Anielllo Fedullo, Vitalii A. Gasanenko
This paper is devoted to the study of the following problem. We have set of diffusion processes with absorption on boundaries in some region at initial time $t=0$. It is required to estimate of number of the unabsorbed processes for the fixed time ~$τ>0$. The number of initial processes is considered as function of $τ$ and $τ\to\infty$.
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We lead the algorithm of expansion of sojourn probability of many-dimensional diffusion processes in small domain. The principal member of this expansion defines normalizing coefficient for special limit theorems.
We discuss some applications of the Mittag-Leffler function and related probability distributions in the theory of renewal processes and continuous time random walks. In particular we show the asymptotic (long time) equivalence of a generic power law waiting time to the Mittag-Leffler waiting time distribution via rescaling and respeeding the clock of time. By a second respeeding (by rescaling the spatial variable) we obtain the diffusion limit of the continuous time random walk under power law regimes in time and in space. Finally, we exhibit the time-fractional drift process as a diffusion l…
We have random number of independent diffusion processes with absorption on boundaries in some region at initial time $t=0$. The initial numbers and positions of processes in region is defined by Poisson random measure. It is required to estimate of number of the unabsorbed processes for the fixed time \~$τ>0$. The Poisson random measure depends on $τ$ and $τ\to\infty$.
We study an inverse first-hitting problem for a one-dimensional, time-homogeneous diffusion $X(t)$ reflected between two boundaries $a$ and $b,$ which starts from a random position $η.$ Let $a \le S \le b$ be a given threshold, such that $P( η\in [a,S])=1,$ and $F$ an assigned distribution function. The problem consists of finding the distribution of $η$ such that the first-hitting time of $X$ to $S$ has distribution $F.$ This is a generalization of the analogous problem for ordinary diffusions, i.e. without reflecting, previously considered by the author.
We propose an approach to compute the boundary crossing probabilities for a class of diffusion processes which can be expressed as piecewise monotone (not necessarily one-to-one) functionals of a standard Brownian motion. This class includes many interesting processes in real applications, e.g., Ornstein-Uhlenbeck, growth processes and geometric Brownian motion with time dependent drift. This method applies to both one-sided and two-sided general nonlinear boundaries, which may be discontinuous. Using this approach explicit formulas for boundary crossing probabilities for certain nonlinear bou…
The aim of this paper is to study the law of the last passage time of a linear diffusion to a curved boundary. We start by giving a general expression for the density of such a random variable under some regularity assumptions. Following Robbins & Siegmund, we then show that this expression may be computed for some implicit boundaries via a martingale method. Finally, we discuss some links between first hitting times and last passage times via time inversion, and present an integral equation (which we solve in some particular cases) satisfied by the density of the last passage time. Many examp…
Two frameworks that have been used to characterize reflected diffusions include stochastic differential equations with reflection and the so-called submartingale problem. We introduce a general formulation of the submartingale problem for (obliquely) reflected diffusions in domains with piecewise C^2 boundaries and piecewise continuous reflection vector fields. Under suitable assumptions, we show that well-posedness of the submartingale problem is equivalent to existence and uniqueness in law of weak solutions to the corresponding stochastic differential equation with reflection. Our result ge…
In this paper, we develop a Monte Carlo based algorithm for estimating the FPT density of a time-homogeneous SDE through a time-dependent frontier. We consider Brownian bridges as well as localized Daniels curve approximations to obtain tractable estimations of the FPT probability between successive points of a simulated path of the process. Under mild assumptions, a (unique) Daniels curve local approximation can easily be obtained by explicitly solving a non-linear system of equations.
