The Existence and Uniqueness of Solutions for Mean-Reverting <i>γ</i>-Process

Empirical studies show that more and more short-term rate models in capturing the dynamics cannot be described by those classic ones. So the mean-reverting γ-process was correspondingly proposed. In most cases, its coefficients do not satisfy the linear growth condition;even they satisfy the local Lipschitz condition. So we still cannot examine its existence of solutions by traditional techniques. This paper overcomes these difficulties. Firstly, through using the function Lyapunov, it has proven the existence and uniqueness of solutions for mean-reverting γ-process when the parameter . Secondly, when , it proves the solution is non-negative. Finally, it proves that there is a weak solution to the mean-reverting γ-process and the solution satisfies the track uniqueness by defining a function ρ. Therefore, the mean-reverting γ-process has the unique solution.

Nonhomogeneous(H,Q)-Process:The Backward and Forward Equations

As for the backward and forward equation of nonhomogeneous(H, Q) -processes,we proof them in a new way. On the base of that, this paper gives the direct computational formalfor one dimensional distribution of the nonhomogeneous(H, Q) -process.

Production sharing contract: An analysis based on an oil price stochastic process

Assuming that oil price follows the stochastic processes of Geometric Brownian Motion （GBM） or the Mean-Reverting Process （MRP）, this paper takes the net present value （NPV） as an economic index and models the PSC in 11 different scenarios by changing the value of each contract element （i.e. royalty, cost oil, profit oil as well as income tax）. Then the NPVs are shown in probability density graphs to investigate the effect of different elements on contract economics. The results show that under oil price uncertainty the influence of profit oil and income tax on NPV are more significant than those of royalty and cost oil, while a tax holiday could improve the contractor＇s financial status remarkably. Results also show that MRP is more appropriate for cases with low future oil price volatility, and GBM is best for high future oil price volatility.

An infinite-dimensional representation of the Ray-Knight theorems

The classical Ray-Knight theorems for the Brownian motion determine the law of its local time process either at the first hitting time of a given value a by the local time at the origin,or at the first hitting time of a given position b by the Brownian motion.We extend these results by describing the local time process jointly for all a and b,by means of the stochastic integral with respect to an appropriate white noise.Our result applies toμ-processes,and has an immediate application:aμ-process is the height process of a Feller continuous-state branching process(CSBP)with immigration(Lambert(2002)),whereas a Feller CSBP with immigration satisfies a stochastic differential equation(SDE)driven by a white noise(Dawson and Li(2012));our result gives an explicit relation between these two descriptions and shows that the SDE in question is a reformulation of Tanaka’s formula.

QNQL processes-- (H, Q)-processes and their applications

As we know,many scholars have been involved in the study of stochastic processes,whichhave a wide range of applications,such as the Markov processes.In the study of Markov pro-cesses,minimal Markov chain(i.e.the minimal homogeneous denumerable Markov process)got most mature.One of the characteristics of this kind of processes is that the stay time

Stochastic PDEs for large portfolios with general mean-reverting volatility processes

We consider a structural stochastic volatility model for the loss from a large portfolio of credit risky assets.Both the asset value and the volatility processes are correlated through systemic Brownian motions,with default determined by the asset value reaching a lower boundary.We prove that if our volatility models are picked from a class of mean-reverting diffusions,the system converges as the portfolio becomes large and,when the vol-of-vol function satisfies certain regularity and boundedness conditions,the limit of the empirical measure process has a density given in terms of a solution to a stochastic initial-boundary value problem on a half-space.The problem is defined in a special weighted Sobolev space.Regularity results are established for solutions to this problem,and then we show that there exists a unique solution.In contrast to the CIR volatility setting covered by the existing literature,our results hold even when the systemic Brownian motions are taken to be correlated.

Uniform dimension results of multi-parameter stable processes

The problem of uniform dimensions for multi-parameter processes, which may not possess the uniform stochastic H?lder condition, is investigated. The problem of uniform dimension for multi-parameter stable processes is solved. That is, ifZ is a stable (N,d, α)-process and αN ?d, then $$\forall E \subseteq \mathbb{R}_ + ^N , \dim Z\left( E \right) = \alpha \cdot \dim E$$ holds with probability 1, whereZ(E) = {x : ?t ∈E,Z t =x} is the image set ofZ onE. The uniform upper bounds for multi-parameter processes with independent increments under general conditions are also given. Most conclusions about uniform dimension can be considered as special cases of our results.

Modelling and Numerical Valuation of Power Derivatives in Energy Markets

In this work we investigate the pricing of swing options in a model where the underlying asset follows a jump diffusion process.We focus on the derivation of the partial integro-differential equation(PIDE)which will be applied to swing contracts and construct a novel pay-off function from a tree-based pay-off matrix that can be used as initial condition in the PIDE formulation.For valuing swing type derivatives we develop a theta implicit-explicit finite difference scheme to discretize the PIDE using a Gaussian quadrature method for the integral part.Based on known results for the classical theta-method the existence and uniqueness of solution to the new implicit-explicit finite difference method is proven.Various numerical examples illustrate the usability of the proposed method and allow us to analyse the sensitivity of swing options with respect to model parameters.In particular the effects of number of exercise rights,jump intensities and dividend yields will be investigated in depth.

Markov skeleton processes

A new class of stochastic processes--Markov skeleton processes is introduced, which have the Markov property on a series of random times. Markov skeleton processes include minimal Q processes, Doob processes, Q processes of order one, semi-Markov processes , piecewise determinate Markov processes , and the input processes, the queuing lengths and the waiting times of the system GI/G/1, as particular cases. First, the forward and backward equations are given, which are the criteria for the regularity and the formulas to compute the multidimensional distributions of the Markov skeleton processes. Then, three important cases of the Markov skeleton processes are studied: the (H, G, Π)-processes, piecewise determinate Markov skeleton processes and Markov skeleton processes of Markov type. Finally, a vast vistas for the application of the Markov skeleton processes is presented.