Levy Area Research Articles

Strong 1.5 order scheme for second-order stochastic differential equations without Levy area

In this work, a finite difference scheme combined with a truncated spectral expansion of white noise is presented for solving second-order stochastic differential equations (SDEs) with additive white noise. Consistency of the approximate equation, which is obtained by replacing the white noise with its spectral truncation, is considered in mean-square sense. Based on it, a full discrete scheme is constructed and the mean-square convergence order of numerical solution is investigated. Numerical examples show that the presented scheme is of 1.5-order convergence in mean-square sense, which verifies the theoretical findings and is a half order higher than converting the equation to a two-dimensional system with additive white noise and using the Euler-Maruyama scheme to solve the resulting problem. It is also illustrated that the piecewise version of the spectral approximation of white noise plus proper time discretization works well for long-time simulation.

Acceleration of automatic differentiation of solutions to parabolic partial differential equations: a higher order discretization

The paper proposes a new automatic/algorithmic differentiation for the solutions to partial differential equations of parabolic type. In particular, we provide a higher order discretization scheme which is a natural extension of the standard automatic differentiation. A Brownian polynomial approach is introduced to avoid the Levy area simulation. The Lie brackets of vector fields associated with stochastic differential equation play an important role in the proposed scheme. The case that the test function is non-smooth but has Gateaux derivative is considered. Numerical examples are shown to confirm the effectiveness

Dividing the Expropriated Areas Based on Multi-Factor Comprehensive Evaluation—A Case Study of Gongcheng Yao Autonomous County

In recent years, with the rapid development of urbanization in China, land acquisition work is particularly important. The division of expropriated area is the primary factor to be considered when evaluating the comprehensive land price of expropriated area. Reasonable division can help to improve the scientific and applicability of the comprehensive land price of the region. How to maximize the protection of farmers’ rights and interests by the division of land requisition is an urgent problem to be solved. Taking Gongcheng Yao Autonomous County as an example, this study adopted the multi-factor comprehensive evaluation method to evaluate the division of land requisition in this area, and carried out the corresponding spatial analysis and data processing based on GIS technology. Gongcheng Yao Autonomous County was divided into levy areas, and the Delphi method was used to screen the impact factors and determine the weight of the levy areas. Reasonable division of land requisition area can provide references for land requisition area, make it more scientific and reasonable, and protect the rights and interests of farmers.

Forests, Cumulants, Martingales

This work is concerned with forest and cumulant type expansions of general random variables on a filtered probability spaces. We establish a “broken exponential martingale” expansion that generalizes and unifies the exponentiation result of Alos, Gatheral, and Radoicic (SSRN’17; [AGR20]) and the cumulant recursion formula of Lacoin, Rhodes, and Vargas (arXiv; [LRV19]). Specifically, we exhibit the two previous results as lower dimensional projections of the same generalized forest expansion, subsequently related by forest reordering. Our approach also leads to sharp integrability conditions for validity of the cumulant formula, as required by many of our examples, including iterated stochastic integrals, Levy area, Bessel processes, KPZ with smooth noise, Wiener-Ito chaos and “rough” stochastic (forward) variance models.

Couplings in $L^{p}$ distance of two Brownian motions and their Lévy area

Dans cet article, nous étudions les couplages co-adaptés de diffusions hypoelliptiques sur le groupe de Heisenberg (sous-riemannien). Ces diffusions canoniques, que nous appelons mouvement brownien sur le groupe de Heisenberg, sont constituées d’un mouvement brownien planaire et de son aire de Lévy. Nous prouvons que, contrairement au cas des variétés riemanniennes à courbure de Ricci positive ou nulle, pour tout couplage co-adapté, deux mouvements browniens partant de deux points donnés ne peuvent rester à distance bornée pour tout temps $t\geq 0$. Nous prouvons en fait le résultat plus fort qu’ils ne peuvent rester bornés dans $L^{p}$ pour $p\geq 2$. Nous établissons également deux résultats positifs. D’une part, nous étudions le couplage par réflexion et montrons qu’il reste borné dans $L^{p}$ pour $0<p<1$. D’autre part, nous construisons un couplage explicite statique (et en particulier non-adapté) entre les lois de deux mouvements browniens qui donne un contrôle en distance $L^{1}$ de Wasserstein uniforme en temps. Enfin, nous expliquons comment nos résultats se généralisent aux groupes de Heisenberg de dimension supérieure.