Consider a one dimensional diffusion process on the diffusion interval $I$ originated in $x_0\in I$. Let $a(t)$ and $b(t)$ be two continuous functions of $t$, $t>t_0$ with bounded derivatives and with $a(t)<b(t)$ and $a(t),b(t)\in I$, $\forall t>t_0$. We study the joint distribution of the two random variables $T_a$ and $T_b$, first hitting times of the diffusion process through the two boundaries $a(t)$ and $b(t)$, respectively. We express the joint distribution of $T_a, T_b$ in terms of $P(T_a<t,T_a<T_b)$ and $P(T_b<t,T_a>T_b)$ and we determine a system of integral equations verified by thes…
We show a probabilistic functional limit result for one-dimensional diffusion processes that are reflected at an elastic boundary which is a function of the reflection local time. Such processes are constructed as limits of a sequence of diffusions which are discretely reflected by small jumps at an elastic boundary, with reflection local times being approximated by $\varepsilon$-step processes. The construction yields the Laplace transform of the inverse local time for reflection. Processes and approximations of this type play a role in finite fuel problems of singular stochastic control.
Martin boundaries of diffusions and random walks on hyperbolic spaces
本文研究双曲空间上特定扩散过程及其在等距群上关联随机游走的Martin边界。
We propose a discrete time discrete space Markov chain approximation with a Brownian bridge correction for computing curvilinear boundary crossing probabilities of a general diffusion process on a finite time interval. For broad classes of curvilinear boundaries and diffusion processes, we prove the convergence of the constructed approximations in the form of products of the respective substochastic matrices to the boundary crossing probabilities for the process as the time grid used to construct the Markov chains is getting finer. Numerical results indicate that the convergence rate for the p…
We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary pro…
Full-Covariance Chemical Langevin Predator--Prey Diffusion with Absorbing Boundaries
本文提出了一种全协方差化学 Langevin 扩散模型，用于描述预捕食者-猎物系统的吸收边界与灭绝结构，通过耦合捕食-转换事件和密度依赖缩放，实现了对事件级耦合的精确建模。
Diffusions with resetting boundaries: ergodic control and renewal optimization
研究带重置边界的一维扩散的遍历优化问题，推导带非局部边界条件的HJB方程，并给出最优平稳马尔可夫控制的验证定理，通过更新结构得到遍历成本的显式公式。
We study the large deviations of current-type observables defined for Markov diffusion processes evolving in smooth bounded regions of $\mathbb{R}^d$ with reflections at the boundaries. We derive for these the correct boundary conditions that must be imposed on the spectral problem associated with the scaled cumulant generating function, which gives, by Legendre transform, the rate function characterizing the likelihood of current fluctuations. Two methods for obtaining the boundary conditions are presented, based on the diffusive limit of random walks and on the Feynman--Kac equation underlyi…
We study perpetual American option pricing problems in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal exercise times are shown to be the first times at which the underlying asset hits certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and norm…
We investigate some asymptotic properties of general Markov processes conditioned not to be absorbed by moving boundaries. We first give general criteria involving an exponential convergence towards the Q-process, that is the law of the considered Markov process conditioned never to reach the moving boundaries. This exponential convergence allows us to state the existence and uniqueness of quasi-ergodic distribution considering either boundaries moving periodically or stabilizing boundaries. We also state the existence and uniqueness of quasi-limit distribution when absorbing boundaries stabil…
On the boundaries, asymptotic law and Bernoulli-Doob representation of homogeneous bounded martingales
研究有界状态空间中齐次扩散鞅，建立其与伯努利 - 杜布鞅的等价关系，证明伯努利极限渐近性，阐明相关边界分类等关系，通过\(\Phi\)鞅等进行理论说明及信用风险建模应用。
We provide sufficient conditions for the continuity of the free-boundary in a general class of finite-horizon optimal stopping problems arising for instance in finance and economics. The underlying process is a strong solution of one dimensional, time-homogeneous stochastic differential equation (SDE). The proof relies on both analytic and probabilistic arguments and it is based on a contradiction scheme inspired by the maximum principle in partial differential equations (PDE) theory. Mild, local regularity of the coefficients of the SDE and smoothness of the gain function locally at the bound…
We develop a new simulation method for multidimensional diffusions with sticky boundaries. The challenge comes from simulating the sticky boundary behavior, for which standard methods like the Euler scheme fail. We approximate the sticky diffusion process by a multidimensional continuous time Markov chain (CTMC), for which we can simulate easily. We develop two ways of constructing the CTMC: approximating the infinitesimal generator of the sticky diffusion by finite difference using standard coordinate directions, and matching the local moments using the drift and the eigenvectors of the covar…
We study the large deviations of time-integrated observables of Markov diffusions that have perfectly reflecting boundaries. We discuss how the standard spectral approach to dynamical large deviations must be modified to account for such boundaries by imposing zero-current conditions, leading to Neumann or Robin boundary conditions, and how these conditions affect the driven process, which describes how large deviations arise in the long-time limit. The results are illustrated with the drifted Brownian motion and the Ornstein-Uhlenbeck process reflected at the origin. Other types of boundaries…
The latter author, together with collaborators, proposed a numerical scheme to calculate the price of barrier options. The scheme is based on a symmetrization of diffusion process. The present paper aims to give a mathematical credit to the use of the numerical scheme for Heston or SABR type stochastic volatility models. This will be done by showing a fairly general result on the symmetrization (in multi-dimension/multi-reflections). Further applications (to time-inhomogeneous diffusions/ to time dependent boundaries/to curved boundaries) are also discussed.
