This paper presents some new results on the conditional joint probability distributions of phase-type under the mixture of right-continuous Markov jump processes with absorption on the same finite state space $\mathbb{S}$ moving at different speeds, where the mixture occurs at a random time. Such mixture was first proposed by Frydman \cite{Frydman2005} and Frydman and Schuermann \cite{Frydman2008} as a generalization of the mover-stayer model of Blumen et at. \cite{Blumen}, and was recently extended by Surya \cite{Surya2018}. When conditioning on all previous and current information $\mathcal{F}_{t,i}=\mathcal{F}_{t-}\cup\{X_t=i\}$, with $\mathcal{F}_{t-}=\{X_s, 0<s\leq t-\}$ and $i\in\mathbb{S}$, of the mixture process $X$, distributional identities are explicit in terms of the Bayesian updates of switching probability, the likelihoods of observing the sample paths, and the intensity matrices of the underlying Markov processes, despite the fact that the mixture itself is non-Markov. They form non-stationary function of time and have the ability to capture heterogeneity and path dependence. When the underlying processes move at the same speed, in which case the mixture reduces to a simple Markov jump process, these features are removed, and the distributions coincide with that of given by Neuts \cite{Neuts1975} and Assaf et al. \cite{Assaf1984}. Furthermore, when conditioning on $\mathcal{F}_{t-}$ and no exit to the absorbing set has been observed at time $t$, the distributions are given explicitly in terms of an additional Bayesian updates of probability distribution of $X$ on $\mathbb{S}$. Examples are given to illustrate the main results.
New results on conditional joint probability distributions of first exit times are presented for a continuous-time stochastic process defined as the mixture of Markov jump processes moving at different speeds on the same finite state space, while the mixture occurs at a random time. Such mixture was first proposed by Frydman \cite{Frydman2005} and Frydman and Schuermann \cite{Frydman2008} as a generalization of the mover-stayer model of Blumen et at. \cite{Blumen}, and was recently extended by Surya \cite{Surya2018}, in which explicit distributional identities of the process are given, in part…
We study the convergence of $N-$particle systems described by SDEs driven by Brownian motion and Poisson random measure, where the coefficients depend on the empirical measure of the system. Every particle jumps with a jump rate depending on its position and on the empirical measure of the system. Jumps are simultaneous, that is, at each jump time, all particles of the system are affected by this jump and receive a random jump height that is centred and scaled in $N^{-1/2}.$ This particular scaling implies that the limit of the empirical measures of the system is random, describing the conditi…
We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched conditional invariance principle for the random walk, under the condition that it remains positive until time $n$. As a corollary of this result, we study the effect of conditioning the random walk to exceed level $n$ before returning to 0 as $n\to \infty$.
We consider the activated random walk (ARW) model on $\mathbb{Z}^d$, which undergoes a transition from an absorbing regime to a regime of sustained activity. In any dimension we prove that the system is in the active regime when the particle density is less than one, provided that the jump distribution is biased and that the sleeping rate is small enough. This answers a question from Rolla and Sidoravicius (2012) and Dickman, Rolla and Sidoravicius (2010) in the case of biased jump distribution. Furthermore, we prove that the critical density depends on the jump distribution.
The purpose of this paper is to study optimal control of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). To this end, we first prove a stochastic Fokker-Planck equation for the conditional law of the solution of such equations. Combining this equation with the original state equation, we obtain a Markovian system for the state and its conditional law. Furthermore, we apply this to formulate an Hamilton-Jacobi-Bellman (HJB) equation for the optimal control of conditional McKean-Vlasov jump diffusions. Th…
We study the problem of optimal stopping of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). We obtain sufficient variational inequalities for a function to be the value function of such a problem and for a stopping time to be optimal. To achieve this, we combine the state equation for the conditional McKean-Vlasov equation with the associated stochastic Fokker-Planck equation for the conditional law of the solution of the state. This gives us a Markovian system which can be handled by using a version of…
We consider a class of continuous time Markov chains on a compact metric space that admit an invariant measure strictly positive on open sets together with absorbing states. We prove the joint large deviation principle for the empirical measure and flow. Due to the lack of uniform ergodicity, the zero level set of the rate function is not a singleton. As corollaries, we obtain the Donsker-Varadhan rate function for the empirical measure and a variational expression of the rate function for the empirical flow.
We consider the Activated Random Walk model in any dimension with any sleep rate and jump distribution and ergodic initial state. We show that the stabilization properties depend only on the average density of particles, regardless of how they are initially located on the lattice.
We consider the so-called $\natural$-model. It is an one-default model which gives the conditional law of a random time with respect to a reference filtration. This model has been studied in the case where the parameters are continuous. In this paper we will establish the $\natural$-model in the case of jump parameters. We then prove the corresponding enlargement of filtration formula and we compute the derivative of the conditional distribution functions of the random time.
