Small Time Asymptotics Research Articles

Small time asymptotics of the entropy of the heat kernel on a Riemannian manifold

We give an asymptotic expansion of the relative entropy between the heat kernel qZ(t,z,w) of a compact Riemannian manifold Z and the normalized Riemannian volume for small values of t and for a fixed element z∈Z. We prove that coefficients in the expansion can be expressed as universal polynomials in the components of the curvature tensor and its covariant derivatives at z, when they are expressed in terms of normal coordinates. We describe a method to compute the coefficients, and we use the method to compute the first three coefficients. The asymptotic expansion is necessary for an unsupervised machine-learning algorithm called the Diffusion Variational Autoencoder.

ROUGH-HESTON LOCAL-VOLATILITY MODEL

In industrial applications it is quite common to use stochastic-volatility models driven by semi-martingale Markov volatility processes. However, in order to fit exactly market volatilities, these models are usually extended by adding a local-volatility term. Here, we consider the case of singular Volterra processes, and we extend them by adding a local-volatility term to their Markov lift by preserving the stylized results implied by these models on plain-vanilla options. In particular, we focus on the rough-Heston model, and we analyze the small-time asymptotics of its implied local-volatility function in order to provide a proper extrapolation scheme to be used in calibration.

On-diagonal asymptotics for heat kernels of a class of inhomogeneous partial differential operators

We consider certain constant-coefficient differential operators on Rd that have positive-definite symbols. Each such operator Λ with symbol P defines a semigroup of operators e−tΛ, t>0, admitting a continuous convolution kernel HPt for which the large-time behavior of HPt(0) cannot be deduced by basic scaling arguments. The simplest example has symbol P(ξ)=(η+ζ2)2+η4, ξ=(η,ζ)∈R2. We devise a method that allows us to determine the large-time behavior of HPt(0) for several classes of examples of this type and we show that these large-time asymptotics are preserved by perturbations of Λ by certain higher-order differential operators. For the P just given, it turns out that HPt(0)∼cPt−5/8 when t tends to infinity. We show how such results are relevant to understand the iterated convolution powers of certain finitely-supported complex functions on Zd. We also discuss how these techniques provide precise small-time asymptotics for HPt(0) in some cases when the operator Λ is not hypoelliptic. The simplest such example Λ has symbol P(ξ)=η2+(η−ξ2)4 and we show that HPt(0)∼cPt−1/2 as t tends to 0 in this case. Our work represents a first basic step towards a good understanding of the semigroups associated with these differential operators. Obtaining meaningful off-diagonal upper bounds for the convolution kernels of these semigroups remains an interesting challenge.

On the small time asymptotics of stochastic predator–prey model with nonlinear functional response

On the small time asymptotics of stochastic predator–prey model with nonlinear functional response

The Heat Kernel on the Diagonal for a Compact Metric Graph

We analyze the heat kernel associated with the Laplacian on a compact metric graph, with standard Kirchhoff–Neumann vertex conditions. An explicit formula for the heat kernel as a sum over loops, developed by Roth and Kostrykin–Potthoff–Schrader, allows for a straightforward analysis of small-time asymptotics. We show that the restriction of the heat kernel to the diagonal satisfies a modified version of the heat equation. This observation leads to an “edge” heat trace formula, expressing the a sum over eigenfunction amplitudes on a single edge as a sum over closed loops containing that edge. The proof of this formula relies on a modified heat equation satisfied by the diagonal restriction of the heat kernel. Further study of this equation leads to explicit formulas for graphs which are symmetric about each vertex.

Upper functions for sample paths of Lévy(-type) processes

We study the small-time asymptotics of sample paths of Lévy processes and Lévy-type processes. Namely, we investigate under which conditions the limit lim supt→0 1 f(t)|Xt−X0| is finite resp. infinite with probability 1. We establish integral criteria in terms of the infinitesimal characteristics and the symbol of the process. Our results apply to a wide class of processes, including solutions to Lévy-driven SDEs and stable-like processes. For the particular case of Lévy processes, we recover and extend earlier results from the literature. Moreover, we present a new maximal inequality for Lévy-type processes, which is of independent interest.

Small-time expansion for the density of a planar (quadratic) Langevin diffusion

ABSTRACT The unscaled small-time asymptotics of the density is approached by means of the Brownian bridge, regarding planar analogues of the Langevin diffusion, which are strictly hypoelliptic. While the non-quadratic case remains open, in the quadratic case a new precision is derived.

