Starting from the potential theoretic definition of the local times of a Markov process - when these exist - we obtain a Tanaka formula for the local times of symmetric Lévy processes. The most interesting case is that of the symmetric $\al$-stable Lévy process (for $\al\in[1,2]$) which is studied in detail. In particular, we determine which powers of such a process are semimartingales. These results complete, in a sense, the works by K. Yamada \cite{yamada02} and Fitzsimmons and Getoor \cite{fitzsimmonsgetoor92a}.
Using the Malliavin calculus with respect to Gaussian processes and the multiple stochastic integrals we derive Itô's and Tanaka's formulas for the $d$-dimensional bifractional Brownian motion.
The objective of this paper is to study the local time and Tanaka formula of symmetric $G$-martingales. We introduce the local time of $G$-martingales and show that they belong to $G$-expectation space $L_{G}^{2}(Ω_{T})$. The bicontinuous modification of local time is obtained. We finally give the Tanaka formula for convex functions of $G$-martingales.
We prove the Ito-Tanaka formula and the existence of pathwise stochastic integrals for a wide class of Gaussian processes. Motivated by financial applications, we define the stochastic integrals as forward-type pathwise integrals introduced by Föllmer and as pathwise generalized Lebesgue-Stieltjes integrals introduced by Zähle. As an application, we illustrate the importance of Ito-Tanaka formula for pricing and hedging of financial derivatives.
In this paper, we shall introduce the Tanaka formula from viewpoint of the Doob-Meyer decomposition. For symmetric Lévy processes, if the local time exists, Salminen and Yor (2007) obtained the Tanaka formula by using the potential theoretic techniques. On the other hand, for asymmetric stable processes with index $α\in (1,2)$, we studied the Tanaka formula by using Itô's stochastic calculus and the Fourier analysis. In this paper, we study the Tanaka formula for asymmetric Lévy processes via the potential theoretic approach. We give several examples for important processes. Our approach also …
For symmetric Lévy processes, if the local times exist, the Tanaka formula has already constructed via the techniques in the potential theory by Salminen and Yor (2007). In this paper, we study the Tanaka formula for arbitrary strictly stable processes with index $α\in (1,2)$ including spectrally positive and negative cases in a framework of Itô's stochastic calculus. Our approach to the existence of local times for such processes is different from Bertoin (1996).
The functional Ito formula, firstly introduced by Bruno Dupire for continuous semimartingales, might be extended in two directions: different dynamics for the underlying process and/or weaker assumptions on the regularity of the functional. In this paper, we pursue the former type by proving the functional version of the Meyer-Tanaka Formula. Following the idea of the proof of the classical time-dependent Meyer-Tanaka formula, we study the mollification of functionals and its convergence properties. As an example, we study the running maximum and the max-martingales of Yor and Obloj.
We construct superprocesses with dependent spatial motion (SDSMs) in Euclidean spaces $R^d$ with $d\ge1$ and show that,even when they start at some unbounded initial positive Radon measure such as Lebesgue measure on $R^d$, their local times exist when $d\le3$. A Tanaka formula of the local time is also derived.
In this paper, we study the notion of local time and Tanaka formula for the G-Brownian motion. Moreover, the joint continuity of the local time of the G-Brownian motion is obtained and its quadratic variation is proven. As an application, we generalize It^o's formula with respect to the G-Brownian motion to convex functions.
In the setting of finite reflection groups, we prove that the projection of a Brownian motion onto a closed Weyl chamber is another Brownian motion normally reflected on the walls of the chamber. Our proof is probabilistic and the decomposition we obtain may be seen as a multidimensional extension of Tanaka's formula for linear Brownian motion. The paper is closed with a description of the boundary process through the local times at zero of the distances from the initial process to the facets.
Let $B_t$ be a one dimensional Brownian motion, and let $α'$ denote the derivative of the intersection local time of $B_t$ as defined in Jay Rosen's work (see references). The object of this paper is to prove the following formula $(1/2)α'_t(x) + (1/2)sgn(x)t = \int_0^t L_s^{B_s - x}dB_s - \int_0^t sgn(B_t - B_u - x) du$ which was given as a formal identity by Rosen without proof.