It is well-known that 0 is the absorbing state for a branching system. Each particle in the system lives a random long time and gives a random number of new particles at its death time. It stops when the system has no particle. This paper is devoted to studying the fixed range crossing numbers until any time t. The joint probability distribution of fixed range crossing numbers of such processes until time t is obtained by using a new method. In particular, the probability distribution of total death number is given for Markov branching processes until time t.
The oil and water model is an interacting particle system with two types of particles and a dynamics that conserves the number of particles, which belongs to the so-called class of Abelian networks. Widely studied processes in this class are sandpiles models and activated random walks, which are known (at least for some choice of the underlying graph) to undergo an absorbing-state phase transition. This phase transition characterizes the existence of two regimes, depending on the particle density: a regime of fixation at low densities, where the dynamics converges towards an absorbing state an…
In this article, we consider time-inhomogeneous diffusive particle systems, whose particles jump from the boundary of a bounded open subset of $\R^d$, $d\geq 1$. We give a sufficient criterion for the family of empirical distributions of such systems to be uniformly tight, independently of the jump location of the particles. As an application, we show that the conditional distribution of a family of time-inhomogeneous and environment-dependent diffusions conditioned not to hit the boundary of a bounded open subset of $\R^d$ is uniformly tight.
Consider a continuous time Markov chain with rates Q in the state space Λ\cup\{0\} with 0 as an absorbing state. In the associated Fleming-Viot process N particles evolve independently in Λwith rates Q until one of them attempts to jump to the absorbing state 0. At this moment the particle comes back to Λinstantaneously, by jumping to one of the positions of the other particles, chosen uniformly at random. When Λis finite, we show that the empirical distribution of the particles at a fixed time converges as N\to\infty to the distribution of a single particle at the same time conditioned on non…
The asymptotic behaviour of empirical measures has plenty of studies. However, the research on conditional empirical measures is limited. Being the development of Wang \cite{eW1}, under the quadratic Wasserstein distance, we investigate the rate of convergence of conditional empirical measures associated to subordinated Dirichlet diffusion processes on a connected compact Riemannian manifold with absorbing boundary. We give the sharp rate of convergence for any initial distribution and prove the precise limit for a large class of initial distributions. We follow the basic idea of Wang, but all…
By using absolutely continuous lower bounds of the Lévy measure, explicit gradient estimates are derived for the semigroup of the corresponding Lévy process with a linear drift. A derivative formula is presented for the conditional distribution of the process at time $t$ under the condition that the process jumps before $t$. Finally, by using bounded perturbations of the Lévy measure, the resulting gradient estimates are extended to linear SDEs driven by Lévy-type processes.
This paper is the first part of a series of papers on filtering for partially observed jump diffusions satisfying a stochastic differential equation driven by Wiener processes and Poisson martingale measures. The coefficients of the equation only satisfy appropriate growth conditions. Some results in filtering theory of diffusion processes are extended to jump diffusions and equations for the time evolution of the conditional distribution and the unnormalised conditional distribution of the unobserved process at time $t$, given the observations until $t$, are presented.
We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched \textit{conditional} invariance principle for the random walk, under the condition that it remains positive until time $n$. As a corollary of this result, we study the effect of conditioning the random walk to exceed level $n$ before returning to 0 as $n\to \infty$. One of the main tools for proving these conditional limit la…
The filtering equations associated to a partially observed jump diffusion model $(Z_t)_{t\in [0,T]}=(X_t,Y_t)_{t\in [0,T]}$, driven by Wiener processes and Poisson martingale measures are considered. Building on results from two preceding articles on the filtering equations, the regularity of the conditional density of the signal $X_t$, given observations $(Y_s)_{s\in [0,t]}$, is investigated, when the conditional density of $X_0$ given $Y_0$ exists and belongs to a Sobolev space, and the coefficients satisfy appropriate smoothness and growth conditions.
