TY  - JOUR
A1  - Conforti, Giovanni
A1  - Pra, Paolo Dai
A1  - Roelly, Sylvie
T1  - Reciprocal Class of Jump Processes
JF  - Journal of theoretical probability
N2  - Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of compound Poisson processes whose jumps belong to a finite set . We propose a characterization of the reciprocal class as the unique set of probability measures on which a family of time and space transformations induces the same density, expressed in terms of the reciprocal invariants. The geometry of plays a crucial role in the design of the transformations, and we use tools from discrete geometry to obtain an optimal characterization. We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state.
KW  - Reciprocal processes
KW  - Stochastic bridges
KW  - Jump processes
KW  - Compound Poisson processes
Y1  - 2015
U6  - https://doi.org/10.1007/s10959-015-0655-3
SN  - 0894-9840
SN  - 1572-9230
VL  - 30
SP  - 551
EP  - 580
PB  - Springer
CY  - New York
ER  - 
TY  - JOUR
A1  - Conforti, Giovanni
A1  - Kosenkova, Tetiana
A1  - Roelly, Sylvie
T1  - Conditioned Point Processes with Application to Levy Bridges
JF  - Journal of theoretical probability
N2  - Our first result concerns a characterization by means of a functional equation of Poisson point processes conditioned by the value of their first moment. It leads to a generalized version of Mecke’s formula. En passant, it also allows us to gain quantitative results about stochastic domination for Poisson point processes under linear constraints. Since bridges of a pure jump Lévy process in Rd with a height a can be interpreted as a Poisson point process on space–time conditioned by pinning its first moment to a, our approach allows us to characterize bridges of Lévy processes by means of a functional equation. The latter result has two direct applications: First, we obtain a constructive and simple way to sample Lévy bridge dynamics; second, it allows us to estimate the number of jumps for such bridges. We finally show that our method remains valid for linearly perturbed Lévy processes like periodic Ornstein–Uhlenbeck processes driven by Lévy noise.
KW  - Ornstein-Uhlenbeck
Y1  - 2019
U6  - https://doi.org/10.1007/s10959-018-0863-8
SN  - 0894-9840
SN  - 1572-9230
VL  - 32
IS  - 4
SP  - 2111
EP  - 2134
PB  - Springer
CY  - New York
ER  - 
TY  - JOUR
A1  - Conforti, Giovanni
A1  - Roelly, Sylvie
T1  - Bridge mixtures of random walks on an Abelian group
JF  - Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability
KW  - random walk on Abelian group
KW  - reciprocal class
KW  - stochastic bridge
Y1  - 2017
U6  - https://doi.org/10.3150/15-BEJ783
SN  - 1350-7265
SN  - 1573-9759
VL  - 23
SP  - 1518
EP  - 1537
PB  - International Statistical Institute
CY  - Voorburg
ER  - 
TY  - JOUR
A1  - Conforti, Giovanni
A1  - Leonard, Christian
A1  - Murr, Rüdiger
A1  - Roelly, Sylvie
T1  - Bridges of Markov counting processes. Reciprocal classes and duality formulas
JF  - Electronic communications in probability
N2  - Processes sharing the same bridges are said to belong to the same reciprocal class. In this article we analyze reciprocal classes of Markov counting processes by identifying their reciprocal invariants and we characterize them as the set of counting processes satisfying some duality formula.
KW  - Counting process
KW  - bridge
KW  - reciprocal class
KW  - duality formula
Y1  - 2015
U6  - https://doi.org/10.1214/ECP.v20-3697
SN  - 1083-589X
VL  - 20
PB  - Univ. of Washington, Mathematics Dep.
CY  - Seattle
ER  -