SDE initiated by It\^o and Gihman-Skorohod. Notably the drift transformation by G.Maruyama 1954, M. Motoo 1960, I.V.Girsanov 1960Contents of $Cho$ ’s bookChap.1 Basic concepts (measurable functions, conditional expectations, independence)Chap.2 Stochastic integrals ( $ba_{\sim}qed$ on L\’evy processes)Chap.3 Transformation formula for stochastic integralsChap.4 Existence theorems of the solutions of stochastic integral equations(Uniqueness and existence, Markov property of the solution and its generator, continuityand differentiability of the solution with respect to the initial position)Chap.5 Linear stochastic integral equationsChap.6 Stability in stochastic equationsExcerpt from Preface of Cho’s book(translated by Daehong Kim)In dealing with random phenomena, a principal feature of stochastic equation is inthat it describes the states of phenomena directly rather than describing them by meansof probability distributiOns indirectly. This resembles the classical differential equationswhich give direct expressions of the changes of states of deterministic phenomena. Dueto this common feature, stochastic equations can be actually regarded as extensions ofdifferential equations.In order to build up a theory of st,$o(hastic$ equations, we first need to introduce theconcept of the stochastic integral and study its properties, which will be the basic contentsof Chapters 2 and 3. $ln$ particular, the transformation formula of stochastic integralsderived in Chapter 3 is a new forinula that can not be found in any other ordinaryintegration theory and will play very import.ant roles throughout the first half of thepresent volume.3 One dimensional diffusions and general Markov pro-cesses[I.6] K. It\^o, Stochastic Processes I, II (in Japanese), Iwanami-Shoten, Tokyo, 1957; (Russiantranslation by A.D. Wentzell), Moscow, 1960, 1963(English translation by Yuji Ito) Essentials of Stochastic Processes, Translations of Mathematical192Monographs Vol.231, AMS, Providence, 2006[I.7] K. It\^o, Lectures on Stochastic Processes, Tata Institute, Bombay 1960[D] E.B. Dynkin, Foundations of the theory of Markov Processes (in Russioan), Moscow 1959[D] E.B. Dynkin, Markov Processes, Moscow 1962; English translation, Springer-Verlag, 1965[IM] K. It\^o and H. P. McKean, Jr., Diffust on Proce sses and Thet $r$ Sample Paths, Springer-Verlag,1965Intrinsic structures of the one-dimensional diffusion revealed by W. Fellera general theory on a Markov process and the stnectures of its additive functionalsIntrinsic generator of a one dimensional diffusion$X=(X_{t}, P_{x})$ : a diffusion (a continuous strong Markov process) on a one dimensionalregular open interval $I$ , quasi left continuous and of no killing inside. The generator $\mathcal{G}$ of$X$ admits the expression$\mathcal{G}u=\frac{d}{dm}\frac{d}{ds}u$where $s$ is a strictly increasing continuous function on $I$ and $m$ is a strictly increasingfunction on $I$ given by$P_{x}( \sigma_{a}<\sigma_{b})=\frac{s(x)-s(a)}{s(b)-s(a)}$ . $a<x<b$ . $m_{J}(x)=- \frac{dE_{x}[\tau_{J}]}{ds(x)},$ $x\in J,$ $\overline{J}\subset I$ .Feller’s saying: $s$ indicates the road map and $m$ indicates the speed of the diffusion travellerX.It\^o-McKean [IM] legitimates Feller’s saying in the following fashion:Let $X=(X_{t}, P_{x})$ be the Brownian motion on $R$ . In this case,$s(x)=2x,$ $m(x)=x$ . For a given strictly increasing function $m_{0}$ , define$A_{t}= \int_{R}\ell(t, .