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Posterior distributions for community structure in sparse planted bi-section models are shown to achieve exact (resp. almost-exact) recovery, with sharp bounds for the sparsity regimes where edge probabilities decrease as $O(\log(n)/n)$ (resp. $O(1/n)$). Assuming posterior recovery, one may interpret credible sets (resp. enlarged credible sets) as asymptotically consistent confidence sets; the diameters of those credible sets are controlled by the rate of posterior concentration. If credible levels are chosen to grow to one quickly enough, corresponding credible sets can be interpreted as frequentist confidence sets without conditions on posterior concentration. In the regimes with $O(1/n)$ edge sparsity, or when within-community and between-community edge probabilities are very close, credible sets may be enlarged to achieve frequentist asymptotic coverage, also without conditions on posterior concentration., 29 pp., 3 fig

We consider a general mathematical model for protein subunit vaccine with a focus on the MF59-adjuvanted spike glycoprotein-clamp vaccine for SARS-CoV-2, and use the model to study immunological outcomes in the humoral and cell-mediated arms of the immune response from vaccination. The mathematical model is fit to vaccine clinical trial data. We elucidate the role of Interferon-𝛾 and Interleukin-4 in stimulating the immune response of the host. Model results, and results from a sensitivity analysis, show that a balance between the T𝐻 1 and T𝐻 2 arms of the immune response is struck, with the T𝐻 1 response being dominant. The model predicts that two-doses of the vaccine at 28 days apart will result in approximately 85% humoral immunity loss relative to peak immunity approximately 6 months post dose 1.

For a discrete time multitype supercritical Galton-Watson process $(Z_n)_{n\in \mathbb{N}}$ and corresponding genealogical tree $\mathbb{T}$, we associate a new discrete time process $(Z_n^{\Phi})_{n\in\mathbb{N}}$ such that, for each $n\in \mathbb{N}$, the contribution of each individual $u\in\mathbb{T}$ to $Z_n^{\Phi}$ is determined by a (random) characteristic $\Phi$ evaluated at the age of $u$ at time $n$. In other words, $Z_n^{\Phi}$ is obtained by summing over all $u\in \mathbb{T}$ the corresponding contributions $\Phi_u$, where $(\Phi_u)_{u\in \mathbb{T}}$ are i.i.d. copies of $\Phi$. Such processes are known in the literature under the name of Crump-Mode-Jagers (CMJ) processes counted with characteristic $\Phi$. We derive a LLN and a CLT for the process $(Z_n^{\Phi})_{n\in\mathbb{N}}$ in the discrete time setting, and in particular, we show a dichotomy in its limit behavior. By applying our main result, we also obtain a generalization of the results in Kesten-Stigum [17]., Comment: a revisited version, which incorporates the referee's suggestions, and several other improvements and explanations. In particular, the section explaining how to recover the results from Kesten-Stigum [17] has been rewritten

For classical finite time horizon stopping problems driven by a Brownian motion \[V(t,x) = \sup_{t\leq\tau\leq0}E_{(t,x)}[g(\tau,W_{\tau})],\] we derive a new class of Fredholm type integral equations for the stopping set. For large problem classes of interest, we show by analytical arguments that the equation uniquely characterizes the stopping boundary of the problem. Regardless of the uniqueness, we use the representation to rigorously find the limit behavior of the stopping boundary close to the terminal time. Interestingly, it turns out that the leading-order coefficient is universal for wide classes of problems. We also discuss how the representation can be used for numerical purposes.

In this article we study multivariate continuous-time autoregressive moving-average (MCARMA) processes with values in convex cones. More specifically, we introduce matrix-valued MCARMA processes with L\'evy noise and present necessary and sufficient conditions for processes from this class to be cone valued. We derive specific hands-on conditions in the following two cases: First, for classical MCARMA on $\mathbb{R}_{d}$ with values in the positive orthant $\mathbb{R}_{d}^{+}$. Second, for MCARMA processes on real square matrices taking values in the cone of symmetric and positive semi-definite matrices. Both cases are relevant for applications and we give several examples of positivity ensuring parameter specifications. In addition to the above, we discuss the capability of positive semi-definite MCARMA processes to model the spot covariance process in multivariate stochastic volatility models. We justify the relevance of MCARMA based stochastic volatility models by an exemplary analysis of the second order structure of positive semi-definite well-balanced Ornstein-Uhlenbeck based models., Comment: 37 pages

We define and study the Dirichlet fractional Gaussian fields on the Sierpinski gasket and show that they are limits of fractional discrete Gaussian fields defined on the sequence of canonical approximating graphs., v2: Accepted version, to appear in Stoch. Proc. Appl

This paper investigates multidimensional extended type-I BSVIEs and infinite families of BSDEs in the case of quadratic generators. We establish existence and uniqueness results in the case of fully quadratic as well as Lipschitz-quadratic quadratic generators. We also present and discuss a type of flow property satisfied by this family of BSVIEs. As a preliminary step, we establish the well-posedness of a class of infinite families of BSDEs, as introduced in Hern{\aa}ndez and Possama\"i [20], which are of interest in their own right. Our approach relies on the strategy developed by Tevsadze [49] for quadratic BSDEs and the treatment of Lipschitz extended type-I BSVIEs in [20]. We motivate their analysis of both of these objects by a series of practical applications.

