Tanaka Formula Research Articles

Measure-Valued Flows Given Consistent Exchangeable Families

Given a Polish space E and a consistent exchangeable family of all k-point motions in E, a universal framework on related measure-valued flow (MVF) describing how probabilities evolve under the consistent exchangeable family is given. And when E=R 1 and 1-point motion is a 1-dimensional diffusion, local time and Tanaka formula for MVFs are studied.

A random walk analogue of Lévy’s Theorem

In this paper we will give a simple symmetric random walk analogue of Lévy’s Theorem. We will give a new definition of a local time of the simple symmetric random walk. We apply a discrete Itô formula to some absolute value like function to obtain a discrete Tanaka formula. Results in this paper rely upon a discrete Skorokhod reflection argument. This random walk analogue of Lévy’s theorem was already obtained by G. Simons ([14]) but it is still worth noting because we will use a discrete stochastic analysis to obtain it and this method is applicable to other research. We note some connection with previous results by Csáki, Révész, Csörgő and Szabados. Finally we observe that the discrete Lévy transformation in the present version is not ergodic. Lastly we give a Lévy-type theorem for simple nonsymmetric random walk using a discrete bang-bang process.

On the Weighted Local Time and the Tanaka Formula for the Multidimensional Fractional Brownian Motion

We determine the weighted local time for the multidimensional fractional Brownian motion from the occupation time formula. We also discuss on the Itô and Tanaka formula for the multidimensional fractional Brownian motion. In these formulas the Skorohod integral is applicable if the Hurst parameter of fractional Brownian motion is greater than 1/2. If the Hurst parameter is less than 1/2, then we use the Skorohod type integral introduced by Nualart and Zakai for the stochastic integral and establish the Itô and Tanaka formulas.

Local Time and a Tanaka Formula for a Class of Discontinuous Measure-Valued Branching Processes

We define the local time of a discontinuous superprocess X in ℝ d with symmetric α-stable motions and (1 + β)-branching as an L 1-limit of approximating local times ∫0 t Xs (ϕ ϵ)ds, where {ϕϵ} is a sequence of smooth functions converging to δ0 in distributional sense as ϵ → 0. We prove that the limit Lt 0 exists if d < (1 + 1/β)α and does not depend of the particular choice of sequence {ϕϵ}. We show that Lt 0 admits a Tanaka formula-like representation which includes a term that incorporates the discontinuities of X. Fleischmann [10] proved that the occupation measure induced by X is absolutely continuous with respect to Lebesgue measure; using the Tanaka formula we give an easy proof of Fleischmann's result.

Local time and Tanaka formula for a multitype Dawson–Watanabe superprocess

AbstractIn [3] Dynkin defined the local time of a continuous superprocess as a stochastic integral and gave a criterion for existence of local time. Here we prove that the conditions in Dynkin's existence criterion are satisfied by the multitype Dawson–Watanabe superprocess, and give a Tanaka formula‐like representation of the local time which is used to show that the occupation measure of the multitype superprocess is absolutely continuous with respect to an appropriate reference measure, and that the corresponding density coincides a.s. with the local time. (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

On The Local Times of Fractional Ornstein–Uhlenbeck Process

In this short note, we study the local times of the fractional Ornstein–Uhlenbeck process XH with Hurst index 1/2<H<1 solving the Langevin equation with fractional noise $$ \hbox{d}X_{t}^{H} = -X_{t}^{H} \hbox{d}t + \nu \hbox{d}B_{t}^{H}, \quad X_{0}^{H} = x, $$ where ν > 0 and BH is a fractional Brownian motion with Hurst index 1/2<H<1. We give Tanaka formula for the process and some properties of local times.

Constrained Stochastic LQ Control with Random Coefficients, and Application to Portfolio Selection

This paper is devoted to the study of a stochastic linear-quadratic (LQ) optimal control problem where the control variable is constrained in a cone, and all the coefficients of the problem are random processes. Employing Tanaka's formula, optimal control and optimal cost are explicitly obtained via solutions to two extended stochastic Riccati equations (ESREs). The ESREs, introduced for the first time in this paper, are highly nonlinear backward stochastic differential equations (BSDEs), whose solvability is proved based on a truncation function technique and Kobylanski's results. The general results obtained are then applied to a mean-variance portfolio selection problem for a financial market with random appreciation and volatility rates, and with short-selling prohibited. Feasibility of the problem is characterized, and efficient portfolios and efficient frontier are presented in closed forms.

