Related papers: Stochastic analysis for obtuse random walks
These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered include the Clark formula and predictable…
We study a particular class of complex-valued random variables and their associated random walks: the complex obtuse random variables. They are the generalization to the complex case of the real-valued obtuse random variables which were…
A random walk-based method is proposed to efficiently compute the solution of a large class of fractional in time linear systems of differential equations (linear F-ODE systems), along with the derivatives with respect to the system…
We introduce a lattice random walk discretisation scheme for stochastic differential equations (SDEs) that samples binary or ternary increments at each step, suppressing complex drift and diffusion computations to simple 1 or 2 bit random…
We construct a pathwise calculus for functionals of integer-valued measures and use it to derive an martingale representation formula with respect to a large class of integer-valued random measures. Using these results, we extend the…
In this paper, we will establish a discrete-time version of Clark(-Ocone-Haussmann) formula, which can be seen as an asymptotic expansion in a weak sense. The formula is applied to the estimation of the error caused by the martingale…
The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walks, time changed by a discrete quadratic variation process. One basis of this is a similar…
We consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice. We determine the probability generating functions, the transition probabilities and the relevant moments. The convergence of the stochastic process to a…
An analytical formula for the occurence probability of Markovian stochastic paths with repeatedly visited and/or equal departure rates is derived. This formula is essential for an efficient investigation of the trajectories belonging to…
We consider a Markov jump process on a general state space to which we apply a time-dependent weak perturbation over a finite time interval. By martingale-based stochastic calculus, under a suitable exponential moment bound for the…
We consider a Poisson process $\eta$ on a measurable space $(\BY,\mathcal{Y})$ equipped with a partial ordering, assumed to be strict almost everwhwere with respect to the intensity measure $\lambda$ of $\eta$. We give a Clark-Ocone type…
We analyze the stochastic proximal subgradient descent in the case where the objective functions are path differentiable and verify a Sard-type condition. While the accumulation set may not be reduced to unique point, we show that the time…
A random walk problem with particles on discrete double infinite linear grids is discussed. The model is based on the work of Montroll and others. A probability connected with the problem is given in the form of integrals containing…
We propose a variety of models of random walk, discrete in space and time, suitable for simulating stable random variables of arbitrary index $\alpha$ ($0< \alpha \le 2$), in the symmetric case. We show that by properly scaled transition to…
We derive Stein approximation bounds for functionals of uniform random variables, using chaos expansions and the Clark-Ocone representation formula combined with derivation and finite difference operators. This approach covers sums and…
We describe stochastic calculus in the context of processes that are driven by an adapted point process of locally finite intensity and are differentiable between jumps. This includes Markov chains as well as non-Markov processes. By…
In this paper we introduce a family of stochastic gradient estimation techniques based of the perturbative expansion around the mean of the sampling distribution. We characterize the bias and variance of the resulting Taylor-corrected…
In the study of dynamical processes on networks, there has been intense focus on network structure -- i.e., the arrangement of edges and their associated weights -- but the effects of the temporal patterns of edges remains poorly…
We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to…
We introduce a new class of estimators for the linear response of steady states of stochastic dynamics. We generalize the likelihood ratio approach and formulate the linear response as a product of two martingales, hence the name…