Ciprian Tudor, Khalifa Es-Sebaiy
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.
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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.
Let $B^{H, K}= \big\{B^{H, K}(t), t \in \R_+ \big\}$ be a bifractional Brownian motion in $\R^d$. We prove that $B^{H, K}$ is strongly locally nondeterministic. Applying this property and a stochastic integral representation of $B^{H, K}$, we establish Chung's law of the iterated logarithm for $B^{H, K}$, as well as sharp Hölder conditions and tail probability estimates for the local times of $B^{H, K}$. We also consider the existence and the regularity of the local times of multiparameter bifractional Brownian motion $B^{\bar{H}, \bar{K}}= \big\{B^{\bar{H}, \bar{K}}(t), t \in \R^N_+ \big\}$ i…
Our aim in this paper is to improve Hölder continuity results for the bifractional Brownian motion (bBm) $(B^{α,β}(t))_{t\in[0,1] }$ with $0<α<1$ and $0<β\leq 1$. We prove that almost all paths of the bBm belong (resp. do not belong) to the Besov spaces $\mathbf{Bes}(αβ,p)$ (resp. $\mathbf{bes}(αβ,p)$) for any $\frac{1}{αβ}<p<\infty$, where $\mathbf{bes}(αβ,p)$ is a separable subspace of $\mathbf{Bes}(αβ,p)$. We also show the Itô-Nisio theorem for the bBm with $αβ>\frac{1}{2}$ in the Hölder spaces $\mathcal{C}^γ$, with $γ<αβ$.
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).
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…
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.
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 …
Stochastic calculus with respect to fractional Brownian motion (fBm) has attracted a lot of interest in recent years, motivated in particular by applications in finance and Internet traffic modeling. Multifractional Brownian motion (mBm) is a Gaussian extension of fBm that allows to control the pointwise regularity of the paths of the process and to decouple it from its long range dependence properties. This generalization is obtained by replacing the constant Hurst parameter H of fBm by a function h(t). Multifractional Brownian motion has proved useful in many applications, including the ones…
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.
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.
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…
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 purpose of this paper is to establish the convergence in law of the sequence of "midpoint" Riemann sums for a stochastic process of the form f'(W), where W is a Gaussian process whose covariance function satisfies some technical conditions. As a consequence we derive a change-of-variable formula in law with a second order correction term which is an Itô integral of f''(W) with respect to a Gaussian martingale independent of W. The proof of the convergence in law is based on the techniques of Malliavin calculus and uses a central limit theorem for q-fold Skorohod integrals, which is a multi…
We develop a stochastic analysis for a Gaussian process $X$ with singular covariance by an intrinsic procedure focusing on several examples such as covariance measure structure processes, bifractional Brownian motion, processes with stationary increments. We introduce some new spaces associated with the self-reproducing kernel space and we define the Paley-Wiener integral of first and second order even when $X$ is only a square integrable process continuous in $L^2$. If $X$ has stationary increments, we provide necessary and sufficient conditions so that its paths belong to the self-reproducin…
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…
A Meyer-Itô Formula for Stable Processes via Fractional Calculus
本文通过分数阶微积分计算一维严格稳定过程无穷小生成元的逆，提出具有非零局部时间项的 Meyer-Itô 定理，推广了 Tanaka 公式并给出非对称常返稳定过程的半鞅分解。
Stochastic integration with respect to Gaussian processes, such as fractional Brownian motion (fBm) or multifractional Brownian motion (mBm), has raised strong interest in recent years, motivated in particular by applications in finance, Internet traffic modeling and biomedicine. The aim of this work to define and develop, using White Noise Theory, an anticipative stochastic calculus with respect to a large class of Gaussian processes, denoted G, that contains, among many other processes, Volterra processes (and thus fBm) and also mBm. This stochastic calculus includes a definition of a stocha…
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