Journal of Theoretical Probability
Journal Prestige (SJR): 0.981 Citation Impact (citeScore): 1 Number of Followers: 3 Hybrid journal (It can contain Open Access articles) ISSN (Print) 15729230  ISSN (Online) 08949840 Published by SpringerVerlag [2467 journals] 
 Moderate Deviation Principle for Multiscale Systems Driven by Fractional
Brownian Motion
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Abstract: Abstract In this paper, we study the moderate deviations principle (MDP) for slow–fast stochastic dynamical systems where the slow motion is governed by small fractional Brownian motion (fBm) with Hurst parameter \(H\in (1/2,1)\) . We derive conditions on the moderate deviations scaling and on the Hurst parameter H under which the MDP holds. In addition, we show that in typical situations the resulting action functional is discontinuous in H at \(H=1/2\) , suggesting that the tail behavior of stochastic dynamical systems perturbed by fBm can have different characteristics than the tail behavior of such systems that are perturbed by standard Brownian motion.
PubDate: 20230116

 Asymptotic Properties of Random Contingency Tables with Uniform Margin

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Abstract: Abstract Let \(C\ge 2\) be a positive integer. Consider the set of \(n\times n\) nonnegative integer matrices whose row sums and column sums are all equal to Cn and let \(X=(X_{ij})_{1\le i,j\le n}\) be uniformly distributed on this set. This X is called the random contingency table with uniform margin. In this paper, we study various asymptotic properties of \(X=(X_{ij})_{1\le i,j\le n}\) as \(n\rightarrow \infty \) .
PubDate: 20230109

 Normal Approximation of Compound Hawkes Functionals

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Abstract: Abstract We derive quantitative bounds in the Wasserstein distance for the approximation of stochastic integrals with respect to Hawkes processes by a normally distributed random variable. In the case of deterministic and nonnegative integrands, our estimates involve only the third moment of the integrand in addition to a variance term using a squared norm of the integrand. As a consequence, we are able to observe a “third moment phenomenon” in which the vanishing of the first cumulant can lead to faster convergence rates. Our results are also applied to compound Hawkes processes, and improve on the current literature where estimates may not converge to zero in large time or have been obtained only for specific kernels such as the exponential or Erlang kernels.
PubDate: 20230107

 A Large Deviation Principle for the Stochastic Heat Equation with General
Rough Noise
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Abstract: Abstract We study the Freidlin–Wentzell large deviation principle for the nonlinear onedimensional stochastic heat equation driven by a Gaussian noise: $$\begin{aligned} \frac{\partial u^{{\varepsilon }}(t,x)}{\partial t}=\frac{\partial ^2 u^{{\varepsilon }}(t,x)}{\partial x^2}+\sqrt{{\varepsilon }}\sigma (t, x, u^{{\varepsilon }}(t,x))\dot{W}(t,x),\quad t> 0,\, x\in \mathbb {R}, \end{aligned}$$ where \(\dot{W}\) is white in time and fractional in space with Hurst parameter \(H\in \left( \frac{1}{4},\frac{1}{2}\right) \) . Recently, Hu and Wang (Ann Inst Henri Poincaré Probab Stat 58(1):379–423, 2022) have studied the wellposedness of this equation without the technical condition of \(\sigma (0)=0\) which was previously assumed in Hu et al. (Ann Probab 45(6):4561–4616, 2017). We adopt a new sufficient condition proposed by Matoussi et al. (Appl Math Optim 83(2):849–879, 2021) for the weak convergence criterion of the large deviation principle.
PubDate: 20230106

 Semiuniform Feller Stochastic Kernels

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Abstract: Abstract This paper studies transition probabilities from a Borel subset of a Polish space to a product of two Borel subsets of Polish spaces. For such transition probabilities it introduces and studies the property of semiuniform Feller continuity. This paper provides several equivalent definitions of semiuniform Feller continuity and establishes its preservation under integration. The motivation for this study came from the theory of Markov decision processes with incomplete information, and this paper provides the fundamental results useful for this theory.
PubDate: 20230106

 Stochastic Dynamics of Generalized Planar Random Motions with Orthogonal
Directions
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Abstract: Abstract We study planar random motions with finite velocities, of norm \(c>0\) , along orthogonal directions and changing at the instants of occurrence of a nonhomogeneous Poisson process with rate function \(\lambda = \lambda (t),\ t\ge 0\) . We focus on the distribution of the current position \(\bigl (X(t), Y(t)\bigr ),\ t\ge 0\) , in the case where the motion has orthogonal deviations and where also reflection is admitted. In all the cases, the process is located within the closed square \(S_{ct}=\{(x,y)\in {\mathbb {R}}^2\,:\, x + y \le ct\}\) and we obtain the probability law inside \(S_{ct}\) , on the edge \(\partial S_{ct}\) and on the other possible singularities, by studying the partial differential equations governing all the distributions examined. A fundamental result is that the vector process (X, Y) is probabilistically equivalent to a linear transformation of two (independent or dependent) onedimensional symmetric telegraph processes with rate function proportional to \(\lambda \) and velocity c/2. Finally, we extend the results to a wider class of orthogonaltype evolutions.
PubDate: 20230104

