arXiv subjects

Johannes Brutsche

Publications and source records attributed to Johannes Brutsche.

6 recordsLinked to original sources

arXiv2026

The level of self-organized criticality in oscillating Brownian motion: $n$-consistency and stable Poisson-type convergence of the MLE

For some discretely observed path of oscillating Brownian motion with level of self-organized criticality $ρ_0$, we prove in the infill asymptotics that the MLE is $n$-consistent, where $n$ denotes the sample size, and derive its limit distribution with respect to stable convergence. As the transition density of this homogeneous Markov process is not even continuous in $ρ_0$, the analysis is highly non-standard. Therefore, interesting and somewhat unexpected phenomena occur: The likelihood function splits into several components, each of them contributing very differently depending on how close the argument $ρ$ is to $ρ_0$. Correspondingly, the MLE is successively excluded to lay outside a compact set, a $1/\sqrt{n}$-neighborhood and finally a $1/n$-neighborhood of $ρ_0$ asymptotically. The crucial argument to derive the stable convergence is to exploit the semimartingale structure of the sequential suitably rescaled local log-likelihood function (as a process in time). Both sequentially and as a process in $ρ$, it exhibits a bivariate Poissonian behavior in the stable limit with its intensity being a multiple of the local time at $ρ_0$.

math.ST↗

arXiv2026

Arbitrage in Estimate Nothing: an example

We give a two-period counterexample to the absence of arbitrage for the posterior-weighted pricing rule in Estimate nothing by Duembgen and Rogers. Both physical models have strictly positive transition densities, and each model is equipped with an equivalent martingale measure. Nevertheless, the mixed price of a single derivative falls deterministically from $5/2$ to $2$ between two trading dates. If these prices are tradable, shorting the derivative and closing the position one period later yields a certain profit. A finite-state appendix also illustrates the failure of recursive consistency.

q-fin.MF↗

arXiv2026

Stable convergence of partial sum processes towards discontinuous limits

We develop a stable convergence theorem for partial sum processes on sample-size dependent stochastic bases. The result allows multidimensional semimartingale limits that have conditionally independent increments and both a continuous and discontinuous martingale part. Motivated by infill asymptotics, it complements classical Gaussian stable limit theorems and supports applications to likelihood based statistical inference.

math.PR↗

arXiv2026

Self-organized regime switching in null-recurrent dynamics

Based on discrete observations $X_0,X_Δ,\dots, X_{nΔ}$ for $Δ=n^{-γ}$ with $γ\in [0,1)$ of the null-recurrent dynamic $dX_t = σ(X_t)dW_t$ with a Brownian motion $W$ and $σ(x)=α\mathbb{1}\{x<ρ\} + β\mathbb{1}\{x\geq ρ\}$, we derive rate of convergence and limiting distribution of the profile MLE for $ρ$. This includes low-frequency asymptotics ($γ=0$) for which the observations form a null-recurrent Markov chain. The derived non-standard limit is the argsup over a doubly stochastic drifted Poisson process explicitly involving the local time of oscillating Brownian motion. Its dependence on $ρ$ as well as the unknown volatility levels $α$ and $β$ is shown to be continuous w.r.t. the topology of weak convergence, enabling statistical inference. Whereas this limit is independent of the sampling frequency, the profile MLE's rate of convergence equals $n^{-(1+γ)/2}$ and is proven to be minimax optimal. The surprising idea of the proof of the limit theorem is to relate the long-term behavior of the null-recurrent Markov chain to the infill asymptotics on a fixed time interval. Indeed, in the very special case that $(X_t)_{t\geq 0}$ is started in the true parameter $X_0=ρ_0$, the process $(X_t-ρ_0)_{t\geq 0}$ is shown to possess a desirable distributional self-similarity. On basis of the strong Markov property, the artificial constallation of starting in $ρ_0$ is finally overcome by a coupling argument.

math.ST↗

arXiv2026

Sharp adaptive nonparametric testing for constant volatility

Based on discrete observations, we develop a test to infer if the volatility function $σ(\cdot)$ within the nonparametric Gaussian white noise model $dY_t = σ(t)dW_t$ is constant. The testing procedure is shown to be minimax-optimal and adaptive for infill asymptotics and these results entail that a deviation from the null hypothesis of constancy is best measured in terms of the ratio of $σ(t)$ and its $L^2$-average. The derivation of optimal constants requires the construction of hypotheses with height $h(b)$, where the parameter $b$ solves $F_n(b)=0$ for given functions $F_n$. Proving this equation to be solvable for each $n\in\mathbb{N}$ and establishing quantitative bounds of the solutions is built upon the implicit function theorem.

math.ST↗

arXiv2024

Sharp adaptive and pathwise stable similarity testing for scalar ergodic diffusions

Within the nonparametric diffusion model, we develop a multiple test to infer about similarity of an unknown drift $b$ to some reference drift $b_0$: At prescribed significance, we simultaneously identify those regions where violation from similiarity occurs, without a priori knowledge of their number, size and location. This test is shown to be minimax-optimal and adaptive. At the same time, the procedure is robust under small deviation from Brownian motion as the driving noise process. A detailed investigation for fractional driving noise, which is neither a semimartingale nor a Markov process, is provided for Hurst indices close to the Brownian motion case.

math.ST↗