No arbitrage and the existence of ACLMMs in general diffusion models
本文研究单资产一般扩散市场模型中无套利（NA）与绝对连续局部鞅测度（ACLMM）存在性的关系，证明在有限时间跨度下NA等价于ACLMM存在加上尺度函数正则性及无反射边界条件，无限时间跨度下二者直接等价，并通过反例证明该刻画是尖锐的。
We study the Bayesian problems of detecting a change in the drift rate of an observable diffusion process with linear and exponential penalty costs for a detection delay. The optimal times of alarms are found as the first times at which the weighted likelihood ratios hit stochastic boundaries depending on the current observations. The proof is based on the reduction of the initial problems into appropriate three-dimensional optimal stopping problems and the analysis of the associated parabolic-type free-boundary problems. We provide closed form estimates for the value functions and the boundar…
Diffusion processes with boundaries are models of transport phenomena with wide applicability across many fields. These processes are described by their probability density functions (PDFs), which often obey Fokker-Planck equations (FPEs). While obtaining analytical solutions is often possible in the absence of boundaries, obtaining closed-form solutions to the FPE is more challenging once absorbing boundaries are present. As a result, analyses of these processes have largely relied on approximations or direct simulations. In this paper, we studied two-dimensional, time-homogeneous, spatially-…
We consider processes that coincide with a given diffusion process except on the boundaries of a finite collection of domains. The behavior on each of the boundaries is asymmetric: the process is much more likely to enter the interior of the domain than to enter the interior of its complement, with a small parameter controlling the trapping mechanism. We describe the limiting behavior of the processes. In particular, if the parameters controlling the boundary behavior have different orders of magnitude for different domains or if the domains are nested, metastable distributions between the tra…
Boundary Effects in the Diffusion of New Products on Cartesian Networks
本文分析了D维笛卡尔网络上离散Bass模型的边界效应，证明边界附近节点采纳率较低且随距离指数衰减，总体采纳水平降幅与节点数M呈$\ rac{1}{M^{1/D}}$关系，并以太阳能采纳数据验证。
We study the random processes with non-local memory and obtain new solutions of the Mori-Zwanzig equation describing non-markovian systems. We analyze the system dynamics depending on the amplitudes $ν$ and $μ_0$ of the local and non-local memory and pay attention to the line in the ($ν$, $μ_0$)-plane separating the regions with asymptotically stationary and non-stationary behavior. We obtain general equations for such boundaries and consider them for three examples of the non-local memory functions. We show that there exist two types of the boundaries with fundamentally different system dynam…
Close encounters of the sticky kind: Brownian motion at absorbing boundaries
本文构建了一维粘性布朗运动接触式模型，计算了粒子位置和占用时间的联合概率密度，以首次跨越随机阈值的占用时间定义吸收事件，并揭示了粘性RTP扩散极限与粘性BM的映射关系。
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