Alexander Sokol

Causal interpretation of stochastic differential equations

We give a causal interpretation of stochastic differential equations (SDEs) by defining the postintervention SDE resulting from an intervention in an SDE. We show that under Lipschitz conditions, the solution to the postintervention SDE is equal to a uniform limit in probability of postintervention structural equation models based on the Euler scheme of the original SDE, thus relating our definition to mainstream causal concepts. We prove that when the driving noise in the SDE is a Levy process, the postintervention distribution is identifiable from the generator of the SDE.

Exponential Martingales and Changes of Measure for Counting Processes

We give sufficient criteria for the Doléans-Dade exponential of a stochastic integral with respect to a counting process local martingale to be a true martingale. The criteria are adapted particularly to the case of counting processes and are sufficiently weak to be useful and verifiable, as we illustrate by several examples. In particular, the criteria allow for the construction of for example nonexplosive Hawkes processes, counting processes with stochastic intensities depending on diffusion processes as well as inhomogeneous finite-state Markov processes.

An Elementary Proof that the First Hitting Time of an Open Set by a Jump Process is a Stopping Time

We give a short and elementary proof that the first hitting time of an open set by the jump process of a cadlag adapted process is a stopping time.

Quantifying identifiability in independent component analysis

We are interested in consistent estimation of the mixing matrix in the ICA model, when the error distribution is close to (but different from) Gaussian. In particular, we consider

A generic model for spouse's pensions with a view towards the calculation of liabilities

n

DEGREES OF FREEDOM FOR NONLINEAR LEAST SQUARES ESTIMATION

independent samples from the ICA model

Revisiting the forward equations for inhomogeneous semi-Markov processes

X = A\epsilon

AN EXTENDED NOVIKOV-TYPE CRITERION FOR LOCAL M ARTINGALES WITH JUMPS

, where we assume that the coordinates of

PROVING EXISTENCE RESULTS IN MARTINGALE THEORY USI NG A SUBSEQUENCE PRINCIPLE

\epsilon

Intervention in Ornstein-Uhlenbeck SDEs

are independent and identically distributed according to a contaminated Gaussian distribution, and the amount of contamination is allowed to depend on