Small Time Asymptotics Research Articles

The motion of elliptic cylinder under free surface

A problem on generation of unsteady nonlinear waves on the surface of an infinitely deep ideal fluid due to the motion of a submerged elliptical cylinder is considered. It is supposed that the cylinder can rotate in addition to translational two-dimensional motion. The initial formulation of the problem is reduced to an integrodifferential system of equations for the functions defining the free surface shape, the normal and tangential components of velocity on the free boundary. The small-time asymptotics of the solution is constructed in the case of the cylinder that moves with a constant acceleration from rest.

The Subelliptic Heat Kernel on the Anti-de Sitter Space

We study the subelliptic heat kernel of the sub-Laplacian on a 2n+1-dimensional anti-de Sitter space ℍ 2n+1 which also appears as a model space of a CR Sasakian manifold with constant negative sectional curvature. In particular we obtain an explicit and geometrically meaningful formula for the subelliptic heat kernel. The key idea is to work in a set of coordinates that reflects the symmetry coming from the Hopf fibration \(\mathbb{S}^{1}\to \mathbb{H}^{2n+1}\). A direct application is obtaining small time asymptotics of the subelliptic heat kernel. Also we derive an explicit formula for the sub-Riemannian distance on ℍ 2n+1.

Understanding the Correlation Skew Modeling: Local Correlation versus Common Jumps Model

The aim of this article is to show how a Correlation Skew can be built in a local correlation model or with common jumps on the underlyings of the basket. Using small time asymptotics (Watanabe's expansion and evolution operator theory), we provide efficient approximations to help understand how the basket implied variance skew is generated.

Well-posedness and the small time large deviations of the stochastic integrable equation governing short-waves in a long-wave model

This paper is concerned with the stochastic integrable equation governing short-waves in a long-wave model. Firstly, the local well-posedness for this system is established by fixed point argument and (bilinear) trilinear estimates. Then the small time asymptotics of the equation is proved. The corresponding results for the stochastic Hunter–Saxton equation can be obtained by the same methods.

A Small Noise Asymptotic Expansion for Young SDE Driven by Fractional Brownian Motion: A Sharp Error Estimate With Malliavin Calculus

This article shows an analytically tractable small noise asymptotic expansion with a sharp error estimate for the expectation of the solution to Young’s pathwise stochastic differential equations (SDEs) driven by fractional Brownian motions with the Hurst index H > 1/2. In particular, our asymptotic expansion can be regarded as small noise and small time asymptotics by the error estimate with Malliavin culculus. As an application, we give an expansion formula in one-dimensional general Young SDE driven by fractional Brownian motion. We show the validity of the expansion through numerical experiments.

Pseudo generators for under-resolved molecular dynamics

Many features of a molecule which are of physical interest (e.g. molecular conformations, reaction rates) are described in terms of its dynamics in configuration space. This article deals with the projection of molecular dynamics in phase space onto configuration space. Specifically, we study the situation that the phase space dynamics is governed by a stochastic Langevin equation and study its relation with the configurational Smoluchowski equation in the three different scaling regimes: Firstly, the Smoluchowski equations in non-Cartesian geometries are derived from the overdamped limit of the Langevin equation. Secondly, transfer operator methods are used to describe the metastable behaviour of the system at hand, and an explicit small-time asymptotics is derived on which the Smoluchowski equation turns out to govern the dynamics of the position coordinate (without any assumptions on the damping). By using an adequate reduction technique, these considerations are then extended to one-dimensional reaction coordinates. Thirdly, we sketch three different approaches to approximate the metastable dynamics based on time-local information only.

