Discrete Itô Formulas and Their Applications to Stochastic Numerics

Exact asymptotic formulae of the stationary distribution of a discrete-time 2d-QBD process: an example and additional proofs

A discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process), $\{\boldsymbol{Y}_n\}=\{(X_{1,n},X_{2,n},J_n)\}$, is a two-dimensional skip-free random walk $\{(X_{1,n},X_{2,n})\}$ on $\mathbb{Z}_+^2$ with a supplemental process $\{J_n\}$ on a finite set $S_0$. The supplemental process $\{J_n\}$ is called a phase process. The 2d-QBD process $\{\boldsymbol{Y}_n\}$ is a Markov chain in which the transition probabilities of the two-dimensional process $\{(X_{1,n},X_{2,n})\}$ vary according to the state of the phase process $\{J_n\}$. This modulation is assumed to be space homogene…

Partition functions of discrete coalescents: from Cayley's formula to Frieze's ζ(3) limit theorem

In these expository notes, we describe some features of the multiplicative coalescent and its connection with random graphs and minimum spanning trees. We use Pitman's proof of Cayley's formula, which proceeds via a calculation of the partition function of the additive coalescent, as motivation and as a launchpad. We define a random variable which may reasonably be called the empirical partition function of the multiplicative coalescent, and show that its typical value is exponentially smaller than its expected value. Our arguments lead us to an analysis of the susceptibility of the Erdős-Rény…

Approximations of the ruin probability in a discrete time risk model

Based on a discrete version of the Pollaczeck-Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber-Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations.

Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises

The Clark-Ocone formula in the theory of discrete-time chaotic calculus holds only for square integrable functionals of discrete-time normal noises. In this paper, we aim at extending this formula to generalized functionals of discrete-time normal noises. Let $Z$ be a discrete-time normal noise that has the chaotic representation property. We first prove a result concerning the regularity of generalized functionals of $Z$. Then, we use the Fock transform to define some fundamental operators on generalized functionals of $Z$, and apply the above mentioned regularity result to prove the continui…

Game theoretic derivation of discrete distributions and discrete pricing formulas

In this expository paper we illustrate the generality of game theoretic probability protocols of Shafer and Vovk (2001) in finite-horizon discrete games. By restricting ourselves to finite-horizon discrete games, we can explicitly describe how discrete distributions with finite support and the discrete pricing formulas, such as the Cox-Ross-Rubinstein formula, are naturally derived from game-theoretic probability protocols. Corresponding to any discrete distribution with finite support, we construct a finite-horizon discrete game, a replicating strategy of Skeptic, and a neutral forecasting st…

离散调和函数的阿尔姆格伦单调性公式

An Almgren monotonicity formula for discrete harmonic functions

研究无限组合图上调和函数的阿尔姆格伦单调性公式，通过证明其离散类似物，探讨了相关应用。

A note on Ito's formula for discrete-time quantum walk

We present an Ito's formula for the one-dimensional discrete-time quantum walk and give some examples including a Tanaka's formula by using the formula. Moreover we discuss integrals for the quantum walk.

On Convergence to SLE$_6$ II: Discrete Approximations and Extraction of Cardy's Formula for General Domains

Following the approach outlined in [17], convergence to SLE$_6$ of the Exploration Processes for the correlated bond-triangular type models studied in [6] is established in [3] and the present work. In this second part, we focus on establishing Cardy's Formula for general domains.

离散时间量子游走的费曼公式

Feynman Formula for Discrete-time Quantum Walks

本文通过费曼公式将离散时间量子游走与四状态马尔可夫加性过程关联，推导谱分解与极限密度的关系，并利用概率表示提出高效的蒙特卡洛计算方法。

Existence of Saddle Points in Discrete Markov Games and Its Application in Numerical Methods for Stochastic Differential Games

This work establishes sufficient conditions for existence of saddle points in discrete Markov games. The result reveals the relation between dynamic games and static games using dynamic programming equations. This result enables us to prove existence of saddle points of non-separable stochastic differential games of regime-switching diffusions under appropriate conditions.

Optimal hedging in discrete time

Building on the work of Schweizer (1995) and Cern and Kallseny (2007), we present discrete time formulas minimizing the mean square hedging error for multidimensional assets. In particular, we give explicit formulas when a regime-switching random walk or a GARCH-type process is utilized to model the returns. Monte Carlo simulations are used to compare the optimal and delta hedging methods.

半离散抛物安德森模型中分区函数的因子化公式

A factorization formula for the partition function in the semi-discrete parabolic Anderson model

研究高温度下半离散抛物安德森模型中分区函数的极限行为，证明其在L²和几乎必然意义下的存在性，并推导出点到点分区函数的因子化公式。

Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations

The main result of this article establishes strong convergence rates on the whole probability space for explicit space-time discrete numerical approximations for a class of stochastic evolution equations with possibly non-globally monotone coefficients such as stochastic Burgers equations with additive trace-class noise. The key idea in the proof of our main result is (i) to bring the classical Alekseev-Gröbner formula from deterministic analysis into play and (ii) to employ uniform exponential moment estimates for the numerical approximations.

An adaptive time-stepping fully discrete scheme for stochastic NLS equation: Strong convergence and numerical asymptotics

In this paper, we propose and analyze an adaptive time-stepping fully discrete scheme which possesses the optimal strong convergence order for the stochastic nonlinear Schrödinger equation with multiplicative noise. Based on the splitting skill and the adaptive strategy, the $H^1$-exponential integrability of the numerical solution is obtained, which is a key ingredient to derive the strong convergence order. We show that the proposed scheme converges strongly with orders $\frac12$ in time and $2$ in space. To investigate the numerical asymptotic behavior, we establish the large deviation prin…

显式时空离散数值近似在二维随机Navier-Stokes方程中的强收敛性

Strong convergence for explicit space-time discrete numerical approximation for 2D stochastic Navier-Stokes equations

本文研究了二维不可压缩随机Navier-Stokes方程在流形上由加性噪声驱动的显式时空离散近似方案的强收敛性，利用现有结果证明了该方法对非全局单调非线性随机偏微分方程的强收敛性。

Discrete Green's functions for products of regular graphs

Discrete Green's functions are the inverses or pseudo-inverses of combinatorial Laplacians. We present compact formulas for discrete Green's functions, in terms of the eigensystems of corresponding Laplacians, for products of regular graphs with or without boundary. Explicit formulas are derived for the cycle, torus, and 3-dimensional torus, as is an inductive formula for the $t$-dimensional torus with $n$ vertices, from which the Green's function can be completely determined in time $O(t n^{2-1/t}\log{n})$. These Green's functions may be used in conjunction with diffusion-like problems on gra…

显式时空离散数值近似对随机Allen-Cahn方程的强收敛性速率

Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations

本文研究了具有超线性增长非线性项的随机偏微分方程（SPDEs）的显式时空离散数值近似方法，证明了对时空白噪声驱动的随机Allen-Cahn方程的强收敛性速率，并建立了强时间与空间近似误差的下界，表明所证明的强收敛速率本质上是尖锐的，通常无法进一步改进。

A unified formulation of Gaussian vs. sparse stochastic processes - Part II: Discrete-domain theory

This paper is devoted to the characterization of an extended family of CARMA (continuous-time autoregressive moving average) processes that are solutions of stochastic differential equations driven by white Levy innovations. These are completely specified by: (1) a set of poles and zeros that fixes their correlation structure, and (2) a canonical infinitely-divisible probability distribution that controls their degree of sparsity (with the Gaussian model corresponding to the least sparse scenario). The generalized CARMA processes are either stationary or non-stationary, depending on the locati…