The derivative of self-intersection local time (DSLT) for Brownian motion was introduced by Rosen and subsequently used by others to study the $L^2$ and $L^3$ moduli of continuity of Brownian local time. A version of the DSLT for fractional Brownian motion (fBm) has also appeared in the literature. In the present work, we further its study by proving a Tanaka-style formula for it. In the course of this endeavor we provide a Fubini theorem for integrals with respect to fBm. The Fubini theorem may be of independent interest, as it generalizes (to Hida distributions) similar results previously se…
Three concepts of local times for deterministic c{à}dl{à}g paths are developed and the corresponding pathwise Tanaka--Meyer formulae are provided. For semimartingales, it is shown that their sample paths a.s. satisfy all three pathwise definitions of local times and that all coincide with the classical semimartingale local time. In particular, this demonstrates that each definition constitutes a legit pathwise counterpart of probabilistic local times. The last pathwise construction presented in the paper expresses local times in terms of normalized numbers of interval crossings and does not de…
Following a hedging based approach to model free financial mathematics, we prove that it should be possible to make an arbitrarily large profit by investing in those one-dimensional paths which do not possess local times. The local time is constructed from discrete approximations, and it is shown that it is $α$-Hölder continuous for all $α<1/2$. Additionally, we provide various generalizations of Föllmer's pathwise Itô formula.
Processes which arise as solutions to stochastic differential equations involving the local time (SDELTs), such as skew Brownian motion, are frequent sources of inspiration in theory and applications. Existence and uniqueness results for such equations rely heavily on the Itô-Tanaka formula. Recent interest in time-inhomogeneous SDELTs indicates the need for comprehensive existence and uniqueness results in the time-dependent case, however, the absence of a suitable time-dependent Itô-Tanaka formula forms a major barrier. Rigorously developing a two-parameter integral with respect to local tim…
In this short note we study homogenization of symmetric $d$-dimensional Lévy processes. Homogenization of one-dimensional pure jump Markov processes has been investigated by Tanaka \emph{et al.} in 1992; their motivation was the work by Benssousan \emph{et al.}\ from 1975 on the homogenization of diffusion processes in $\mathbb{R}^d$. We investigate a similar problem for a class of symmetric pure-jump Lévy processes on $\mathbb{R}^d$ and we identify -- using Mosco convergence -- the limit process.
These notes contains an introduction to the theory of Brownian and diffusion local time, as well as its relations to the Tanaka Formula, the extended Ito-Tanaka formula for convex functions, the running maximum process, and the theory of regulated stochastic differential equations. The main part of the exposition is very pedestrian in the sense that there is a considerable number of intuitive arguments, including the use of the Dirac delta function, rather than formal proofs. For completeness sake we have, however, also added a section where we present the formal theory and give full proofs of…
In the present paper, Ito formula and Tanaka formula for a special kind of symmetric random walk in the complex plain are studied. The random walk is called Parisian walk, and its local time is defined to be the number of exit from some regions.
We study a continuous pathwise local time of order p for continuous functions with finite p-th variation along a sequence of time partitions, for even integers p >= 2. With this notion, we establish a Tanaka-type change of variable formula, as well as Tanaka-Meyer formulae. We also derive some identities involving this high-order pathwise local time, each of which generalizes a corresponding identity from semimartingale theory. We then use collision local times between multiple functions of arbitrary regularity, to study the dynamics of ranked continuous functions of arbitrary regularity. We p…
The aim of this work is to define and perform a study of local times of all Gaussian processes that have an integral representation over a real interval (that maybe infinite). Very rich, this class of Gaussian processes, contains Volterra processes (and thus fractional Brownian motion), multifractional Brownian motions as well as processes, the regularity of which varies along the time. Using the White Noise-based anticipative stochastic calculus with respect to Gaussian processes developed in [Leb17], we first establish a Tanaka formula. This allows us to define both weighted and non-weighted…
We introduce two natural notions for the occupation measure of a function $V$ with finite variation. The first yields a signed measure, and the second a positive measure. By comparing two versions of the change-of-variables formula, we show that both measures are absolutely continuous with respect to Lebesgue measure. Occupation densities can be thought of as local times of $V$, and are described by a Meyer-Tanaka like formula.
The existence of self-intersection local time (SILT), when the time diagonal is intersected, of the $(α,d,β)$-superprocess is proved for $d/2<α$ and for a renormalized SILT when $d/(2+(1+β)^{-1})<α\leq d/2$. We also establish Tanaka-like formula for SILT.
We present an Ito's formula for the one-dimensional discrete-time quantum walk and give some examples including a Tanaka's formula by using the formula. Moreover we discuss integrals for the quantum walk.
