Characterization Of Brownian Motion Research Articles

Itô's rule and Lévy's theorem in vector lattices

The change of variable formula, or Itô's rule, is studied in a Dedekind complete vector lattice E with weak order unit E. Using the functional calculus we prove that for a Hölder continuous semimartingale Xt=Xa+Mt+Bt,t∈J, and a twice continuously differentiable function f, the formula(0.1)f(Xt)=f(Xa)+∫0tf′(Xs)dMs+∫0tf′(Xs)dBs+12∫0tf″(Xs)d〈M〉s,0≤s≤t∈J holds. The first integral in the formula is an Itô integral with reference to the local martingale M and the second and third integrals are Dobrakov-type integrals of a vector valued function with reference to a vector valued measure. Using the formula, we prove Lévy's characterization of Brownian motion as being a continuous martingale with compensator tE. The proof of this result yields a concrete description of abstract Brownian motion defined in vector lattices.

Kinetic description of quantum Brownian motion

We stress the relevance of the two features of translational invariance and atomic nature of the gas in the quantum description of the motion of a massive test particle in a gas, corresponding to the original picture of Einstein used in the characterization of Brownian motion. The master equation describing the reduced dynamics of the test particle is of Lindblad form and complies with the requirement of covariance under translations.

Martingales, detrending data, and the efficient market hypothesis

We discuss martingales, detrending data, and the efficient market hypothesis (EMH) for stochastic processes x( t) with arbitrary diffusion coefficients D( x, t). Beginning with x-independent drift coefficients R( t) we show that martingale stochastic processes generate uncorrelated, generally non-stationary increments. Generally, a test for a martingale is therefore a test for uncorrelated increments. A detrended process with an x-dependent drift coefficient is generally not a martingale, and so we extend our analysis to include the class of ( x, t)-dependent drift coefficients of interest in finance. We explain why martingales look Markovian at the level of both simple averages and 2-point correlations. And while a Markovian market has no memory to exploit and presumably cannot be beaten systematically, it has never been shown that martingale memory cannot be exploited in 3-point or higher correlations to beat the market. We generalize our Markov scaling solutions presented earlier, and also generalize the martingale formulation of the EMH to include ( x, t)-dependent drift in log returns. We also use the analysis of this paper to correct a misstatement of the ‘fair game’ condition in terms of serial correlations in Fama's paper on the EMH. We end with a discussion of Levy's characterization of Brownian motion and prove that an arbitrary martingale is topologically inequivalent to a Wiener process.

A characterization of brownian motion in a lipschitz domain by its killing distributions

Let (Y t, Qx) be a strong Markov process in a bounded Lipschitz domainD with continuous paths up to its lifetime ζ, and let (X t, Px) be a Brownian motion inD. IfY ζ− exists in ∂D andQ x(Yζ− ∈C)=Px(Xζ− ∈C) for all Borel subsetsC of ∂D and allx, thenY is a time change ofX.

On the Independence of Linear Functionals of Linear Processes

A class of continuous parameter random processes is defined whose members, $x(t)$, are integrals of a nonstationary random “filter” function, $h(t,\lambda )$, with respect to an independent increments process $Z(\lambda )$. Linear functionals l of $x(t)$ expressible as integrals of a random function $\phi (\lambda )$ with respect to $Z(\lambda )$ are considered. When $Z(\lambda )$ is Gaussian and $\phi _1 (\lambda )$ and $\phi _2 (\lambda )$ are nonrandom, the linear functionals $l_1 $ and $l_2 $ are independent if and only if they are uncorrelated. For general $Z(\lambda )$ and the cases in which $\phi _1 (\lambda )$ and $\phi _2 (\lambda )$ are independent or jointly Gaussian with zero mean, much stronger necessary and sufficient conditions are obtained. These conditions are applied to the Karhunen–Loéve expansion coefficients of an independent increments process to obtain a new characterization of Brownian motion. The conditions are also applied to a class of wide-sense stationary linear processes with rational spectral densities to obtain a characterization of the subclass of Gaussian processes. Finally, it is shown that most random sampling of band-limited white Gaussian noise produces dependent samples.

On a Characterization of Brownian Motion

Let $X(t)$ be a homogeneous random process with independent increments, and $a(t)$, $b(t)$ certain functions that are continuous in the closed interval a, b. Then the following theorem is true:If\[ \int_a^b {[a(t)b(t)]^2 dt \ne 0,} \] the integrals\[ Y = \int_a^b {a(t)dX(t)} {\textit{ and }}Z\int_a^b {b(t)dX(t)} \]are independent, and if at least one of the integrals\[ \int_a^b {\frac{{a^2 (t)}} {{b^2 (t)}}dt\,} ,\int_a^b {\frac{{b^2 (t)}} {{a^2 (t)}}dt\,} \]exists, then the process$X(t)$is a Brownian motion.A similar theorem is proved for infinite linear forms\[ \sum\limits_{k = 1}^\infty {a_k X_k } ,\sum\limits_{k = 1}^\infty {b_k X_k } .\]