This paper extends a class of discount problem of singular
stochastic control with stopping time. We extend the state process and cost function
to general case. By stochastic analysis and optimal control theory, the "fail-stop"
control strategy is its optimal control. The conditions of the "fail-stop" strategy
and optimal control function and control method are given. The conclusion in this
paper has a fairly deep application.

In classical credibility theory, the claim amounts of different
insurance policies in a portfolio are assumed to be independent and the premiums are derived
under squared-error loss function. Wen et al. (2012) studied the credibility models with a
dependence structure among the claim amounts of one insurance policy that is called time
changeable effects and obtained the credibility formula. In this paper, we generalized this
dependence structure called time changeable effects to the claim amounts of different
insurance policies in a portfolio. Credibility premiums are obtained for Buhlmann and
Buhlmann-Straub credibility models with dependence structure under balanced loss function.

In this paper, we present a logarithm representation of operator
scaling stable random fields which in particular contains a class of Log-fractional stable
motion , and investigate the related sample paths regularity.

We consider a Markov switching exponential Levy model in which the
underlying economy switches between a finite number of states. The switching is modeled by a
hidden Markov chain. We explore the link between options prices in Markov switching exponential
Levy models and the related partial integro-differential equations in the case of European
options.

Algebraic convergence in -sense is studied for the reflecting
diffusion processes on noncompact manifold with non-convex boundary. A series of sufficient
and necessary conditions for the algebraic convergence are presented.

The paper considers a risk model with two dependent classes of
insurance business. In this model, the two claim number processes are partly sparsely
correlated through an Erlang(2) process. By introducing an auxiliary model, we obtain the
integral equations for ultimate ruin probabilities, and discuss the asymptotic property of
ruin probabilities by renewal approach. We also get the linear differential equations of
ruin probabilities of the model and the corresponding auxiliary model when claims follow
the exponential distributions, and show how solves the linear differential equations by a
specific example.

Recurrent event data usually occur in long-term studies which concern
recurrence rates of the disease. In studies of medical sciences, patients who have infected
with the disease, like cancer, were conventionally regarded as impossible to be cured. However,
with the development of medical sciences, recently those patients were found to be possibly
recovered from the disease. The recurrence rate of the events, which is of primary interest,
may be affected by the cure rate that may exist. Therefore, we proposed semiparametric
statistical analysis for recurrent event data with subjects possibly being cured. In our
approach, we present a proportional rate model for recurrence rate with the cure rate adjusted
through a Logistic regression model, and develop some estimating equations for estimation of
the regression parameters, with their large sample properties, including consistency and
asymptotic normality established. Numerical studies under different settings were conducted
for assessing the proposed methodology and the results suggest that they work well for
practical situations. The approach is applied to a bladder cancer dataset which motivated our
study.

In this paper we describe the excursions from a set explicitly for
recurrent Markov chain with discrete time. A new exit system is presented through using a
law conditioned by specifying the starting point and ending point of excursions. In a simple
case, we verify that our conditioned excursion law is a discrete approximation for that of
a diffusion.

Let , , ,
be all independent PRHR variables. Firstly, we show that implies . Secondly, we consider the comparison of
convolutions of independent heterogeneous PRHR variables with respect to the usual stochastic
ordering. Suppose  and , we prove that  implies ,
for all . The results established here strengthen some of the results known in
the literature.

Let ,  be two independent,
-dimensional sub-fractional Brownian motions with respective indices .
Assume . Our principal results are the necessary and sufficient condition for the
existence and smoothness of the collision local time and the intersection local time of
 and  through chaos expansion and elementary inequalities.