The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka--Rosen--Yor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-i…
For $x\in R^d- \{0\}$, in dimension $d=3$, we study the asymptotic behavior of the local time $L_t^x$ of super-Brownian motion $X$ starting from $δ_0$ as $x \to 0$. Let $ψ(x)=((1/2π^2) \log (1/|x|))^{1/2}$ be a normalization, Theorem 1 implies that $(L_t^x-(1/2π|x|))/ψ(x)$ converges in distribution to a standard normal distributed random variable as $x\to 0$. For dimension $d=2$, Theorem 2 implies that $L^x_t-(1/π)\log(1/|x|)$ is $L^1$ bounded as $x\to 0$. To do this, we prove a Tanaka formula for the local time which refines a result in Barlow, Evans and Perkins.
We study a notion of local time for a continuous path, defined as a limit of suitable discrete quantities along a general sequence of partitions of the time interval. Our approach subsumes other existing definitions and agrees with the usual (stochastic) local times a.s. for paths of a continuous semimartingale. We establish pathwise version of the Itô-Tanaka, change of variables and change of time formulae. We provide equivalent conditions for existence of pathwise local time. Finally, we study in detail how the limiting objects, the quadratic variation and the local time, depend on the choic…
The classical Ray-Knight theorems for Brownian motion determine the law of its local time process either at the first hitting time of a given value a by the local time at the origin, or at the first hitting time of a given position b by Brownian motion. We extend these results by describing the local time process jointly for all a and all b, by means of stochastic integral with respect to an appropriate white noise. Our result applies to $μ$-processes, and has an immediate application: a $μ$-process is the height process of a Feller continuous-state branching process (CSBP) with immigration (L…
We give a proof of the replica symmetric formula for the free energy of the Sherrington-Kirkpatrick model in high temperature which is based on the TAP formula. This is achieved by showing that the conditional annealed free energy equals the quenched one, where the conditioning is given by an appropriate σ-field with respect to which the TAP solutions are measurable.
In this paper, we will prove that the local time of a Lévy process is of finite $p$-variation in the space variable in the classical sense, a.s. for any $p>2$, $t\geq 0$, if the Lévy measure satisfies $\int_{R\setminus \{0\}}(|y|^{3\over 2}\wedge 1)n(dy)<\infty$, and is a rough path of roughness $p$ a.s. for any $2<p<3$ under a slightly stronger condition for the Lévy measure. Then for any function $g$ of finite $q$-variation ($1\leq q <3$), we establish the integral $\int_{-\infty}^{\infty}g(x)dL_t^x$ as a Young integral when $1\leq q<2$ and a Lyons' rough path integral when $2\leq q<3$. We t…
We develop a formula for the power-law decay of various sets for symmetric stable random vectors in terms of how many vectors from the support of the corresponding spectral measure are needed to enter the set. One sees different decay rates in "different directions", illustrating the phenomenon of hidden regular variation. We give several examples and obtain quite varied behavior, including sets which do not have exact power-law decay.
In 1964 R.Gangolli published a Lévy-Khintchine type formula which characterised $K$ bi-invariant infinitely divisible probability measures on a symmetric space $G/K$. His main tool was Harish-Chandra's spherical functions which he used to construct a generalisation of the Fourier transform of a measure. In this paper we use generalised spherical functions (or Eisenstein integrals) and extensions of these which we construct using representation theory to obtain such a characterisation for arbitrary infinitely divisible probability measures on a non-compact symmetric space. We consider the examp…
We construct a pathwise integration theory, associated with a change of variable formula, for smooth functionals of continuous paths with arbitrary regularity defined in terms of the notion of $p$-th variation along a sequence of time partitions. For paths with finite $p$-th variation along a sequence of time partitions, we derive a change of variable formula for $p$ times continuously differentiable functions and show pointwise convergence of appropriately defined compensated Riemann sums. Results for functions are extended to regular path-dependent functionals using the concept of vertical d…
Institute of Mathematics, University of Zurich（苏黎世大学数学研究所）
We study the free energy of a most used deep architecture for restricted Boltzmann machines, where the layers are disposed in series. Assuming independent Gaussian distributed random weights, we show that the error term in the so-called replica symmetric sum rule can be optimised as a saddle point. This leads us to conjecture that in the replica symmetric approximation the free energy is given by a min max formula, which parallels the one achieved for two-layer case.
