arXiv subjects

Tristan Guillaume

Publications and source records attributed to Tristan Guillaume.

9 recordsLinked to original sources

arXiv2026

The joint exit time and exit location law of a planar Ornstein-Uhlenbeck process on annular sectors and principal axis rectangles

We study the first exit of an isotropic planar Ornstein-Uhlenbeck process from an annular sector, the region bounded by two concentric circular arcs and two radial segments, and obtain explicit eigenfunction expansions for the associated exit functionals. After the ground-state transformation that renders the generator self-adjoint, polar separation reduces the problem to a radial confluent hypergeometric equation whose two independent (Whittaker) solutions of generally non-integral order are combined through a two-radius determinant that fixes the spectrum. In this way we obtain the survival probability, the density and moments of the exit time, and -the principal contribution -the joint law of the exit time and the exit boundary, which resolves both the instant of exit and the boundary piece through which it occurs. As a companion separable case, we also treat a genuinely correlated, reversible planar Ornstein-Uhlenbeck process on a principal-axis rectangle, where the radial functions are replaced by parabolic cylinder functions: the survival probability factorizes into one-dimensional problems, while the joint law of the exit time and the exit side does not. The expansions require only one-dimensional root-finding and quadrature and are validated against Monte Carlo simulation. An application to an optically trapped colloidal particle is discussed, a setting in which the annular geometry arises naturally.

math.PR↗

arXiv2026

Holes in planar parallel sets: An integrated Betti-number bound and its pointwise failure

Let A be a nonempty compact subset of the plane and let A (r) be its parallel set at distance r. We prove that the number of holes of A (r) the number of bounded components of its complement, which is its rst Betti number satises $\infty$ r 0 $β$1(A (r) ) dr $\le$ 4050 (diam A) 4 r -3 0 for every r0 > 0, the integrand vanishing for r $\ge$ diam A/ $\sqrt$ 3. The proof rests on Fu's theorem that the critical values of the distance function of a planar compact set form a set of vanishing half-dimensional Hausdor measure, on two lemmas of Rataj, Spodarev and Meschenmoser, and on a square-root summability estimate for the gaps of the critical-value set, of which we give a complete proof. We show by an explicit family of curves two combs facing each other that no analogous bound can hold at a xed radius: a connected curve of bounded length, diameter, oscillation count, parallel-set area and parallel-set perimeter can have arbitrarily many holes at one radius, so the integrated estimate cannot be replaced by a xed-radius bound in terms of these coarse geometric quantities. We also bound the hole count uniformly in the radius by the number of components of local maxima of the distance function, and record a bound on the boundary length of a parallel set by its area. The results supply the deterministic input for limit theorems on the persistent homology of the Wiener sausage.

cs.CG↗

arXiv2026

Spectral-capacitary bounds for homological recovery from reflected Brownian trajectories

Can the topology of an unknown reflecting domain be recovered from one reflected Brownian trajectory, and how long must it be observed? We formulate this as a minimax homological inference problem based on the reflected Wiener sausage. The optimal observation time is, up to constants, the sum of three terms: a spectral-access or burn-in time set by the Neumann spectral gap; an inverse-capacity detection time for small topological features, showing that Brownian capacity rather than volume controls discovery; and a logarithmic factor in the number of features to be resolved. Matching minimax lower bounds prove that all three contributions are necessary. The proof combines a finite-target spectral hitting estimate, expressed through killed eigenvalues, with a uniform small-hole eigenvalue theorem comparing the principal killed eigenvalue of an epsilon-scale feature with its Brownian capacity, uniformly over interior and boundary targets. The resulting framework distinguishes full reconstruction, obtained from an intrinsic epsilon-net and yielding two-sided homology isomorphisms, from faster feature detection by witness regions, which gives the corresponding surjective homological recovery.

math.PR↗

arXiv2026

Persistent Homology of the Planar Wiener Sausage: Brownian Scaling and a Logarithmic Expectation Law

