20,788 research outputs found
Jump Type Stochastic Differential Equations with Non-Lipschitz Coefficients: Non Confluence, Feller and Strong Feller Properties, and Exponential Ergodicity
This paper considers multidimensional jump type stochastic differential
equations with super linear growth and non-Lipschitz coefficients. After
establishing a sufficient condition for nonexplosion, this paper presents
sufficient non-Lipschitz conditions for pathwise uniqueness. The non confluence
property for solutions is investigated. Feller and strong Feller properties
under non-Lipschitz conditions are investigated via the coupling method.
Sufficient conditions for irreducibility and exponential ergodicity are
derived. As applications, this paper also studies multidimensional stochastic
differential equations driven by L\'evy processes and presents a Feynman-Kac
formula for L\'evy type operators.Comment: J. Differential Equations, to appea
On the Martingale Problem and Feller and Strong Feller Properties for Weakly Coupled L\'evy Type Operators
This paper considers the martingale problem for a class of weakly coupled
L\'{e}vy type operators. It is shown that under some mild conditions, the
martingale problem is well-posed and uniquely determines a strong Markov
process . The process , called a regime-switching
jump diffusion with L\'evy type jumps, is further shown to posses Feller and
strong Feller properties under non-Lipschitz conditions via the coupling
method
Arithmetic intersection on GSpin Rapoport-Zink spaces
We prove an explicit formula for the arithmetic intersection number of
diagonal cycles on GSpin Rapoport-Zink spaces in the minuscule case. This is a
local problem arising from the arithmetic Gan-Gross-Prasad conjecture for
orthogonal Shimura varieties. Our formula can be viewed as an orthogonal
counterpart of the arithmetic-geometric side of the arithmetic fundamental
lemma proved by Rapoport-Terstiege-Zhang in the minuscule case.Comment: Comments welcom
Remarks on the arithmetic fundamental lemma
W. Zhang's arithmetic fundamental lemma (AFL) is a conjectural identity
between the derivative of an orbital integral on a symmetric space with an
arithmetic intersection number on a unitary Rapoport-Zink space. In the
minuscule case, Rapoport-Terstiege-Zhang have verified the AFL conjecture via
explicit evaluation of both sides of the identity. We present a simpler way for
evaluating the arithmetic intersection number, thereby providing a new proof of
the AFL conjecture in the minuscule case.Comment: Minor revisons, to appear in Algebra Number Theor
Learning Deep Generative Models with Doubly Stochastic MCMC
We present doubly stochastic gradient MCMC, a simple and generic method for
(approximate) Bayesian inference of deep generative models (DGMs) in a
collapsed continuous parameter space. At each MCMC sampling step, the algorithm
randomly draws a mini-batch of data samples to estimate the gradient of
log-posterior and further estimates the intractable expectation over hidden
variables via a neural adaptive importance sampler, where the proposal
distribution is parameterized by a deep neural network and learnt jointly. We
demonstrate the effectiveness on learning various DGMs in a wide range of
tasks, including density estimation, data generation and missing data
imputation. Our method outperforms many state-of-the-art competitors
Utility Maximization of an Indivisible Market with Transaction Costs
This work takes up the challenges of utility maximization problem when the
market is indivisible and the transaction costs are included. First there is a
so-called solvency region given by the minimum margin requirement in the
problem formulation. Then the associated utility maximization is formulated as
an optimal switching problem. The diffusion turns out to be degenerate and the
boundary of domain is an unbounded set. One no longer has the continuity of the
value function without posing further conditions due to the degeneracy and the
dependence of the random terminal time on the initial data. This paper provides
sufficient conditions under which the continuity of the value function is
obtained. The essence of our approach is to find a sequence of continuous
functions locally uniformly converging to the desired value function. Thanks to
continuity, the value function can be characterized by using the notion of
viscosity solution of certain quasi-variational inequality
Max-Mahalanobis Linear Discriminant Analysis Networks
A deep neural network (DNN) consists of a nonlinear transformation from an
input to a feature representation, followed by a common softmax linear
classifier. Though many efforts have been devoted to designing a proper
architecture for nonlinear transformation, little investigation has been done
on the classifier part. In this paper, we show that a properly designed
classifier can improve robustness to adversarial attacks and lead to better
prediction results. Specifically, we define a Max-Mahalanobis distribution
(MMD) and theoretically show that if the input distributes as a MMD, the linear
discriminant analysis (LDA) classifier will have the best robustness to
adversarial examples. We further propose a novel Max-Mahalanobis linear
discriminant analysis (MM-LDA) network, which explicitly maps a complicated
data distribution in the input space to a MMD in the latent feature space and
then applies LDA to make predictions. Our results demonstrate that the MM-LDA
networks are significantly more robust to adversarial attacks, and have better
performance in class-biased classification
Regime-Switching Jump Diffusions with Non-Lipschitz Coefficients and Countably Many Switching States: Existence and Uniqueness, Feller, and Strong Feller Properties
This work focuses on a class of regime-switching jump diffusion processes,
which is a two component Markov processes , where
is a component representing discrete events taking values in a
countably infinite set. Considering the corresponding stochastic differential
equations, our main focus is on treating those with non-Lipschitz coefficients.
We first show that there exists a unique strong solution to the corresponding
stochastic differential equation. Then Feller and strong Feller properties are
investigated
borealis - A generalized global update algorithm for Boolean optimization problems
Optimization problems with Boolean variables that fall into the
nondeterministic polynomial (NP) class are of fundamental importance in
computer science, mathematics, physics and industrial applications. Most
notably, solving constraint-satisfaction problems, which are related to
spin-glass-like Hamiltonians in physics, remains a difficult numerical task. As
such, there has been great interest in designing efficient heuristics to solve
these computationally difficult problems. Inspired by parallel tempering Monte
Carlo in conjunction with the rejection-free isoenergetic cluster algorithm
developed for Ising spin glasses, we present a generalized global update
optimization heuristic that can be applied to different NP-complete problems
with Boolean variables. The global cluster updates allow for a wide-spread
sampling of phase space, thus considerably speeding up optimization. By
carefully tuning the pseudo-temperature (needed to randomize the
configurations) of the problem, we show that the method can efficiently tackle
optimization problems with over-constraints or on topologies with a large
site-percolation threshold. We illustrate the efficiency of the heuristic on
paradigmatic optimization problems, such as the maximum satisfiability problem
and the vertex cover problem.Comment: 19 pages, 7 figures, 1 tabl
Certain Properties Related to Well Posedness of Switching Diffusions
This work is devoted to switching diffusions that have two components (a
continuous component and a discrete component). Different from the so-called
Markovian switching diffusions, in the setup, the discrete component (the
switching) depends on the continuous component (the diffusion process). The
objective of this paper is to provide a number of properties related to the
well posedness. First, the differentiability with respect to initial data of
the continuous component is established. Then, further properties including
uniform continuity with respect to initial data, and smoothness of certain
functionals are obtained. Moreover, Feller property is obtained under only
local Lipschitz continuity. Finally, an example of Lotka-Voterra model under
regime switching is provided as an illustration.Comment: 27 page
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