A Second Order Weak Approximation of SDEs Using Markov Chain Without Levy Area Simulation

This paper proposes a new Markov chain approach to second order weak approximation of stochastic differential equations driven by d-dimensional Brownian motion. The scheme is explicitly constructed by polynomials of Brownian motions up to second order and any discrete moment matched random variables or Levy area simulation method are not used. The number of required random variables is still d in one-step simulation on the implementation of the scheme. In the Markov chain, a correction term with Lie bracket of vector fields associated with SDEs appears as the cost of not using moment matched random variables.

$\varepsilon$-Strong simulation for multidimensional stochastic differential equations via rough path analysis

Consider a multidimensional diffusion process $X=\{X(t):t\in [0,1]\}$. Let $\varepsilon>0$ be a deterministic, user defined, tolerance error parameter. Under standard regularity conditions on the drift and diffusion coefficients of $X$, we construct a probability space, supporting both $X$ and an explicit, piecewise constant, fully simulatable process $X_{\varepsilon}$ such that \[\sup_{0\leq t\leq1}\Vert X_{\varepsilon}(t)-X(t)\Vert_{\infty}<\varepsilon\] with probability one. Moreover, the user can adaptively choose $\varepsilon^{\prime}\in (0,\varepsilon )$ so that $X_{\varepsilon^{\prime}}$ (also piecewise constant and fully simulatable) can be constructed conditional on $X_{\varepsilon}$ to ensure an error smaller than $\varepsilon^{\prime}$ with probability one. Our construction requires a detailed study of continuity estimates of the Ito map using Lyons’ theory of rough paths. We approximate the underlying Brownian motion, jointly with the Levy areas with a deterministic $\varepsilon$ error in the underlying rough path metric.

Efficient almost-exact Lévy area sampling

Abstract We present a new method for sampling the Levy area for a two-dimensional Wiener process conditioned on its endpoints. An efficient sampler for the Levy area is required to implement a strong Milstein numerical scheme to approximate the solution of a stochastic differential equation driven by a two-dimensional Wiener process whose diffusion vector fields do not commute. Our method is simple and complementary to those of Gaines–Lyons and Wiktorsson, and amenable to quasi-Monte Carlo implementation. It is based on representing the Levy area by an infinite weighted sum of independent Logistic random variables. We use Chebyshev polynomials to approximate the inverse distribution function of sums of independent Logistic random variables in three characteristic regimes. The error is controlled by the degree of the polynomials, we set the error to be uniformly 1 0 − 12 . We thus establish a strong almost-exact Levy area sampling method. The complexity of our method is square logarithmic. We indicate how it can contribute to efficient sampling in higher dimensions.

Positive Curvature Property for Sub-Laplacian on Nilpotent Lie Group of Rank Two

In this note, we concentrate on the sub-Laplace operator on the nilpotent Lie group of rank two, which is the infinitesimal generator of the diffusion generated by n Brownian motions and their \(\frac{n(n-1)}2\) Levy area processes, which is the simple extension of the sub-Laplacian on the Heisenberg group ℍ. In order to study contraction properties of the associated heat kernel, we show that, as in the cases of the Heisenberg group and the three Brownian motions model, the restriction of the sub-Laplace operator acting on radial functions (see Definition 3.5) satisfies a positive Ricci curvature condition (more precisely a CD(0, ∞ ) inequality), see Theorem 4.5, whereas the operator itself does not satisfy any CD(r, ∞ ) inequality. From this we may deduce some useful, sharp gradient bounds for the associated heat kernel. It can be seen a generalization of the paper (Qian, Bull Sci Math 135:262–278, 2011).

