The bi-Poisson process: a quadratic harness

Quadratic Harnesses, q-commutations, and orthogonal martingale polynomials

We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a q-commutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical constants. Explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest.

Stitching pairs of Levy processes into harnesses

We consider natural exponential families of Levy processes with randomized parameter. Such processes are Markov, and under suitable assumptions, pairs of such processes with shared randomization can be stitched together into a single harness. The stitching consists of deterministic reparametrization of the time for both processes, so that they run on adjacent time intervals, and of the choice of the appropriate law at the boundary. Processes in the Levy-Meixner class have an additional property that they are quadratic harnesses, and in this case stitching constructions produce quadratic harnes…

Askey--Wilson polynomials, quadratic harnesses and martingales

We use orthogonality measures of Askey--Wilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances. Askey--Wilson polynomials are orthogonal martingale polynomials for these processes.

Wilson's 6-j laws and stitched Markov processes

We show how to insert time into the parameters of the Wilson's 6-j laws to construct discrete Markov chains with these laws. By a quadratic transformation we convert them into Markov processes with linear regressions and quadratic conditional variances. Further conversion into the "standard form" gives "quadratic harnesses" with "classical" value of parameter gamma. A random-parameter-representation of the original Markov chain allows us to stitch together two copies of the process, extending time domain of the quadratic harness from (0,1) to all t>0.

Asymmetric Simple Exclusion Process with open boundaries and Quadratic Harnesses

We show that the joint probability generating function of the stationary measure of a finite state asymmetric exclusion process with open boundaries can be expressed in terms of joint moments of Markov processes called quadratic harnesses. We use our representation to prove the large deviations principle for the total number of particles in the system. We use the generator of the Markov process to show how explicit formulas for the average occupancy of a site arise for special choices of parameters. We also give similar representations for limits of stationary measures as the number of sites t…

Asymptotic Distribution of Bernoulli Quadratic Forms

Consider the random quadratic form $T_n=\sum_{1 \leq u < v \leq n} a_{uv} X_u X_v$, where $((a_{uv}))_{1 \leq u, v \leq n}$ is a $\{0, 1\}$-valued symmetric matrix with zeros on the diagonal, and $X_1,$ $X_2, \ldots, X_n$ are i.i.d. $\mathrm{Ber}(p_n)$. In this paper, we prove various characterization theorems about the limiting distribution of $T_n$, in the sparse regime, where $0 < p_n \ll 1$ such that $\mathbb E(T_n)=O(1).$ The main result is a decomposition theorem showing that distributional limits of $T_n$ is the sum of three components: a mixture which consists of a quadratic function o…

Robustness of quadratic hedging strategies in finance via backward stochastic differential equations with jumps

We consider a backward stochastic differential equation with jumps (BSDEJ) which is driven by a Brownian motion and a Poisson random measure. We present two candidate-approximations to this BSDEJ and we prove that the solution of each candidate- approximation converges to the solution of the original BSDEJ in a space which we specify. We use this result to investigate in further detail the consequences of the choice of the model to (partial) hedging in incomplete markets in finance. As an application, we consider models in which the small variations in the price dynamics are modeled with a Poi…

脉冲神经元网络的泊松假设与大种群极限

Poisson Hypothesis and large-population limit for networks of spiking neurons

本文证明泊松假设对具有线性和二次内在动力学的Galves-Löcherbach脉冲神经元网络的副本平均场极限成立，并进一步证明其大种群极限为受随机重置影响的适定神经场模型。

Infinitesimal generators for a class of polynomial processes

We study the infinitesimal generators of evolutions of linear mappings on the space of polynomials, which correspond to a special class of Markov processes with polynomial regressions called quadratic harnesses. We relate the infinitesimal generator to the unique solution of a certain commutation equation, and we use the commutation equation to find an explicit formula for the infinitesimal generator of free quadratic harnesses.

多重泊松-维纳积分与泊松U-统计量的指数不等式及重对数律

Exponential inequalities and laws of the iterated logarithm for multiple Poisson--Wiener integrals and Poisson $U$-statistics

本文针对泊松空间上的多重随机积分和泊松U-统计量建立尾概率与矩不等式，并据此在泊松过程强度趋于无穷时证明其重对数律，进而改进和推广了几何随机图、泊松k-平面交、Ornstein-Uhlenbeck Lévy过程二次泛函及标记过程U-统计量等经典泛函的测度集中结果。

Markov processes, polynomial martingales and orthogonal polynomials

We study general properties for the family of stochastic processes with polynomial regression property, that is that every conditional moment of the process is a polynomial. It turns out that then there exists a family of polynomial martingales $\left\{ M_{n}(X_{t},t)\right\}_{n\geq1}$ that contains complete information on the distribution (both marginal and transitional) of the process. We specify conditions expressed in terms of $M_{n}^{\prime}s$ under which a given process has independent increments and further is a Levy process, contains reversed martingales, is a harness or quadratic harn…

泊松点云上Dirichlet能量的渐近行为

Asymptotic behavior of the Dirichlet energy on Poisson point clouds

该研究针对平面泊松云上的二次对相互作用，通过尺度变换与离散到连续极限分析，证明其几乎必然可由确定性常数乘Dirichlet能量近似，并提出适配的粗粒度函数收敛概念以规避异常区域影响。

具有受控跳跃大小的机制切换下受约束随机线性二次控制

Constrained stochastic linear quadratic control under regime switching with controlled jump size

本文研究了机制切换和泊松跳跃下含双控制约束的随机线性二次控制问题，通过Meyer-Itô公式与广义平方补全技术显式推导出最优值和反馈控制，并利用多维BSDEJs比较定理证明其唯一非负解。

最优调谐泊松重启下随机过程的普适性质研究

On the universal properties of stochastic processes under optimally tuned Poisson restart

本文研究最优调谐泊松重启下随机过程的普适性质，发现了新的普适关系，并推导了多完成场景下完成时间二阶矩的普适不等式，拓展了随机过程重启优化的理论成果。