We derive characteristic function identities for conditional distributions of an r-trimmed Levy process given its r largest jumps up to a designated time t. Assuming the underlying Levy process is in the domain of attraction of a stable process as t goes to 0, these identities are applied to show joint convergence of the trimmed process divided by its large jumps to corresponding quantities constructed from a stable limiting process. This generalises related results in the 1-dimensional subordinator case developed in Kevei & Mason (2014) and produces new discrete distributions on the infinite …
Central limit theorems play an important role in the study of statistical inference for stochastic processes. However, when the nonparametric local polynomial threshold estimator, especially local linear case, is employed to estimate the diffusion coefficients of diffusion processes, the adaptive and predictable structure of the estimator conditionally on the $σ-$field generated by diffusion processes is destroyed, the classical central limit theorem for martingale difference sequences can not work. In this paper, we proved the central limit theorems of local polynomial threshold estimators fo…
This article studies the quasi-stationary behaviour of absorbed one-dimensional diffusions. We obtain necessary and sufficient conditions for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. An important tool is provided by one dimensional strict local martingale diffusions coming down from infinity. We prove under mild assumptions that their expectation at any positive time is uniformly bounded with respect to the initial position. We provide several examples and extensions, including the sticky Brown…
The typical central limit theorems in high-frequency asymptotics for semimartingales are results on stable convergence to a mixed normal limit with an unknown conditional variance. Estimating this conditional variance usually is a hard task, in particular when the underlying process contains jumps. For this reason, several authors have recently discussed methods to automatically estimate the conditional variance, i.e. they build a consistent estimator from the original statistics, but computed at various different time scales. Their methods work in several situations, but are essentially restr…
We study a class of combinations of second order Riesz transforms on Lie groups that are multiply connected, composed of a discrete abelian component and a compact connected component. We prove sharp $L^{p}$ estimates for these operators, therefore generalising previous results. We construct stochastic integrals with jump components adapted to functions defined on our semi-discrete set. We show that these second order Riesz transforms applied to a function may be written as conditional expectation of a simple transformation of a stochastic integral associated with the function. The analysis sh…
We prove the existence and uniqueness of solutions of degenerate linear stochastic evolution equations driven by jump processes in a Hilbert scale using the variational framework of stochastic evolution equations and the method of vanishing viscosity. As an application of this result, we derive the existence and uniqueness of solutions of degenerate parabolic linear stochastic integro-differential equations (SIDEs) in the Sobolev scale. The SIDEs that we consider arise in the theory of non-linear filtering as the equations governing the conditional density of a degenerate jump-diffusion signal…
We consider a piecewise-deterministic Markov process governed by a jump intensity function, a rate function that determines the behaviour between jumps, and a stochastic kernel describing the conditional distribution of jump sizes. We study the point process of upcrossings of a level $b$ by the Markov process. Our main result shows that, under a suitable scaling $ν(b)$, the point process converges, as $b$ tends to infinity, weakly to a geometrically compound Poisson process. We also prove a version of Rice's formula relating the stationary density of the process to level crossing intensities. …
Combinations of neural ODEs with recurrent neural networks (RNN), like GRU-ODE-Bayes or ODE-RNN are well suited to model irregularly observed time series. While those models outperform existing discrete-time approaches, no theoretical guarantees for their predictive capabilities are available. Assuming that the irregularly-sampled time series data originates from a continuous stochastic process, the $L^2$-optimal online prediction is the conditional expectation given the currently available information. We introduce the Neural Jump ODE (NJ-ODE) that provides a data-driven approach to learn, co…
This paper proposes a conditional Curie-Weiss model as a model for opinion formation in a society polarized along two opinions, say opinions 1 and 2. The model comes with interaction strength $β>0$ and bais $h$. Here the population in question is divided into three main groups, namely: Group one consisting of individuals who have decided on opinion 1. Let the proportion of this group be given by $s$. Group two consisting of individauls who have chosen opinion 2. Let $r$ be their proportion. Group three consisting of individuals who are yet to decide and they will decide based on their environm…
We consider the intensity-based approach for the modeling of default times of one or more companies. In this approach the default times are defined as the jump times of a Cox process, which is a Poisson process conditional on the realization of its intensity. We assume that the intensity follows the Cox-Ingersoll-Ross model. This model allows one to calculate survival probabilities and prices of defaultable bonds explicitly. In this paper we assume that the Brownian motion, that drives the intensity, is not observed. Using filtering theory for point process observations, we are able to derive …
In this paper, we derive a representation for the value process associated to the solutions of FBSDEs in a jump-diffusion setting under multiple probability measures. Motivated by concrete financial problems, the latter representations are then applied to devise a generalization of the change of numéraire technique allowing to obtain recursive pricing formulas in the presence of multiple interest rates and collateralization.
We consider an irreducible pure jump Markov process with rates Q=(q(x,y)) on Λ\cup\{0\} with Λcountable and 0 an absorbing state. A quasi-stationary distribution (qsd) is a probability measure νon Λthat satisfies: starting with ν, the conditional distribution at time t, given that at time t the process has not been absorbed, is still ν. That is, ν(x) = νP_t(x)/(\sum_{y\inΛ}νP_t(y)), with P_t the transition probabilities for the process with rates Q. A Fleming-Viot (fv) process is a system of N particles moving in Λ. Each particle moves independently with rates Q until it hits the absorbing sta…
Nonparametric Filtering, Estimation and Classification using Neural Jump ODEs
Department of Mathematics ETH Zurich（数学系，苏黎世联邦理工学院）；Department of Statistics UC Berkeley（统计系，伯克利大学）；Department of Mathematics Albert-Ludwigs-Universität Freiburg（数学系，弗赖堡阿尔伯特-路易斯大学）