Rough path stability of (semi-)linear SPDEs

We give meaning to linear and semi-linear (possibly degenerate) parabolic partial differential equations with (affine) linear rough path noise and establish stability in a rough path metric. In the case of enhanced Brownian motion (Brownian motion with its Levy area) as rough path noise the solution coincides with the standard variational solution of the SPDE.

A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion

In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second order Taylor expansion, where the usual Levy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds for the discretisation of the Levy area terms.

Stochastic Analysis

Stochastic Analysis

Levy area for the free Brownian motion: existence and non-existence

This note extends the work of Capitaine (J. Funct. Anal. 179 (1) (2001) 153) on the Levy area process for the free Brownian motion in two directions. First, we reprove that a Levy area for the Free Brownian motion exists in the Von Neumann tensor product, by exhibiting a non-commutative Burkholder–Davis–Gundy type inequality. Then, we show that there does not exist a Levy area in the projective tensor product.

On the Convergence of Stochastic Integrals Driven by Processes Converging on account of a Homogenization Property

We study the limit of functionals of stochastic processes for which an homogenization result holds. All these functionals involve stochastic integrals. Among them, we consider more particularly the Levy area and those giving the solutions of some SDEs. The main question is to know whether or not the limit of the stochastic integrals is equal to the stochastic integral of the limit of each of its terms. In fact, the answer may be negative, especially in presence of a highly oscillating first-order differential term. This provides us some counterexamples to the theory of good sequence of semimartingales.

Path-wise solutions of stochastic differential equations driven by Lévy processes

In this paper we show that a path-wise solution to the following integral equation Yt = ?0t f(Yt) dXt, Y0 = a I Rd, exists under the assumption that Xt is a Levy process of finite p-variation for some p = 1 and that f is an a-Lipschitz function for some a > p. We examine two types of solution, determined by the solution's behaviour at jump times of the process X, one we call geometric, the other forward. The geometric solution is obtained by adding fictitious time and solving an associated integral equation. The forward solution is derived from the geometric solution by correcting the solution's jump behaviour. Levy processes, generally, have unbounded variation. So we must use a pathwise integral different from the Lebesgue-Stieltjes integral. When X has a finite p-variation almost surely for p 1. When p > 2 we use the integral of Lyons. In order to use this integral we construct the Levy area of the Levy process and show that it has finite (p/2)-variation almost surely.

Conditional Exponential Moments for Iterated Wiener Integrals

We provide sharp exponential moment bounds for (Stratonovich) iterated stochastic integrals under conditioning by certain small balls, including balls in certain Holder-like norms of exponent greater than 1 /3. The proof uses a control of the variation of the Levy area,under conditioning. The results are applied to the computation of the Onsager–Machlup functional of diffusion processes with constant diffusion matrix.

Stochastic Jacobi fields and vector fields induced by varying area on path spaces

We study two classes of vector fields on the path space over a closed manifold with a Wiener Riemannian measure. By adopting the viewpoint of Yang-Mills field theory, we study a vector field defined by varying a metric connection. We prove that the vector field obtained in this way satisfies a Jacobi field equation which is different from that of classical one by taking in account that a Brownian motion is invariant under the orthogonal group action, so that it is a geometric vector field on the space of continuous paths, and induces a quasi-invariant solution flow on the path space. The second object of this paper is vector fields obtained by varying area. Here we follow the idea that a continuous semimartingale is indeed a rough path consisting of not only the path in the classical sense, but also its Levy area. We prove that the vector field obtained by parallel translating a curve in the initial tangent space via a connection is just the vector field generated by translating the path along a direction in the Cameron-Martin space in the Malliavin calculus sense, and at the same time changing its Levy area in an appropriate way. This leads to a new derivation of the integration by parts formula on the path space.