On the Small Time Asymptotics of Quasilinear Parabolic Stochastic Partial Differential Equations

In this paper, we establish small time large deviation principles for the quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are neither monotone nor locally monotone.

On the small time asymptotics of scalar stochastic conservation laws

In this paper, we established a small time large deviation principles for scalar stochastic conservation laws driven by multiplicative noise. The doubling variables method plays a key role.

Diffusion processes associated with elliptic operators with measure-valued coefficients

In this paper, we give a survey on diffusion processes associated with elliptic operators with measure-valued coefficients. We first introduce the existence and uniqueness of Brownian motion and reflected Brownian with measure-valued drifts as well as the Gaussian type estimates of the transition functions. We then study the Dirichlet and Neumann boundary value problems for the associated generators of the diffusion processes and obtain the probabilistic representations for the solutions. we also present the small time asymptotics and large deviations for the Brownian motions with drifts.

Small-time asymptotics of hypoelliptic heat kernels near the diagonal, nilpotentization and related results

We establish small-time asymptotic expansions for heat kernels of hypoelliptic Hormander operators in a neighborhood of the diagonal, generalizing former results obtained in particular by Metivier and by Ben Arous. The coefficients of our expansions are identified in terms of the nilpotentization of the underlying sub-Riemannian structure. Our approach is purely analytic and relies in particular on local and global subelliptic estimates as well as on the local nature of small-time asymptotics of heat kernels. The fact that our expansions are valid not only along the diagonal but in an asymptotic neighborhood of the diagonal is the main novelty, useful in view of deriving Weyl laws for subelliptic Laplacians. In turn, we establish a number of other results on hypoelliptic heat kernels that are interesting in themselves, such as Kac's principle of not feeling the boundary, asymptotic results for singular perturbations of hypoelliptic operators, global smoothing properties for selfadjoint heat semigroups.

A smoothly curved body skimming on shallow water

We investigate the dynamics of a smoothly curved convex body skimming on a layer of shallow water, with the phenomenon of skipping stones as a primary modelling motivation but also industrial applications in mind. The skimming process of such an object may consist of two successive stages: an impact stage and a conditional planing stage. The focus here is on explaining the change from one stage to the other, with the body movement responding freely to the fluid flow pressures and vice versa. We first introduce a water impact model and analyse the conditions under which a smooth body may rapidly transit to a planing motion via small time asymptotics. A planing model is then introduced, and in particular we investigate the effects of pressure conditions in the separation flow; we demonstrate under certain weak adverse pressure gradient conditions a smooth body’s planing motion at early times can be seen as undergoing three successive transitory phases.

The equivalent constant-elasticity-of-variance (CEV) volatility of the stochastic-alpha-beta-rho (SABR) model

This study presents new analytic approximations of the stochastic-alpha-beta-rho (SABR) model. Unlike existing studies that focus on the equivalent Black–Scholes (BS) volatility, we instead derive the equivalent constant-elasticity-of-variance (CEV) volatility. Our approach effectively reduces the approximation error in a way similar to the control variate method because the CEV model is the zero vol-of-vol limit of the SABR model. Moreover, the CEV volatility approximation yields a finite value at a zero strike and thus conveniently leads to a small-time asymptotics for the mass at zero. The numerical results compare favorably with the BS volatility approximations in terms of the approximation accuracy, small-strike volatility asymptotics, and no-arbitrage region.

Small-time fluctuations for the bridge of a sub-Riemannian diffusion

We consider small-time asymptotics for diffusion processes conditioned by their initial and final positions, under the assumption that the diffusivity has a sub-Riemannian structure, not necessarily of constant rank. We show that, if the endpoints are joined by a unique path of minimal energy, and lie outside the sub-Riemannian cut locus, then the fluctuations of the conditioned diffusion from the minimal energy path, suitably rescaled, converge to a Gaussian limit. The Gaussian limit is characterized in terms of the bicharacteristic flow, and also in terms of a second variation of the energy functional at the minimal path, the formulation of which is new in this context.

