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Stability of solutions of BSDEs with random terminal time

Sandrine Toldo (2006)

ESAIM: Probability and Statistics

In this paper, we study the stability of the solutions of Backward Stochastic Differential Equations (BSDE for short) with an almost surely finite random terminal time. More precisely, we are going to show that if (Wn) is a sequence of scaled random walks or a sequence of martingales that converges to a Brownian motion W and if ( τ n ) is a sequence of stopping times that converges to a stopping time τ, then the solution of the BSDE driven by Wn with random terminal time τ n converges to the solution...

Stable random fields and geometry

Shigeo Takenaka (2010)

Banach Center Publications

Let (M,d) be a metric space with a fixed origin O. P. Lévy defined Brownian motion X(a); a ∈ M as 0. X(O) = 0. 1. X(a) - X(b) is subject to the Gaussian law of mean 0 and variance d(a,b). He gave an example for M = S m , the m-dimensional sphere. Let Y ( B ) ; B ∈ ℬ ( S m ) be the Gaussian random measure on S m , that is, 1. Y(B) is a centered Gaussian system, 2. the variance of Y(B) is equal of μ(B), where μ is the uniform measure on S m , 3. if B₁ ∩ B₂ = ∅ then Y(B₁) is independent of Y(B₂). 4. for B i , i = 1,2,..., B i ∩ B j = ∅ , i ≠ j, we...

Stable-1/2 bridges and insurance

Edward Hoyle, Lane P. Hughston, Andrea Macrina (2015)

Banach Center Publications

We develop a class of non-life reserving models using a stable-1/2 random bridge to simulate the accumulation of paid claims, allowing for an essentially arbitrary choice of a priori distribution for the ultimate loss. Taking an information-based approach to the reserving problem, we derive the process of the conditional distribution of the ultimate loss. The "best-estimate ultimate loss process" is given by the conditional expectation of the ultimate loss. We derive explicit expressions for the...

State estimation under non-Gaussian Lévy noise: A modified Kalman filtering method

Xu Sun, Jinqiao Duan, Xiaofan Li, Xiangjun Wang (2015)

Banach Center Publications

The Kalman filter is extensively used for state estimation for linear systems under Gaussian noise. When non-Gaussian Lévy noise is present, the conventional Kalman filter may fail to be effective due to the fact that the non-Gaussian Lévy noise may have infinite variance. A modified Kalman filter for linear systems with non-Gaussian Lévy noise is devised. It works effectively with reasonable computational cost. Simulation results are presented to illustrate this non-Gaussian filtering method.

Stationary distribution of absolute autoregression

Jiří Anděl, Pavel Ranocha (2005)

Kybernetika

A procedure for computation of stationary density of the absolute autoregression (AAR) model driven by white noise with symmetrical density is described. This method is used for deriving explicit formulas for stationary distribution and further characteristics of AAR models with given distribution of white noise. The cases of Gaussian, Cauchy, Laplace and discrete rectangular distribution are investigated in detail.

Stationary distributions for jump processes with memory

K. Burdzy, T. Kulczycki, R. L. Schilling (2012)

Annales de l'I.H.P. Probabilités et statistiques

We analyze a jump processes Z with a jump measure determined by a “memory” process S . The state space of ( Z , S ) is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of ( Z , S ) is the product of the uniform probability measure and a Gaussian distribution.

Stationary gaussian random fields on hyperbolic spaces and on euclidean spheres

S. Cohen, M. A. Lifshits (2012)

ESAIM: Probability and Statistics

We recall necessary notions about the geometry and harmonic analysis on a hyperbolic space and provide lecture notes about homogeneous random functions parameterized by this space. The general principles are illustrated by construction of numerous examples analogous to Euclidean case. We also give a brief survey of the fields parameterized by Euclidean spheres. At the end we give a list of important open questions in hyperbolic case.

Stationary Gaussian random fields on hyperbolic spaces and on Euclidean spheres∗∗∗

S. Cohen, M. A. Lifshits (2012)

ESAIM: Probability and Statistics

We recall necessary notions about the geometry and harmonic analysis on a hyperbolic space and provide lecture notes about homogeneous random functions parameterized by this space. The general principles are illustrated by construction of numerous examples analogous to Euclidean case. We also give a brief survey of the fields parameterized by Euclidean spheres. At the end we give a list of important open questions in hyperbolic case.

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