We compute the loss of power in likelihood ratio tests when we test the
original parameter of a probability density extended by the first Lehmann
alternative.

We apply results of Malliavin-Thalmaier-Watanabe for strong and weak Taylor
expansions of solutions of perturbed stochastic differential equations (SDEs).
In particular, we work out weight expressions for the Taylor coefficients of
the expansion. The results are applied to LIBOR market models in order to deal
with the typical stochastic drift and with stochastic volatility. In contrast
to other accurate methods like numerical schemes for the full SDE, we obtain
easily tractable expressions for accurate pricing. In particular, we present an
easily tractable alternative to “freezing the drift” in LIBOR market models,
which has an accuracy similar to the full numerical scheme. Numerical examples
underline the results.

We establish an unusual second-order almost sure limit theorem for the
minimal position in a one-dimensional super-critical branching random walk, and
also prove a martingale convergence theorem which answers a question of Biggins
and Kyprianou [7]. Our method applies furthermore to the study of directed
polymers on a disordered tree. In particular, we give a rigorous proof of a
phase transition phenomenon for the partition function (from the point of view
of convergence in probability), already described by Derrida and Spohn [14].
Surprisingly, this phase transition phenomenon disappears in the sense of upper
almost sure limits.

Percolation theory has become a useful tool for the analysis of large-scale
wireless networks. We investigate the fundamental problem of characterizing the
critical density $\lambda_c^{(d)}$ for $d$-dimensional Poisson random geometric
graphs in continuum percolation theory. By using a probabilistic analysis which
incorporates the clustering effect in random geometric graphs, we develop a new
class of analytical lower bounds for the critical density $\lambda_c^{(d)}$ in
$d$-dimensional Poisson random geometric graphs. The lower bounds are the
tightest known to date. In particular, for the two-dimensional case, the
analytical lower bound is improved to $\lambda^{(2)}_c \geq 0.7698…$. For the
three-dimensional case, we obtain $\lambda^{(3)}_c \geq 0.4494…$

The rotor-router model is a deterministic analogue of random walk. It can be
used to define a deterministic growth model analogous to internal DLA. We prove
that the asymptotic shape of this model is a Euclidean ball, in a sense which
is stronger than our earlier work. For the shape consisting of $n=\omega_d r^d$
sites, where $\omega_d$ is the volume of the unit ball in $\R^d$, we show that
the inradius of the set of occupied sites is at least $r-O(\log r)$, while the
outradius is at most $r+O(r^\alpha)$ for any $\alpha > 1-1/d$. For a related
model, the divisible sandpile, we show that the domain of occupied sites is a
Euclidean ball with error in the radius a constant independent of the total
mass. For the classical abelian sandpile model in two dimensions, with $n=\pi
r^2$ particles, we show that the inradius is at least $r/\sqrt{3}$, and the
outradius is at most $(r+o(r))/\sqrt{2}$. This improves on bounds of Le Borgne
and Rossin. Similar bounds apply in higher dimensions.

We construct a new random probability measure on the sphere and on the unit
interval which in both cases has a Gibbs structure with the relative entropy
functional as Hamiltonian. It satisfies a quasi-invariance formula with respect
to the action of smooth diffeomorphism of the sphere and the interval
respectively. The associated integration by parts formula is used to construct
two classes of diffusion processes on probability measures (on the sphere or
the unit interval) by Dirichlet form methods. The first one is closely related
to Malliavin’s Brownian motion on the homeomorphism group. The second one is a
probability valued stochastic perturbation of the heat flow, whose intrinsic
metric is the quadratic Wasserstein distance. It may be regarded as the
canonical diffusion process on the Wasserstein space.