Heat content asymptotics for sub-Riemannian manifolds

We study the small-time asymptotics of the heat content of smooth non-characteristic domains of a general rank-varying sub-Riemannian structure, equipped with an arbitrary smooth measure. By adapting to the sub-Riemannian case a technique due to Savo, we establish the existence of the full asymptotic series:QΩ(t)=∑k=0∞aktk/2,as t→0. We compute explicitly the coefficients up to order k=5, in terms of sub-Riemannian invariants of the domain. Furthermore, we prove that every coefficient can be obtained as the limit of the corresponding one for a suitable Riemannian extension.As a particular case we recover, using non-probabilistic techniques, the order 2 formula recently obtained by Tyson and Wang in the Heisenberg group [48]. A consequence of our fifth-order analysis is the evidence for new phenomena in presence of characteristic points. In particular, we prove that the higher order coefficients in the asymptotics can blow-up in their presence. A key tool for this last result is an exact formula for the distance from a specific surface with an isolated characteristic point in the Heisenberg group, which is of independent interest.

On the Small Time Asymptotics of the Dynamical Φ41 Model

In this paper, we establish a small time large deviation principle (small time asymptotics) for the dynamical Φ14 model, which not only involves study of the space-time white noise with intensity $$\sqrt \varepsilon $$ , but also the investigation of the effect of the small (with e) nonlinear drift.

Asymptotic behavior and calibration of short-time option prices under the normal tempered stable model

In recent years, small-time asymptotics of option prices have received much attention. In this paper, we derive short-time first-order expansions for price and implied volatility of at-the-money call options under the exponential Normal Tempered Stable model in two methods. The first method is similar to the approach used by Figueroa-López and Forde for the CGMY model. The other is based on the explicit form of characteristic functions of the Normal Tempered Stable processes. Our numerical results show that the expansions herein remarkably approximate the prices obtained by Fourier transform, even for options with maturities as long as a few years. For this reason, our expansions can be used for parameter calibration and model testing in a simple and efficient way.

Rough volatility and CGMY jumps with a finite history and the Rough Heston model – small-time asymptotics in the regime

A small-time Edgeworth expansion is established for near-the-money European options under a general rough stochastic volatility (RSV) model driven by a Riemann-Liouville (RL) process plus an additional generalised tempered stable Lévy process with when (this relaxes the more complicated and restrictive condition which appeared in an earlier version of the article.), in the regime where log-moneyness as for z fixed, conditioned on a finite volatility history. This can be viewed as a more practical variant of Theorem 3.1 in Fukasawa [Short-time at-the-money skew and rough fractional volatility. Quant. Finance, 2017, 17(2), 189–198] (Fukasawa [Short-time at-the-money skew and rough fractional volatility. Quant. Finance, 2017, 17(2), 189–198] does not allow for jumps or a finite history and uses the somewhat opaque Muravlev representation for fractional Brownian motion), or if we turn off the rough stochastic volatility, the expansion is a variant of the main result in Mijatovic and Tankov [A new look at short-term implied volatility in asset price models with jumps. Math. Finance, 2016, 26(1), 149–183] and Theorem 3.2 in Figueroa-López et al. [Third-order short-time expansions for close-to-the-money option prices under the CGMY model. J. Appl. Math. Finance, 2017, 24, 547–574]. The regime is directly applicable to FX options where options are typically quoted in terms of delta (.10,.25 and .50) not absolute strikes, and we also compute a new prediction formula for the Riemann-Liouville process, which allows us to express the history term for the Edgeworth expansion in a more useable form in terms of the volatility process itself. We later relax the assumption of bounded volatility, and we also compute a formal small-time expansion for implied volatility in the Rough Heston model in the same regime (without jumps) which includes an additional at-the-money, convexity and fourth order correction term, and we outline how one can go to even higher order in the three separate cases , and . (We thank Masaaki Fukasawa and Hongzhong Zhang for fruitful discussions, Adam Hesse for help with Matlab computations and Vian Dinh for help with IT issues.)

The Subelliptic Heat Kernel of the Octonionic Hopf Fibration

We study the sub-Laplacian of the 15-dimensional unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the octonionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of a related sub-Laplacian. As a byproduct we also obtain the spectrum of the sub-Laplacian, the small-time asymptotics of the heat kernel and explicitly compute the sub-Riemannian distance.

Small-Time smile for the multifactor volatility heston model

AbstractWe extend the existing small-time asymptotics for implied volatilities under the Heston stochastic volatility model to the multifactor volatility Heston model, which is also known as the Wishart multidimensional stochastic volatility model (WMSV). More explicitly, we show that the approaches taken in Forde and Jacquier (2009) and Forde, Jacqiuer and Lee (2012) are applicable to the WMSV model under mild conditions, and obtain explicit small-time expansions of implied volatilities.