Modern Stochastics: Theory and Applications

We establish an exact martingale Φ-entropy balance on a Brownian–Poisson product filtration with deterministic jump compensator. For terminal variables with positive lower and finite upper bounds, the balance separates the Gaussian energy from the nonlinear jump remainder on an explicit representation domain. We formulate the Brownian–Poisson predictable representation and ${L^{2}}$-Clark projection formulas in the form needed for this identity. Marked predictable projections provide compatible versions of the add-one difference and preserve the positivity required by the entropy remainder; finite-window arguments justify the passage between integrable and square-integrable representations. This first part supplies the entropy-balance framework used for the evolution-family formulas and concentration estimates in Part II.

We determine the quadratic variational cost that yields the large-deviation upper bound for the mean exit time from $(-1,1)$ in the small-noise regime for an autoregressive process driven by a continuous two-slope map f lying strictly below the diagonal on $(0,1)$. The Klebaner–Liptser reduction expresses this constant as an infimum, over all finite horizons, of quadratic path actions. Högnäs and Jung studied several nonlinear and piecewise-linear autoregressive maps and evaluated the corresponding variational problem for the two-slope family in the boundary cases in which one of the slopes vanishes; the case in which both slopes p and q are nonzero was left open. We solve this remaining case over the full admissible parameter range, including the finite-horizon attainment question, and give an exact parameter classification of the cost. We also settle the remaining boundary regimes $q=1$ and $p=0,\hspace{2.5pt}q\gt 1$, as well as the limiting case $f(1)=1$. Together with the earlier boundary formulas, this completes the exact variational analysis of continuous two-slope maps satisfying $f(x)\lt x$ for $0\lt x\lt 1$.

In this note, we derive a large-deviation-type scaling limit for the discretization error of a Brownian stochastic integral. Somewhat surprisingly, and to the best of our knowledge, this result has not been previously obtained in the literature.

This paper studies Lévy bases on Riemannian manifolds and related processes built from them. We argue that the canonical choice of intensity measure is the Riemannian measure on the manifold, and show how this allows for transporting properties of Lévy bases on ${\mathbb{R}^{d}}$ to the manifold setting via metatimes and subordination constructions. We show how this links to time-changed Lévy processes and how to naturally extend the setting to manifolds with (possibly stochastically) evolving metrics via generalized time-changes. As applications, we introduce so-called M-trawl processes, which generalize the classical trawl processes by letting the trawl operation be induced by a geometric flow, and we discuss the interpretation of Lévy bases on manifolds as Lévy white noise for stochastic partial differential equations on manifolds with Lévy-type forcing.

For the heat equations driven by a stochastic measure on $[-L,L]$ with the Dirichlet boundary condition, we prove that solutions tend to the solution of the heat equations defined on $\mathbb{R}$ as $L\to \mathrm{\infty }$. The estimate of the convergence rate is obtained. For a stochastic measure, we assume the σ-additivity in probability only.

This paper introduces a novel stochastic partial differential equation (SPDE) driven by space-only fractional Lévy noise ${\dot{L}_{x}^{d}}$ with $d\in (0,1/2)$, governed by a second-order elliptic operator with piecewise constant coefficients. We establish a new framework for analyzing diffusion in heterogeneous media, proving the existence, uniqueness, and equivalence of mild and weak solutions for the resulting non-Gaussian random field. A key original contribution is the explicit characterization of the covariance kernel in terms of transmission-reflection paths induced by the interface. We provide sharp estimates for the sample path regularity, proving that the solution is almost surely Hölder continuous of order $\gamma \lt d$ in space and $\beta \lt \frac{2d+1}{4}$ in time. Our results demonstrate that the fractional order of the noise robustly dictates path regularity, which remains preserved despite the singular transmission conditions and the jump structure of the driving noise.