We study degree-one persistent homology of the planar Wiener-sausage filtration generated by standard Brownian motion without drift. In the drifted case, regeneration along the drift direction leads to linear-in-time laws for persistent-homological observables. In the recurrent zero-drift case, this renewal structure disappears. The organizing mechanism is instead Brownian self-similarity: the persistence diagram at time $T$ is equal in law to the image of the unit-time diagram under spatial dilation by $\sqrt T$. Consequently, large-time questions on fixed radius windows are transformed into small-radius questions for the unit-time Brownian trace. Let $B$ be standard planar Brownian motion, let $K_T=B\left(\left[0,T\right]\right)$, and let $K_T^{\left(r\right)}$ be the radius-$r$ Wiener sausage. Since $K_T^{\left(r\right)}$ is connected, its first Betti number $β_1^T\left(r\right)$ is the number of bounded complementary components of $K_T^{\left(r\right)}$. For a bounded nonnegative Borel function $ψ$ supported in a compact interval $\left[a,b\right]\subset\left(0,\infty\right)$, we consider the smoothed Betti-curve observable $\left[r_0,r_1\right] \mathrmΦ_ψ\left(T\right) = \int_{r_0}^{r_1} β_1^T \left( r \right) ψ\left( r \right) dr$. We prove that there exist absolute constants 0

math.PR↗

arXiv2026

Persistent Homology of the Wiener Sausage II: A Central Limit Theorem for Drifted Planar Brownian Motion

Let $X_t = B_t + μt$, $t \geq 0$, be planar Brownian motion with nonzero drift, and let $K_t^r = \{x \in \mathbb{R}^2 : {\rm dist}(x, X[0,t]) \leq r\}$ be the radius-$r$ Wiener sausage up to time $t$. For a bounded Borel function $ψ$ supported in a compact interval $[r_0, r_1] \subset (0,\infty)$, consider the smoothed Betti-curve functional $Φ_ψ(t) := \int_{r_0}^{r_1} β_1^t(r)\,ψ(r)\,dr$, where $β_1^t(r)$ denotes the number of holes of $K_t^r$. In a previous paper, a regeneration scheme along the drift direction was used to prove a law of large numbers for $Φ_ψ(t)$. In the present paper we prove the corresponding central limit theorem. More precisely, there exist a deterministic constant $ρ_ψ$ and a variance $σ_ψ^2 \geq 0$ such that $(Φ_ψ(t) - ρ_ψt)/\sqrt{t} \xrightarrow{d}_{t \to \infty} \mathcal{N}(0, σ_ψ^2)$. We also obtain the finite-dimensional Gaussian limit for finitely many test functions. The proof preserves the regenerative structure of the law of large numbers, but requires a new $L^2$ analysis of the topological interface terms created at regeneration cuts. The key input is a finite-time polynomial moment bound for integrated hole counts of the Wiener sausage. This yields square-integrability of cycle increments, within-cycle oscillations, and the last incomplete-cycle remainder, which in turn allows one to combine a standard central limit theorem for stationary $1$-dependent sequences with a renewal time-change argument.

math.PR↗

arXiv2026

Extrema, Barrier Options, and Semi-Analytic Leverage Corrections in Stochastic-Clock Volatility Models

Barrier derivatives depend on extrema and first-passage events and are therefore highly sensitive to volatility dynamics -- especially to the instantaneous return-volatility correlation $ρ$, often called ``leverage''. This sensitivity makes accurate and fast pricing under realistic stochastic-volatility specifications difficult: two-dimensional PDE solvers are expensive inside calibration loops, while Monte Carlo methods converge slowly when barrier hits are rare and discretely monitored. In equity markets in particular, the pronounced implied-volatility skew motivates factoring in a negative return-volatility correlation. We study a class of continuous-path stochastic-clock volatility models in which the log-price is represented as a Brownian motion run on a random increasing clock. In the baseline independent-clock case (ρ=0), a broad family of barrier-relevant objects-maximum distributions, survival probabilities, and killed joint laws-reduces to one-dimensional quantities determined by the Laplace transform of the terminal clock. This yields transform-only pricing formulas for single- and double-barrier contracts that are fast and numerically stable once the clock transform is available, notably for affine and quadratic clocks. To incorporate leverage without forfeiting tractability, we develop a systematic small-ρexpansion around the ρ=0 backbone. The expansion produces a hierarchy of forced problems whose forcing terms are semi-analytic and computable from baseline barrier objects. We provide two implementable leverage-correction routes\,: forced PDEs and a Duhamel-type Monte Carlo representation, and we show how Pad{é} acceleration can extend practical accuracy to equity-like correlations. Calibration then proceeds by\,: (i) fitting clock parameters from vanillas using only one-dimensional transforms, (ii) precomputing the ρ=0 barrier backbone once, and (iii) iterating on ρ(and any remaining parameters) using the fast semi-analytic corrections-optionally Pad{é}-accelerated-inside a standard least-squares loop.

q-fin.CP↗

arXiv2026

First Passage through a Continuous Barrier: Pathwise Decomposition, Random-Time Structure, and Compensators