 On the Probabilistic Representation of the Free Effective Resistance of
Infinite Graphs
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Abstract: Abstract We completely characterize when the free effective resistance of an infinite graph whose vertices have finite degrees can be expressed in terms of simple hitting probabilities of the random walk on the graph.
PubDate: 20230104

 Strong Local Nondeterminism and Exact Modulus of Continuity for Isotropic
Gaussian Random Fields on Compact TwoPoint Homogeneous Spaces
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Abstract: Abstract This paper is concerned with sample path properties of realvalued isotropic Gaussian fields on compact twopoint homogeneous spaces. In particular, we establish the property of strong local nondeterminism of an isotropic Gaussian field and then exploit this result to establish an exact uniform modulus of continuity for its sample paths.
PubDate: 20230103

 Selfsimilar Free Additive Processes and Freely Selfdecomposable
Distributions
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Abstract: Abstract In the paper by Fan (Inf Dim Anal Quant Probab Rel Topics 9:451–469, 2006), he introduced the marginal selfsimilarity of noncommutative stochastic processes and proved that the marginal distributions of selfsimilar processes with freely independent increments are freely selfdecomposable. In this paper, we firstly introduce a new definition, stronger than Fan’s in general, of selfsimilarity via linear combinations of noncommutative stochastic processes, although the two definitions are equivalent for noncommutative stochastic processes with freely independent increments. We secondly prove the converse of Fan’s result, to complete the relationship between selfsimilar free additive processes and freely selfdecomposable distributions. Furthermore, we construct stochastic integrals with respect to free additive processes for representing the background driving free Lévy processes of freely selfdecomposable distributions. A relationship between freely selfdecomposable distributions and their background driving free Lévy processes in terms of their free cumulant transforms is also given, and several examples are discussed.
PubDate: 20221229

 Error distribution of the Euler approximation scheme for stochastic
Volterra equations
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Abstract: Abstract The purpose of this paper is to establish the convergence in distribution of the normalized error in the Euler approximation scheme for stochastic Volterra equations driven by a standard Brownian motion, with a kernel of the form \((ts)^\alpha \) , where \(\alpha \in \left( \frac{1}{2}, \frac{1}{2}\right) \) .
PubDate: 20221223

 Limit Theorems for Deviation Means of Independent and Identically
Distributed Random Variables
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Abstract: Abstract We derive a strong law of large numbers, a central limit theorem, a law of the iterated logarithm and a large deviation theorem for socalled deviation means of independent and identically distributed random variables. (For the strong law of large numbers, we suppose only pairwise independence instead of (total) independence.) The class of deviation means is a special class of Mestimators or more generally extremum estimators, which are well studied in statistics. The assumptions of our limit theorems for deviation means seem to be new and weaker than the known ones for Mestimators in the literature. In particular, our results on the strong law of large numbers and on the central limit theorem generalize the corresponding ones for quasiarithmetic means due to de Carvalho (Am Stat 70(3):270–274, 2016) and the ones for Bajraktarević means due to Barczy and Burai (Aequ Math 96(2):279–305, 2022).
PubDate: 20221223

 Ergodic Theorems with Random Weights for Stationary Random Processes and
Fields
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Abstract: Abstract Let X(t) be an ergodic stationary random process or an ergodic homogeneous random field on \({\mathbb {R}}^m,m\ge 2\) , and let M(B) be a mixing homogeneous locally finite random Borel measure with mean density \(\gamma \) on \({\mathbb {R}}^m,m\ge 1\) . We assume that X and M are independent and possess finite expectations. If \(\{T_n\}\) is an increasing sequence of bounded convex sets, containing balls of radii \(r_n\rightarrow \infty \) , then $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{\lambda (T_n)}\int _{T_n}X(t)M(\textrm{d}t,w)=\gamma E[X(0)]\text { a.s. and in } L^1. \end{aligned}$$ Special cases are ergodic theorems with averages over finite random sets. Example: If S is an independentofX Poisson random set in \({\mathbb {R}}^m\) with mean density \(\gamma \) , then $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{ {\lambda (T_n)}}\sum _{t\in S \cap T_n}X(t)=\gamma E[X(0 )] \ \text {a.s. and in } L^1 \ \ (\text {card} (S\cap T_n)<\infty \ \text {a.s.}). \end{aligned}$$ These theorems offer a universal way of constructing consistent estimators using observations on finite sets.
PubDate: 20221223