On Tanaka formulae for (α, d, β)-superprocesses

In this paper, Tanaka formulae for ( α , d , β )-superprocesses in the dimensions where the local time exists are established under the optimal initial condition.

Skorohod integration and stochastic calculus beyond the fractional Brownian scale

We extend the Skorohod integral, allowing integration with respect to Gaussian processes that can be more irregular than any fractional Brownian motion. This is done by restricting the class of test random variables used to define Skorohod integrability. A detailed analysis of the size of this class is given; it is proved to be non-empty even for Gaussian processes which are not continuous on any closed interval. Despite the extreme irregularity of these stochastic integrators, the Skorohod integral is shown to be uniquely defined, and to be useful: an Ito formula is established; it is employed to derive a Tanaka formula for a corresponding local time; linear additive and multiplicative stochastic differential equations are solved; an analysis of existence for the stochastic heat equation is given.

Tanaka Formula for Multidimensional Brownian Motions

We study the Tanaka formula for multidimensional Brownian motions in the framework of generalized Wiener functionals. More precisely, we show that the submartingale U(Bt−x) is decomposed in the sence of generalized Wiener functionals into the sum of a martingale and the Brownian local time, U being twice of the kernel of Newtonian potential and Bt the multidimensional Brownian motion. We also discuss on an aspect of the Tanaka formula for multidimensional Brownian motions as the Doob–Meyer decomposition.

Tanaka Formulae for ( , d, β)-Superprocesses

We establish Tanaka like formulae for the local time of (α, d, β)-superprocess in the dimensions where the local time exists. The result generalizes the result of Adler, Lewin who proved existence of Tanaka formulae for a class of super-processes with finite variance. The fact that we abandon the finite variance assumption, requires using an L1+θ convergence argument (with 0<θ<β≤1) rather than L2 convergence, for the derivation of the Tanaka formulae.

Bornitude et continuité de la transformation de Lévy en analyse

In our previous papers (Adv. in Math. 138 (1) (1998) 182; Potential Anal. 12 (2000) 419), we have obtained a decomposition of | f | , where f is a function defined on R n , that is analogous to the one proved by H. Tanaka for martingales (the so-called “Tanaka formula”). More precisely, the decomposition has the form | f |= f ̃ +D ∗ 0( f ) , where D ∗ 0( f ) is (a variant of ) the density of the area integral associated with f. This functional (introduced by R.F. Gundy in his 1983 paper (The density of area integral, Conference on Harmonic Analysis in Honor of Antoni Zygmund. Wadsworth, Belmont, CA, 1983, pp. 138–149.)) can be viewed as the counterpart of the local time in Euclidean harmonic analysis. In this paper, we are interested in boundedness and continuity properties of the mapping f↦ f ̃ (which we call the Lévy transform in analysis) on some classical function or distribution spaces. As was shown in [4,5], the above (non-linear) decomposition is bounded in L p for every p∈[1,+∞[, i.e. one has || f ̃ || p⩽C p|| f|| p , where C p is a constant depending only on p. Nevertheless our methods (roughly speaking, the Calderón–Zygmund theory in [4], stochastic calculus and martingale inequalities in [5]) both gave constants C p whose order of magnitude near 1 is O(1/( p−1)). The aim of this paper is two-fold: first, we improve the preceding result and we answer a natural question, by proving that the best constants C p are bounded near 1. Second, we prove that the Lévy transform f↦ f ̃ is continuous on the Hardy spaces H p with p> n/( n+1).

An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter

We consider fractional Brownian motions B t H with arbitrary Hurst coefficients 0< H<1 and prove the following results: (i) An integral representation of the fractional white noise as generalized Wiener integral; (ii) an Itô formula for generalized functionals of B t H ; (iii) an analogue of Tanaka's formula; (iv) a Clark–Ocone formula for Donsker's delta function of B t H ; (v) an integral representation of the local time of B t H .

Tanaka formula for the fractional Brownian motion

In this paper we find the Wiener chaos expansion for the local time of the fractional Brownian motion with Hurst parameter H and we derive a Tanaka formula in the case H> 1 3 . As an application we deduce an Itô's formula for convex functions.