On the windings of complex-valued Ornstein–Uhlenbeck processes driven by a Brownian motion and by a stable process

We deal with a complex-valued Ornstein–Uhlenbeck (OU) process with parameter starting from a point different from 0 and the way that it winds around the origin. The starting point of this paper is the skew product representation for an OU process which is associated with the skew product representation of its driving planar Brownian motion (BM) under a new deterministic time scale. We present the stochastic differential equations for the radial and for the winding process. Moreover, we obtain the large time (analogue of Spitzer's Theorem for BM in the complex plane) and the small time asymptotics for the winding and for the radial process, and we explore the exit time from a cone for a 2-dimensional OU process. Some limit theorems concerning the angle of the cone (when our process winds in a cone) and the parameter are also presented. Furthermore, we discuss the decomposition of the winding process of a complex-valued OU process in ‘small’ and ‘big’ windings, where, for the ‘big’ windings, we use some results already obtained by J. Bertoin and W. Werner [Asymptotic windings of planar Brownian motion revisited via the Ornstein-Uhlenbeck process. Sém. Prob., XXVIII, Lect. Notes in Mathematics 1583, Springer, Berlin Heidelberg New York, 1994, pp. 138–152] and we show that only the ‘small’ windings contribute in the large time limit. Finally, we study the windings of a complex-valued OU process driven by a stable process and we obtain similar results for its (well-defined) winding and radial process.

Kusuoka–Stroock gradient bounds for the solution of the filtering equation

We obtain sharp gradient bounds for perturbed diffusion semigroups. In contrast with existing results, the perturbation is here random and the bounds obtained are pathwise. Our approach builds on the classical work of Kusuoka and Stroock [13,14,16,17], and extends their program developed for the heat semi-group to solutions of stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The analysis allows us to derive pathwise gradient bounds for the un-normalised conditional distribution of a partially observed signal. It uses a pathwise representation of the perturbed semigroup following Ocone [22]. The estimates we derive have sharp small time asymptotics.

Heat Content and Small Time Asymptotics for Schrödinger Operators on ℝ d ${\mathbb {R}}^{d}$

This paper studies the heat content for Schrodinger operators of the fractional Laplacian \({(-{\Delta })^{\frac {\alpha }{2} }}\), 0<α≤2 in \({\mathbb {R}}^{d}\), d≥1. Employing probabilistic and analytic techniques, a small time asymptotic expansion formula is given and the “heat content invariants” are identified. To the best of our knowledge, these results are new even in the case of the Laplacian, α = 2.

Small-time asymptotics of stopped Lévy bridges and simulation schemes with controlled bias

We characterize the small-time asymptotic behavior of the exit probability of a L\'evy process out of a two-sided interval and of the law of its overshoot, conditionally on the terminal value of the process. The asymptotic expansions are given in the form of a first-order term and a precise computable error bound. As an important application of these formulas, we develop a novel adaptive discretization scheme for the Monte Carlo computation of functionals of killed L\'evy processes with controlled bias. The considered functionals appear in several domains of mathematical finance (e.g., structural credit risk models, pricing of barrier options, and contingent convertible bonds) as well as in natural sciences. The proposed algorithm works by adding discretization points sampled from the L\'evy bridge density to the skeleton of the process until the overall error for a given trajectory becomes smaller than the maximum tolerance given by the user.

The Subelliptic Heat Kernels of the Quaternionic Hopf Fibration

The main goal of this work is to study the sub-Laplacian of the unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the quaternionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of the conformal sub-Laplacian and small-time asymptotics. As a byproduct of our study we also obtain several results related to the sub-Laplacian of a projected Hopf fibration.

The Small Time Asymptotics of SPDEs with Reflection

We study stochastic partial differential equations with singular drifts and with reflection, driven by space-time white noise with nonconstant diffusion coefficients under periodic boundary conditions. The existence and uniqueness of invariant measures is established under appropriate conditions. As a byproduct, the Hölder continuity of the solution is obtained. The strong Feller property is also obtained. Moreover, we show large deviation principle.

Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry

In this paper, we study the small time asymptotics for the heat kernel on a sub-Riemannian manifold, using a perturbative approach. We explicitly compute, in the case of a 3D contact structure, the first two coefficients of the small time asymptotics expansion of the heat kernel on the diagonal, expressing them in terms of the two basic functional invariants χand κ defined on a 3D contact structure.

Unsteady motion of an elliptic cylinder under a free surface

A problem of generation of nonlinear unsteady waves on the surface of an ideal liquid in an infinitely deep fluid due to the motion of a submerged elliptic cylinder is considered. The initial formulation of the problem is reduced to an integrodifferential system of equations for the function defining the free surface shape and for the normal and tangential components of velocity on the free surface. The small-time asymptotics of the solution is constructed for the case of cylinder motion with a constant acceleration from the state at rest.