Let t be the first-passage time of a continuous barrier by a c{à}dl{à}g adapted process. We show that t admits a canonical fourfold pathwise decomposition into continuous contact, contact from the left followed by an upward jump, exact hit by jump, and strict overshoot by jump from below. This refinement is more informative than the classical contact-versus-overshoot dichotomy for random-time purposes, because it separates modes with different predictability properties. In particular, the left-contact component always defines an accessible stopping time and becomes predictable under a no-premature-left-contact condition, which we prove to be both sufficient and necessary for the canonical running-supremum announcing sequence to work. On the gap side, under a structural exclusion of predictable gap-crossings, the corresponding restricted time is totally inaccessible. In the semimartingale setting, we obtain a sharp compensator criterion for the predictable-side condition, explicit compensator formulas for the jump-driven crossing modes, and a decomposition of the compensator of the default indicator into its predictable jump part and continuous part. As an application, for a mean-reverting affine jump-diffusion with upward exponential jumps, we derive the boundary-value problem governing the overshoot mode, prove that the differentiated third-order ODE is equivalent to the original problem only when a boundary compatibility condition is retained, and establish verification and uniqueness for the discounted problem. This yields an explicit Green-Volterra representation, a first-order small-q expansion expansion, and, in the undiscounted case, closed formulas for the overshoot and creeping probabilities

math.PR↗

arXiv2026

Exit times from time-dependent random domains: continuity, weak convergence, and exit-time profiles Draft -currently under review at Stochastic Processes and their Applications

We study exit times from time-dependent domains under joint perturbations of the trajectory and the domain. Representing a moving domain by a continuous barrier $Φ$ on space-time, we reduce the exit problem to a one-dimensional first-passage problem for the scalarised path $y(t) := Φ(t,x(t))$. Our first main result is a deterministic continuity theorem: the exit-time functional is continuous, under local Skorokhod $J_1$ convergence of the path and local uniform convergence of the barrier, at every configuration satisfying an explicit non-tangency condition (NT). We show that NT is sharp in the sense that it characterises the continuity set of the functional. As a direct consequence, weak convergence of exit times follows from joint weak convergence of paths and barriers whenever the limiting pair satisfies NT almost surely; no independence or structural restrictions between trajectory and domain are required. Our second main result is a functional limit theorem: the exit-time profile $u\mapstoτ(u)$, viewed as a càdlàg function of the barrier level, converges in the Skorokhod $M_1$ topology under the same hypotheses, with a concrete example showing that $J_1$ convergence can fail. Concrete verification routes for NT are provided, including a non-characteristic/Itô criterion for diffusions, and the full framework is illustrated through a worked Donsker-type example.

math.PR↗

arXiv2026

Persistence of the Wiener Sausage: Sampling Stability and a Law of Large Numbers for Drifted Planar Brownian Motion DRAFT -CURRENTLY UNDER REVIEW

We study the persistent homology of the offset filtration generated by the range of a planar Brownian motion with constant nonzero drift. The members of this filtration are the Wiener sausages of increasing radius, and the degree-one persistence diagram records the birth and death of holes in the thickened trace as the radius varies. Our first result is a sampling theorem: for any continuous path in R d observed on a time grid $π$n the bottleneck distance between the persistence diagram of the continuous offset filtration and that of the sampled point cloud is bounded by the pathwise modulus of continuity $ω$X (|$π$n|). For Brownian motion this yields the almost-sure rate O |$π$n| log(1/|$π$n|) . Our second and main result is a law of large numbers for the drifted planar case. For every bounded Borel weight $ψ$ supported on a compact radius window [r0, r1] with r0 > 0, the smoothed persistence functional $Φ$ $ψ$ (T ), where $β$ T 1 (r) counts the holes in the radius-r sausage at time T , satisfies $Φ$ $ψ$ (T )/T $\rightarrow$ $ρ$ $ψ$ almost surely and in L 1 for a deterministic constant $ρ$ $ψ$ . This yields a finite positive intensity measure on the radius axis that governs the linear growth of topological complexity. The proof introduces a regeneration scheme along the drift direction: projecting the planar path onto the drift axis produces a one-dimensional Brownian motion with positive drift, whose ladder hits and bounded-backtracking events generate i.i.d. path blocks. The non-additivity of topology under concatenation is controlled by a Boundary Lemma, which combines a deterministic Mayer-Vietoris estimate with a geometric bound relating integrated Betti numbers to sausage area via the coarea formula. A Betti-curve representation converts the two-parameter persistence problem into a one-parameter family of fixed-radius hole counts, making the regeneration argument possible.

math.PR↗