On Some Inequalities of Local Times for Azéma Martingales

Let ( u t,cal G t) t ≥0 be Azema martingale and its filtration, and let ( λ t x;x∈R,t≥0) be the local times of the Azema martingale defined by the following Tanaka formula: u t 1 { u t>x}=∫ 0 t1 { u s - >x}du s+1 2 λ t x. Then, for every ( cal G t) t ≥0 stopping time T and every p>0, there exist two universal constants cp, Cp >0 depending only on p, such that c p ∥T 1 /2∥ p≤∥λ T *∥ p≤C p∥T 1 /2∥ p , where λ t * =sup x ∈Rλ t x .

Mouvement brownien et formule de Tanaka en analyse

In our previous paper [5], we have obtained a decomposition of ∣f∣, where f is a function defined on Rd, that is analogous to the one proved by H. Tanaka in the early sixties for Brownian martingales (the so-called ‘Tanaka formula’). The original proofs use purely analytic methods (e.g. the Calderon–Zygmund theory, etc.). In this paper, we give a new proof of our `Tanaka formula in analysis’, that is based on probabilistic arguments. The main tools here are Brownian motion, stochastic calculus and Burkholder–Gundy inequalities for martingales. These methods allow us to improve somewhat our previous results, by proving that some significant constants do not depend on the dimension d.

Closed-Form Equations for Room Smoke Filling During an Initial Fire.

Closed-form equations were derived to calculate the smoke layer height and temperature in the room of fire origin. First of all, a conventional zone type model was introduced and verified against the full-scale experimental data. Then, by using the calculation model, virtual experimental data were created varying the building design parameters such as fire growth rate, floor area and ceiling height. Regression equation was derived for smoke layer temperature using the virtual experimental data that correlates the dependence on building design parameters. As to the smoke layer height, an analytical formula (Tanaka's formula) was verified against virtual experimental data. This type of simple, hand-calculable formula would be useful for practitioners in fire safety engineering design of building for evacuation safety from smoke.

Local Sojourn Time of Diffusion and Degenerating Processeson a Mobile Surface

The existence of the local sojourn time on the surface is established for multi-dimensional Ito processes, and equations are derived for probability distributions. An explicit formula of the type of the Tanaka formula is obtained for local time. Local time continuity is established. The limiting properties of the local time are investigated for degenerating diffusion.

Optimal stopping and maximal inequalities for geometric Brownian motion

Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτE(max0≤t≤τXt−cτ), whereX= (Xt)t≥0is geometric Brownian motion with drift μ and volatility σ &gt; 0, and the supremum is taken over all stopping times forX. The payoff is shown to be finite, if and only if μ &lt; 0. The optimal stopping time is given by τ*= inf {t&gt; 0 |Xt=g*(max0≤t≤sXs)} wheres↦g*(s) is themaximalsolution of the (nonlinear) differential equationunder the condition 0 &lt;g(s) &lt;s, where Δ = 1 − 2μ / σ2andK= Δ σ2/ 2c. The estimate is establishedg*(s) ∼ ((Δ − 1) /KΔ)1 / Δs1−1/Δass→ ∞. Applying these results we prove the following maximal inequality:where τ may be any stopping time forX. This extends the well-known identityE(supt&gt;0Xt) = 1 − (σ2/ 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.

Optimal stopping and maximal inequalities for geometric Brownian motion

Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτ E (max0≤t≤τ X t − c τ), where X = (X t ) t≥0 is geometric Brownian motion with drift μ and volatility σ &amp;gt; 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, if and only if μ &amp;lt; 0. The optimal stopping time is given by τ* = inf {t &amp;gt; 0 | X t = g * (max0≤t≤s X s )} where s ↦ g *(s) is the maximal solution of the (nonlinear) differential equation under the condition 0 &amp;lt; g(s) &amp;lt; s, where Δ = 1 − 2μ / σ2 and K = Δ σ2 / 2c. The estimate is established g *(s) ∼ ((Δ − 1) / K Δ)1 / Δ s 1−1/Δ as s → ∞. Applying these results we prove the following maximal inequality: where τ may be any stopping time for X. This extends the well-known identity E (sup t&amp;gt;0 X t ) = 1 − (σ 2 / 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.