 Existence and Uniqueness of Solutions for Multidimensional Reflected
Backward Stochastic Differential Equations with Diagonally Quadratic
Generators
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Abstract: Abstract In this paper, we study multidimensional reflected backward stochastic differential equations (BSDEs) with diagonally quadratic generators. Using the comparison theorem for diagonally quadratic BSDEs established recently in Luo (Disc Contin Dyn Syst 41(6):2543–2557, 2021), we obtain the existence and uniqueness of a solution by a penalization method. Moreover, we provide a comparison theorem.
PubDate: 20221219

 WassersteinType Distances of TwoType ContinuousState Branching
Processes in Lévy Random Environments
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Abstract: Abstract Under natural conditions, we prove exponential ergodicity in the \( L_1\) Wasserstein distance of twotype continuousstate branching processes in Lévy random environments with immigration. Furthermore, we express precisely the parameters of the exponent. The coupling method and the conditioned branching property play an important role in the approach. Using the tool of superprocesses, ergodicity in total variation distance is also proved.
PubDate: 20221219

 Support Theorem for Lévydriven Stochastic Differential Equations

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Abstract: Abstract We provide a support theorem for the law of the solution to a stochastic differential equation (SDE) with jump noise. This theorem applies to quite general Lévydriven SDEs and is illustrated by examples with rather degenerate jump noises, where the theorem leads to an informative description of the support.
PubDate: 20221217

 Resolution of SigmaFields for Multiparticle FiniteState Action
Evolutions with Infinite Past
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Abstract: Abstract For multiparticle finitestate action evolutions, we prove that the observation \( \sigma \) field admits a resolution involving a third noise which is generated by a random variable with uniform law. The Rees decomposition from semigroup theory and the theory of infinite convolutions are utilized in our proofs.
PubDate: 20221217

 Using Stein’s Method to Analyze Euler–Maruyama Approximations of
RegimeSwitching Jump Diffusion Processes
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Abstract: Abstract For a kind of regimeswitching jump diffusion process \((X_t,Z_t)_{t\ge 0}\) , under some conditions, it is exponentially ergodic under the weighted total variation distance with ergodic measure \(\mu \) . We use the Euler–Maruyama scheme of the process \((X_t,Z_t)_{t\ge 0}\) which has an ergodic measure \(\mu _{\eta }\) ( \(\eta \) is the step size of the Euler–Maruyama scheme) to approximate the ergodic measure \(\mu \) . Furthermore, we use Stein’s method to prove that the convergence rate of \(\mu _{\eta }\) to \(\mu \) is \(\eta ^{\frac{1}{2}}\) in terms of some functionclass distance \(d_{{\mathcal {G}}}(\mu ,\mu _{\eta })\) .
PubDate: 20221215

 Equivalence–Singularity Dichotomy in Markov Measures

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Abstract: Abstract We establish an equivalence–singularity dichotomy for a large class of onedimensional Markov measures. Our approach is new in that we deal with onesided and twosided chains simultaneously and in that we do not appeal to a 01 law. In fact, we deduce a new 01 law from the dichotomy.
PubDate: 20221202

 SecondOrder Tail Behavior for Stochastic Discounted Value of Aggregate
Net Losses in a DiscreteTime Risk Model
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Abstract: Abstract Consider a discretetime risk model, in which an insurer makes both riskfree and risky investments. Within period k, the net loss is denoted by a realvalued random variable \(X_k\) , and the stochastic discount factor is a bounded positive random variable \(Y_k\) . Assume that \((X_k,Y_k), k\in {\mathbb {N}}\) , form a sequence of independent and identically distributed random pairs following a common bivariate Farlie–Gumbel–Morgenstern distribution with marginal distributions F on \({\mathbb {R}}\) and G on [a, b], respectively, for some \(0<a\le b<\infty \) . Under the condition that F is secondorder subexponential, we establish a secondorder expansion for the tail probability of the stochastic discounted value of aggregate net losses. Compared with the firstorder one, our secondorder asymptotic result is more precise, which is demonstrated by numerical studies.
PubDate: 20221201

 HigherOrder Error Estimates of the DiscreteTime Clark–Ocone
Formula
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Abstract: Abstract In this article, we investigate the convergence rate of the discretetime Clark–Ocone formula provided by Akahori–Amaba–Okuma (J Theor Probab 30: 932–960, 2017). In that paper, they mainly focus on the \(L_{2}\) convergence rate of the firstorder error estimate related to the tracking error of the delta hedge in mathematical finance. Here, as two extensions, we estimate “the higher order error” for Wiener functionals with an integrability index 2 and “an arbitrary differentiability index.”
PubDate: 20221201