We analyse the problem of online parameter estimation for a stochastic McKean–Vlasov equation, and the associated system of weakly interacting particles. We propose an online estimator for the parameters of the McKean–Vlasov SDE, or the interacting particle system, which is based on a continuous-time stochastic gradient ascent scheme with respect to the asymptotic log-likelihood of the interacting particle system. We characterise the asymptotic behaviour of this estimator in the limit as � → ∞ , and also in the joint limit as � → ∞ and � → ∞ . In these two cases, we obtain almost sure or � 1 convergence to the stationary points of a limiting contrast function, under suitable conditions which guarantee ergodicity and uniform-in-time propagation of chaos. We also establish, under the additional condition of global strong concavity, � 2 convergence to the unique maximiser of the asymptotic log-likelihood of the McKean–Vlasov SDE, with an asymptotic convergence rate which depends on the learning rate, the number of observations, and the dimension of the non-linear process. Our theoretical results are supported by two numerical examples, a linear mean field model and a stochastic opinion dynamics model.

We consider a diffusion in $\mathbb{R}^n$ whose coordinates each behave as one-dimensional Brownian motions, that behave independently when apart, but have a sticky interaction when they meet. The diffusion in $\mathbb{R}^n$ can be viewed as the $n$-point motion of a stochastic flow of kernels. We derive the Kolmogorov backwards equation and show that for a specific choice of interaction it can be solved exactly with the Bethe ansatz. We then use our formulae to study the behaviour of the flow of kernels for the exactly solvable choice of interaction., Comment: 43 pages

We derive the Marchenko-Pastur (MP) law for sample covariance matrices of the form $V_n=\frac{1}{n}XX^T$, where $X$ is a $p\times n$ data matrix and $p/n\to y\in(0,\infty)$ as $n,p \to \infty$. We assume the data in $X$ stems from a correlated joint normal distribution. In particular, the correlation acts both across rows and across columns of $X$, and we do not assume a specific correlation structure, such as separable dependencies. Instead, we assume that correlations converge uniformly to zero at a speed of $a_n/n$, where $a_n$ may grow mildly to infinity. We employ the method of moments tightly: We identify the exact condition on the growth of $a_n$ which will guarantee that the moments of the empirical spectral distributions (ESDs) converge to the MP moments. If the condition is not met, we can construct an ensemble for which all but finitely many moments of the ESDs diverge. We also investigate the operator norm of $V_n$ under a uniform correlation bound of $C/n^{\delta}$, where $C,\delta>0$ are fixed, and observe a phase transition at $\delta=1$. In particular, convergence of the operator norm to the maximum of the support of the MP distribution can only be guaranteed if $\delta>1$. The analysis leads to an example for which the MP law holds almost surely, but the operator norm remains stochastic in the limit, and we provide its exact limiting distribution.

Nonparametric estimation for semilinear SPDEs, namely stochastic reaction-diffusion equations in one space dimension, is studied. We consider observations of the solution field on a discrete grid in time and space with infill asymptotics in both coordinates. Firstly, we derive a nonparametric estimator for the reaction function of the underlying equation. The estimate is chosen from a finite-dimensional function space based on a least squares criterion. Oracle inequalities provide conditions for the estimator to achieve the usual nonparametric rate of convergence. Adaptivity is provided via model selection. Secondly, we show that the asymptotic properties of realized quadratic variation based estimators for the diffusivity and volatility carry over from linear SPDEs. In particular, we obtain a rate-optimal joint estimator of the two parameters. The result relies on our precise analysis of the H\"older regularity of the solution process and its nonlinear component, which may be of its own interest. Both steps of the calibration can be carried out simultaneously without prior knowledge of the parameters.

