Related papers: Two-dimensional Kac-Rice formula. Application to s…

We study the problem of estimating the score function of an unknown probability distribution $\rho^*$ from $n$ independent and identically distributed observations in $d$ dimensions. Assuming that $\rho^*$ is subgaussian and has a…

We present two explicit rational formulae for Bachelier, or normal, implied volatility. The formulae take the option price, forward, strike, and expiry as inputs and return the implied normal volatility without iteration. They follow the…

A loss function measures the discrepancy between the true values (observations) and their estimated fits, for a given instance of data. A loss function is said to be proper (unbiased, Fisher consistent) if the fits are defined over a unit…

In this paper, we analyze the drift-implicit (or backward) Euler numerical scheme for a class of stochastic differential equations with unbounded drift driven by an arbitrary $\lambda$-H\"older continuous process, $\lambda\in(0,1)$. We…

In this paper, we consider an inference problem for the first order autoregressive process driven by a long memory stationary Gaussian process. Suppose that the covariance function of the noise can be expressed as $\abs{k}^{2H-2}$ times a…

Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where "almost everywhere" refers to the Lebesgue measure. In this paper we prove a differentiability result of similar type,…

Let $G/K$ be an irreducible symmetric space where $G$ is a non-compact, connected Lie group and $K$ is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$…

Sine-Wiener noise is increasingly adopted in realistic stochastic modeling for its bounded nature. However, many features of the SW noise are still unexplored. In this paper, firstly, the properties of the SW noise and its integral process…

We consider a special class of mean field SDEs with common noise which depend on the image of the solution (i.e. the conditional distribution given noise). The strong well-posedness is derived under a monotone condition which is weaker than…

We show that a computable function $f:\mathbb R\rightarrow\mathbb R$ has Luzin's property (N) if and only if it reflects $\Pi^1_1$-randomnes, if and only if it reflects $\Delta^1_1(\mathcal O)$-randomness, and if and only if it reflects…

We study the probability distribution $F(u)$ of the maximum of smooth Gaussian fields defined on compact subsets of $\R^d$ having some geometric regularity. Our main result is a general formula for the density of $F$. Even though this is an…

In this note we prove convergence of Green functions with Neumann boundary conditions for the random walk to their continuous counterparts. Also a few Beurling type hitting estimates are obtained for the random walk on discretizations of…

We establish the interior and exterior Gauss-Green formulas for divergence-measure fields in $L^p$ over general open sets, motivated by the rigorous mathematical formulation of the physical principle of balance law via the Cauchy flux in…

We derive fundamental sampling bounds for smooth signals in continuous settings without sparsity assumptions. By introducing the Fourier ratio as a measure of spectral compressibility induced by smoothness, we obtain explicit, deterministic…

Let $S \subset \Bbb R^n$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\Bbb R^n$. If $X$ contains a ball of each…

The purpose of this article is to give a rather thorough understanding of the compact support property for measure-valued processes corresponding to semi-linear equations of the form \[ \begin{aligned}& u_t=Lu+\beta u-\alpha u^p \text{in}…

The Kirchhoff integral is a fundamental integral in scattering theory, appearing in both the Kirchhoff approximation, as well as the small slope approximation. In this work, a functional Taylor series approximation to…

Let $E$ be a measurable subset in a segment $[0,r]$ in the positive part of the real axis in the complex plane, and $U=u-v$ be the difference of subharmonic functions $u\not\equiv -\infty$ and $v\not\equiv-\infty$ on the complex plane. An…

We investigate the equation $(u_t + (f(u))_x)_x = f''(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave…

We establish formulas that give the intrinsic volumes, or curvature measures, of sublevel sets of functions defined on Riemannian manifolds as integrals of functionals of the function and its derivatives. For instance, in the Euclidean…