\epsilon)dn\tau_{0}(x)$ , $\tau_{t}=A_{t}^{-1}$ (1)where $\ell(t, x)$ is L\’evy’s local time of $X$ at $x\in R$ . Then the time changed process $Y=$$(X_{\tau_{t}}, P_{x})$ is a diffusion process on $R$ corresponding to$s(x)=2x,$ $m(x)=m_{0}(x)$ , that is to say, a time change of $X$ amounts a replacement ofthe measure $m$ keeping the road map $s$ invariant.The above defined functional $\{A_{t};t\geq 0\}$ is the most general expression of the positivecontinuous additive functional (PCAF in abbreviation) of the one-dimensional Brownianmotion. But such expression does not hold in general.Symmetry and Dirichlet form of a one dimensional diffusionLet $X$ be a diffusion on a regular $0$ ] $)en$ interval $I=(r_{1}, r_{2})$ as before.Then its generator is given by (1).Since the resolvent $\{G_{\alpha}\}$ of $X$ is known to have a symmetric density kernel with respectto the speed measure $m$ ([I.6]), $X$ is $\uparrow\gamma\not\supset-\iota*1^{n1Inet}$ ri $($ Define$\mathcal{E}^{(s)}(u. \iota)=\int\frac{du}{ds}\frac{dv}{ds}ds$ .193By making use of a detailed boundary behaviors of the resolvent proved in [I.6], anintegration by parts gives$- \int \mathcal{G}u\cdot vdm=\mathcal{E}^{(s)}(u.v)$ , $u=G_{o}f)v=G_{C\supset}g,$ $f,$ $g\in C_{0}(I)$Actually the Dinchlet form $(\mathcal{E}, \mathcal{F})$ of $X$ on $L^{2}(I;m)$ is given by$\mathcal{F}=$ $\{u\in L^{2}(I;m)$ : absolute]y continuous in $s$ ,$\mathcal{E}^{(s)}(u, u)<\infty$ . $tA(r_{t})=0$ , if $r_{t}$ is approachable $\}$ ,$\mathcal{E}(u, v)=\mathcal{E}^{(s)}(t1.1’)$ . $u,$ $v\in \mathcal{F}$ , (2)where $r_{i}$ is said to be approachable if $s$ has a finite limit there. $(\mathcal{E}, \mathcal{F})$ is a regular, stronglylocal Dirichlet form on $L^{2}(I;r\tau)$ . This identification is proved only recently inM. Fukushima, From one dimensional diffusions to symmetric Markov processes, a volume ‘Trib-ute to Professor Kiyosi It\^o’ of SPA, to appear.Thus the one dimensional absorbing diffusion on a regular open interval, its possiblestochastic transformation and its possible symmetric extensions can be handled entirelyin the framework of Dirichlet forms.4 From one dimensional diffusions to symmetric Huntprocesses$E$ : alocally compact separable metric space$X=(X_{t}, P_{x})$ : a Hunt process (a right continuous strong Markov process with quasi leftcontinuity) on $E$An extended real valued function $A_{t}(w)$ of $t\geq 0,$ $\omega\in\Omega$ , is called an additive functional(AFin abbreviation) of $X$ if it it right continuous in $t$ , has a left limit and $A_{s+t}(\omega)=$$A_{s}(\omega)+A_{t}(\theta_{s}\omega)$ . A $[0, \infty]$ -valued continuous AF is called a $PCAF$. The totality of PCAF’sof $X$ is denoted by $A_{c}^{+}$Let $X$ be symmetric with respect to a positive Radon measure $m$ on $E$ of full supportwith the associated Dirichlet from $(\mathcal{E}, \mathcal{F})$ ( $\mathcal{F}$ is the domain of the form $\mathcal{E}$ ) being regular.The totality of the smooth measures $(\sigma$-finite positive measures charging no set of zerocapacity) is denoted by $S$ .There is a one-to-one correspendence between $S$ and $A_{c}^{+}$ characterized by the relationthat $\mu\in S$ is the Revuz measure of $A\in A_{c}^{+}$ in the sense that$\lim_{t\downarrow 0}\frac{1}{t}E_{m}[\int_{0}^{t}f(X_{s})dA_{s}]=\int_{E}fd\mu$, $\forall f\in \mathcal{B}_{+}(E)$The Revuz measure of $A\in A_{c^{\backslash }}^{+}$ will be denoted $b\gamma\mu_{A}$ .Under the above setting, it was shown in[F] M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland/Kodansha, 