Large deviation principles for the stochastic quasi-geostrophic equations

In this paper we establish the large deviation principle for the stochastic quasi-geostrophic equation with small multiplicative noise in the subcritical case. The proof is mainly based on the weak convergence approach. Some analogous results are also obtained for the small time asymptotics of the stochastic quasi-geostrophic equation.

Small-time heat kernel asymptotics at the sub-Riemannian cut locus

For a sub-Riemannian manifold provided with a smooth volume, we relate the small-time asymptotics of the heat kernel at a point y of the cut locus from $x$ with roughly "how much" $y$ is conjugate to $x$. This is done under the hypothesis that all minimizers connecting $x$ to $y$ are strongly normal, i.e. all pieces of the trajectory are not abnormal. Our result is a refinement of the one of Leandre $4t \log p_t(x, y) \to −d^2(x, y)$ for $t \to 0$, in which only the leading exponential term is detected. Our results are obtained by extending an idea of Molchanov from the Riemannian to the sub-Riemannian case, and some details we get appear to be new even in the Riemannian context. These results permit us to obtain properties of the sub-Riemannian distance starting from those of the heat kernel and vice versa. For the Grushin plane endowed with the Euclidean volume, we get the expansion $p_t(x, y) \sim t^{−5/4} \exp(−d^2(x, y)/4t)$ where $y$ is reached from a Riemannian point $x$ by a minimizing geodesic which is conjugate at $y$.

Small-time asymptotics for fast mean-reverting stochastic volatility models

In this paper, we study stochastic volatility models in regimes where the maturity is small, but large compared to the mean-reversion time of the stochastic volatility factor. The problem falls in the class of averaging/homogenization problems for nonlinear HJB-type equations where the "fast variable" lives in a noncompact space. We develop a general argument based on viscosity solutions which we apply to the two regimes studied in the paper. We derive a large deviation principle, and we deduce asymptotic prices for out-of-the-money call and put options, and their corresponding implied volatilities. The results of this paper generalize the ones obtained in Feng, Forde and Fouque [SIAM J. Financial Math. 1 (2010) 126-141] by a moment generating function computation in the particular case of the Heston model.

Short-time asymptotics of heat kernels of hypoelliptic Laplacians on unimodular Lie groups

We consider the problem of computing heat kernel small-time asymptotics for hypoelliptic Laplacians associated to left-invariant sub-Riemannian structures on unimodular Lie groups of type I. We use the non-commutative Fourier transform of the Lie group together with perturbation theory for semigroups of operators in deriving these asymptotics. We illustrate our approach on the example of the Heisenberg group, and, as an application, we compute the short-time behaviour of the hypoelliptic heat kernel on the step 3 nilpotent Cartan and Engel groups, for which no closed-form expression for the hypoelliptic heat kernel is yet known.

Asymptotiques en temps petit du noyau de la chaleur des métriques riemanniennes et sous-riemanniennes

. We provide the small-time asymptotics of the heat kernel at the cut locus in three cases: generic Riemannian manifolds in dimension less or equal to 5, generic 3D contact and 4D quasi-contact sub-Riemannian manifolds (close to the starting point). As a byproduct we show that, for generic Riemannian manifolds of dimension less or equal to 5, the only possible singularities of the exponential map along a minimizing geodesic are A 3 and A 5 .

Small-Time Asymptotics of Option Prices and First Absolute Moments

We study the leading term in the small-time asymptotics of at-the-money call option prices when the stock price process S follows a general martingale. This is equivalent to studying the first centered absolute moment of S. We show that if S has a continuous part, the leading term is of order √T in time T and depends only on the initial value of the volatility. Furthermore, the term is linear in T if and only if S is of finite variation. The leading terms for pure-jump processes with infinite variation are between these two cases; we obtain their exact form for stable-like small jumps. To derive these results, we use a natural approximation of S so that calculations are necessary only for the class of Lévy processes.