We introduce and study some backward Kolmogorov equations associated to stochastic filtering problems. Measure-valued processed arise naturally in the context of stochastic filtering and one can formulate two stochastic differential equations, called Zakai and Kushner-Stratonovitch equation, that are satisfied by a positive measure and a probability measure-valued process respectively. The associated Kolmogorov equations have been intensively studied, mainly assuming that the measure-valued processes admit a density and then by exploiting stochastic calculus techniques in Hilbert spaces. Our approach differs from this since we do not assume the existence of a density and we work directly in the context of measures. We first formulate two Kolmogorov equations of parabolic type, one on a space of positive measures and one on a space of probability measures, and then we prove existence and uniqueness of classical solutions. In order to do that, we prove some intermediate results of independent interest. In particular, we prove It\^o formulas for the composition of measure-valued filtering processes and real-valued functions. Moreover we study the regularity of the solution to the filtering equations with respect to the initial datum. In order to achieve these results, proper notions of derivatives on space of positive measures have been introduced and discussed.

We extend the theoretical results for any FOU(p) processes for the case in which the Hurst parameter is less than 1/2 and we show theoretically and by simulations that under some conditions on T and the sample size n it is possible to obtain consistent estimators of the parameters when the process is observed in a discretized and equispaced interval [0, T ]. Also we will show that the FOU(p) processes can be used to model a wide range of time series varying from short range dependence to large range dependence with similar results as the ARMA or ARFIMA models, and in several cases outperforms those. Lastly, we give a way to obtain explicit formulas for the auto-covariance function for any FOU(p) and we present an application for FOU(2) and FOU(3).

We obtain well-posedness results for a class of ODE with a singular drift and additive fractional noise, whose right-hand-side involves some bounded variation terms depending on the solution. Examples of such equations are reflected equations, where the solution is constrained to remain in a rectangular domain, as well as so-called perturbed equations, where the dynamics depend on the running extrema of the solution. Our proof is based on combining the Catellier-Gubinelli approach based on Young nonlinear integration, with some Lipschitz estimates in $p$-variation for maps of Skorokhod type, due to Falkowski and S{\l}ominski. An important step requires to prove that fractional Brownian motion, when perturbed by sufficiently regular paths (in the sense of $p$-variation), retains its regularization properties. This is done by applying a variant of the stochastic sewing lemma., Comment: accepted in SPA

To deal with stochastic hybrid systems with general state-dependent switching, we propose an approximation method by a sequence of stochastic hybrid systems with piecewise constant type switching. The convergence rate in the Wasserstein distance is estimated in terms of the difference between transition rate matrices. Our method is based on an elaborate construction of coupling processes in terms of Skorokhod's representation theorem for jumping processes. Moreover, we establish explicit criteria on the ergodicity and stability for stochastic hybrid systems with piecewise constant type switching. Some examples are given to illustrate the sharpness of these criteria., Comment: 26 pages

In this work, four problems for stochastic fractional pseudo-parabolic containing bounded and unbounded delays are investigated. The fractional derivative and the stochastic noise we consider here are the Caputo operator and the fractional Brownian motion. For the two problems involving bounded delays, we aim at establishing global existence, uniqueness, and regularity results under integral Lipschitz conditions for the non-linear source terms. Such behaviors of mild solutions are also analyzed in the unbounded delay cases but under globally and locally Lipschitz assumptions. We emphasize that our results are investigated in the novel spaces C([−r, T];L p (Ω, Wl,q(D))), Cµ((−∞, T];L p (Ω, Wl,q(D))), and the weighted space F ε µ((−∞, T];L p (Ω, Wl,q(D))), instead of usual ones C([−r, T];L 2 (Ω, H)), Cµ((−∞, T];L 2 (Ω, H)). The main technique allowing us to overcome the rising difficulties lies on some useful Sobolev embeddings between the Hilbert space H = L 2 (D) and Wl,q(D), and some well-known fractional tools. In addition, we also study the H¨older continuity for the mild solutions, which can be considered as one of the main novelties of this paper. Finally, we consider an additional result connecting delay stochastic fractional pseudo-parabolic equations and delay stochastic fractional parabolic equations. We show that the mild solution of the first model converges to the mild solution of the second one, in some sense, as the diffusion parameter β → 0 +.

In this paper, we study the connections between three concepts - the reverse H\"older inequality for matrix-valued martingales, the well-posedness of linear BSDEs with unbounded coefficients, and the well-posedness of quadratic BSDE systems. In particular, we show that a linear BSDE with bmo (bounded mean oscillation) coefficients is well-posed if and only if the stochastic exponential of a related matrix-valued martingale satisfies a reverse H\"older inequality. Furthermore, we give structural conditions under which these two equivalent conditions are satisfied. Finally, we apply our results on linear equations to obtain global well-posedness results for two new classes of non-Markovian quadratic BSDE systems with special structure.

We prove an extension of a seminal result of Bertini and Giacomin. Namely we consider weakly asymmetric exclusion processes with several distinct initial data simultaneously, then run according to the basic coupling, and we show joint convergence to the solution of the KPZ equation with the same driving noise in the limiting equation. Along the way, we analyze fine properties of nontrivially coupled solutions-in-law of KPZ-type equations., Comment: 22 pages