1980that a time change of $X$ by a fully supported $A\in A_{c}^{+}$ amounts to a replacement of $m$with $\mu_{A}$ by keeping the (extended) Dirichlet form $\mathcal{E}$ invariant, generalizing the It\^o-McKeantheorem for the one-dimensional diHusion.1945Motoo-Watanabe theory on MAF of a Hunt pro-cessFor a general Hunt process $X$ on $E$ , an AF $\Lambda f_{t}$ of $X$ is said to be a martingale $AF(MAF$in abbreviation) if$E_{x}[M_{t}^{2}]<\infty$ , $E_{x}[\Lambda f_{t}]=0$ , $\forall t\geq 0,$ $x\in E$ .The totality of MAF’s is denoted by M. The structure of the space $\mathcal{M}$ was explored intwo papers, which marked a starting point of the modern theory of stochastic calculus inMarkov processes:[MW] M. Motoo and S. Watanabe, On a class of additive functionals of Markov processes, $J$.Math. Kyoto Univ. 4(1965), 429-469[W] S. Watanabe, On discontinuous additive functionals and L\’evy measures of a Markov process,Japanese J. Math. 34 (1964), 53-701. For any $M\in \mathcal{M}$ , there exists a unique PCAF $\{\Lambda f\rangle\in A_{c}^{+}$ such that $E_{x}[M_{t}^{2}]=E_{x}[\{M\}_{t}]$for any $t\geq 0,$ $x\in E$ . Let { $M,$ $L \rangle=\frac{1}{4}\{\langle M+L\rangle-\{M-L\rangle\}$ for $M,$ $L\in \mathcal{M}$ .2. For any $M\in \mathcal{M}$ and any function $f$ on $E$ with $E_{x}[ \int_{0}^{t}|f(X_{s})|d\{M\}_{s}]<\infty$ , thereexists a unique $f\cdot M\in \mathcal{M}$ such that$\langle f\cdot M,$ $L \}_{t}=\int_{0}^{t}f(X_{s})d(\Lambda f,$ $L\}_{s}$ $\forall t\geq 0$ . (3)$f\cdot M$ is called the stochastic integral.3. Any $M\in M$ adimts a unique decomposition $\Lambda f=M^{c}+M^{d},$ $M^{c}\in \mathcal{M}_{c},$ $M^{d}\in \mathcal{M}_{d}$ ,where$\mathcal{M}_{c}=$ { $M\in \mathcal{M}$ : $\lrcorner\# f_{t}$ is continuous}, $\mathcal{M}_{d}=\{M\in \mathcal{M}$ : $\{\Lambda f, L\rangle=0\forall L\in \mathcal{M}_{c}\}$4. A Levy system for $X$ is a pair $(N, H)$ of a kernel $N(x, dy)$ on $(E_{\triangle}, \mathcal{B}(E_{\Delta}))$ and a PCAF$H\in A_{c}^{+}$ such that$E_{x}[ \sum_{s\leq l}f(X_{s-}, X_{s})]=E_{\gamma}[\cdot/0^{t}(\int_{E_{\Delta}}f(X_{s}, y)N(X_{s}, dy))dH_{s}]$ (4)for any non-negative Borel function $f$ on $E\cross E$ vanishing on the diogonal.A L\’evy system exists.Any $M\in \mathcal{M}_{d}$ can be represented ut ing the L\’evy system as a difference of functionalsappearing in both sides of (4).6 Stochastic calculus for semi-martingales[KW] H. Kunita and S. Watanabe, On square integrable martingales, Nagoya Math. J. 30(1967),209-245195[M] P. A. Meyer, Int\’egrals stochastiques ( $\rfloor$ expos\’es), in Seminaire de Probabilites $I$, LNM 39,Springer, 1967, pp 72-162The space $\mathcal{M}$ of MAF’s of a Hunt process is replaced by the space of general squareintegrable martingales $M$ . Due to the Doob-Meyer decomposition theorem of a submartin-gale, $\langle M\rangle$ is well defined as a predi $\mathfrak{c}\cdot t,able$ inc$\cdot$reasing process and the stochastic integral isdefined analogously to (3).It\^o’s formula is established for a general semimartingale (local martingale $+$ process ofbounded variation). Semimaringale theory is further developed in[DM] C. Dellacherie and P. A. Meyer, $Prob_{l},b_{7}l;t\acute{e}sel$ Potentiel, Chap. I-IV, 1975; Chap. V- VIII,1980; Chap. IX-XI, 1983; Chap. XII-XVI, 1987, Hermann, Paris.7 Decomposition of AF of a symmetric Hunt processLet $X=(X_{t}, P_{x})$ be a Hunt process on $E$ synimetric with respect to a fully supportedpositive Radon measure $m$ on $E$ with the associated Dirichlet form $(\mathcal{E}, \mathcal{F})$ on $L^{2}(E;m)$being regular.For $u\in \mathcal{F}$ , the composite process $u(X_{f})$ is not necesarily a seinimartingale. Neverthe-less, it admits a unique decomposition ([F])$u(X_{t})-u(X_{0})=\lrcorner ll_{t}^{1\iota\iota]}+N_{t}^{1u]}$ , $t\geq 0$where $M^{1u]}$ is a MAF of finite energy aiid $N^{1u]}$ is a continuous AF of zero energy. Aenergy $e(A)$ of an AF is defined by $e(A)=\lim_{t\downarrow 0}\frac{1}{t}E_{m}[A_{t}^{2}]$ . A CAF of zero energy is notnecessarily of bounded variation but its quadratic variation is zero with respect to $P_{m}$ .8 Stochastic derivation of the Beurling-Deny formula[FOT] M. Fukushima, $Y$ , Oshima and M. Takeda, Symmetric Markov processes and Dinchlet$fo s$, Walter de Gruyter, 1994[FT] M. Fukushima and M. Takeda, Markov Processes, (in Japanese) Baifukan, 2008[CF] Z.Q. Chen and M. Fukushima, Symmetnc Markov Processes, Time Changes and BoundaryTheory, a Book ManuscriptAny $A\in A_{c}^{+}$ admits its Revuz nieasure $l^{4_{A}}\in S$ .Let $(N, H)$ be a L\’evy system of $X$ . Using the Revuz measure $\mu_{H}$ of $H\in A_{c}^{+}$ , definethe jumping measure and the killing meas $ure$ of $X$ by$J(dx, dy)=N(x, dy)_{l}\iota_{H}(d\tau)$ $t_{t}(dc)=N(x, \{\triangle\})_{l}\iota_{H}(dx)$For any $u\in \mathcal{F}$ , the following $identit_{V}$ holds:$\mathcal{E}(u. u)=\frac{1}{2}l_{\langle\Lambda/}^{l}|\tau’|\rangle(E)+\frac{1}{2}\int_{F_{\lrcorner}}u(x)^{2}\wedge’(dx)$ .Let $M^{[u]}=\Lambda f^{[u],c}+M^{|u|,d}$ be the Motoo-Watanabe decompostion of $\Lambda f^{[u]}\in \mathcal{M}$ and$\mathcal{E}^{(c)}(u, u)=\frac{1}{2}\mu_{(M)}|_{A}|\ulcorner(E)$ .196A computation of $\frac{1}{2}\mu_{\langle M\rangle}1\tau r|d(E)$ using the L\’evv syst,em formula (5) then yield$\mathcal{E}(u, u)=\mathcal{E}^{(c)}(u, u)+\frac{1}{2}\int_{E\cross E}(u(x)-u(y))^{2}J(dx, dy)+\int_{E}u(x)^{2}\kappa(dx)$ . (5)$\mathcal{E}^{(c)}(u, v)=\frac{1}{4}\{\mathcal{E}^{(c)}(u+v, u+v)-\mathcal{E}^{(c)}(u-v, u-v)\}$ has the strongly local property: if$u\cdot m$ has a compact support and $v$ is constant on a neighbourhood of it, then $\mathcal{E}(u, v)=0$ .The formula (5) was first announced by[BD] A. Beurling and J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A 45 (1959), 208-215and its analytic proof was given in 1974.L\’evy system is well defined for any Hunt process and special standard process as well.Revuz measure of a PCAF is well defined for any right process relative to any excessivemeasure. Any right process admits a weak dual moderate Markov process with respectto a given excessive measure.[FG] P. J. Fitzsimmons and R. K. Getoor, Smooth measures and continuous additive functionalsof right Markov processes. In $It\hat{o}$ ’s stochastic calculus and probobability theory. Springer, Tokyo,1996, $pp31- 49$has established the one-to-one Revuz correspondence between PCAF’s and smooth mea-sures ( $\sigma$-finite measures charging no semi-polar sets).Those suggest some possibility to extend the above calculus to non-symmetric Markovprocesses (Z. M. Ma et.al., G. Trutnau).197

A BRIEF SURVEY ON STOCHASTIC CALCULUS IN MARKOV PROCESSES (Stochastic Analysis of Jump Processes and Related Topics)

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A BRIEF SURVEY ON STOCHASTIC CALCULUS IN MARKOV PROCESSES (Stochastic Analysis of Jump Processes and Related Topics)

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