完备市场下的均衡和财富优化

针对跳扩散模型中的优化与均衡问题,利用鞅方法和随机点过程理论,建立了跳扩散模型下的均衡市场,分析了市场中的财富优化问题,给出了均衡大宗商品现货价格、最优财富过程、最优投资组合及最优消费过程.     

一类扩散过程轨道的excursion分解

本文对于一类扩散过程的轨道作了excursion分解。应用Maisonnenve给出的exitsystem,得到了通过扩散的转移密度表出的相应泊松点过程的特征测度。作为例子,给出了Ornstein-Uhlenbeek过程的一个随机积分表示,最后用Geoor的方法计算出了熟知的该过程的一个不变测度。     

扩散过程的极性的一个充分条件

本文研究非退化多维扩散过程的极性，得到了一个充分条件。     

跳跃扩散过程的期权定价模型

假定股票价格的跳过程为计数过程,建立了股票价格服从跳扩散过程的行为模型.运用随机分析中的鞅方法,推导出了股票价格的跳过程为计数过程的欧式期权定价公式,推广了已有的结果.     

基于跳扩散过程的数据选择权定价

本文在风险中性原理下研究基于跳扩散过程的数据选择权定价问题,推导了标的资产价格服从跳扩散过程的数据选择权的定价公式。     

跳扩散过程下的保险商偿债率模型研究

本文研究了在有金融困境成本的情况下,带有跳扩散过程的保险商偿债率(SR)模型的问题.利用Girsanov定理进行测度变换的方法以及跳扩散过程下的看涨期权定价公式,获得了保险商终期收益的现值的结果.推广了不带跳扩散过程的保险商偿债率模型的结果.     

非紧Riemann流形上平稳扩散过程的熵产生率

给出了具有有界Nelson速度流的平稳扩散过程的熵产生率的封闭形式,讨论了一列反射扩散的熵产生率的极限。     

高维扩散过程扩散系数估计的中偏差

本文考虑高维扩散过程的大偏差.对于高维扩散过程dX(t)=σ(t)dB(t),(其中σ(t)未知),我们讨论其平方变差过程[X]t=∫0t(σσ*)(s)ds的估计的大偏差及中偏差.通过利用Gartner-Ellis定理,得到了上述估计在固定时刻t=1时的中偏差;同时通过计算其对数矩生成函数的Fenchel-Legendre变换,得到其速率函数的显式表达.     

主丛上的扩散过程与配丛截面空间的微分算子

本文研究扩散过程在主丛上的提升及微分算子在配丛截面空间上的提升。我们证明了提升得到的扩散过程的生成元可作为配丛截面空间上的二阶微分算子,它就是原扩散过程的无穷小生成元的提升。由此我们给出了协变的Feynman-Kac公式,这是文献[4]结论在非平凡主丛上的推广。应用这些结果,我们给出了Riemann流形上Girsanov-Cameron-Martin定理的几何形式的证明。     

二维扩散过程的旋转数

本文定义了二维扩散过程的旋转数,证明了遍历过程旋转数的存在性和正规性。     

The approximation of additive functionals of diffusion processes

Let us consider a diffusion process in R_d . Around each point x one may consider a ring of size ? and a process which counts the crossings over the ring. Integrating with respect to a measure μ(dx) and letting ?→ 0 one gets an additive functional. This is a natural generalization of the approximation theorem of the local time of one dimensional Brownian motion by means of “downcrossings”. For multidimensional Brownian motion the result was established by Bally. The present paper introduces a new method which allows us to handle general diffusions     

Phase Space Transformations of Gaussian Diffusions

It is shown that for Gaussian diffusions, the transformation back to Brownian motion, usually accomplished via the Girsanov (or Feynman–Kac) formula and time-shift, can be obtained by a classical canonical, i.e. symplectic, transformation in phase space. The method is based on constants of motion, in this case the Wronskian. Similar transformations for general diffusions are briefly discussed.     

Fake exponential Brownian motion

We construct a fake exponential Brownian motion, a continuous martingale different from classical exponential Brownian motion but with the same marginal distributions, thus extending results of Albin and Oleszkiewicz for fake Brownian motions. The ideas extend to other diffusions.     

Réflexion entre deux diffusions conjuguées

We observed, in a previous work, that Brownian motion reflected on an independent time-reversed Brownian motion is again Brownian motion. We present the generalisation of this result to pairs of conjugate diffusions (which are also dual, in the sense of Siegmund). To cite this article: F. Soucaliuc, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1119–1124.     

Asymptotically One-Dimensional Diffusion on the Sierpinski Gasket and Multi-Type Branching Processes with Varying Environment

Asymptotically one-dimensional diffusions on the Sierpinski gasket constitute a one parameter family of processes with significantly different behaviour to the Brownian motion. Due to homogenization effects they behave globally like the Brownian motion, yet locally they have a preferred direction of motion. We calculate the spectral dimension for these processes and obtain short time heat kernel estimates in the Euclidean metric. The results are derived using branching process techniques, and we give estimates for the left tail of the limiting distribution for a supercritical multi-type branching process with varying environment.     

Distributions of functionals of diffusions with jumps stopped at the location of the maximum or minimum

The paper deals with methods of computation of distributions of integral functionals of diffusions with jumps at time moments at which the maximal and minimal values of diffusions are achieved. As an example, we obtain closed-form expressions for the Laplace transform of joint locations of the minimum and maximum of a process that equals the sum of a Brownian motion and the compound Poisson process. Bibliography: 7 titles.     

Winding numbers for 2-dimensional,positive recurrent diffusions

A well-known theorem by Spitzer states that the winding number of a standard Brownian motion around the origin is asymptotically Cauchy-distributed. A similar result is derived for positive recurrent diffusions in the plane given by a non-degenerate stochastic equation.     

An occupation time related potential measure for diffusion processes

In this paper, for homogeneous diffusion processes, the approach of Y. Li and X. Zhou [Statist. Probab. Lett., 2014, 94: 48–55] is adopted to find expressions of potential measures that are discounted by their joint occupation times over semi-infinite intervals (-∞, α) and (α, ∞): The results are expressed in terms of solutions to the differential equations associated with the diffusions generator. Applying these results, we obtain more explicit expressions for Brownian motion with drift, skew Brownian motion, and Brownian motion with two-valued drift, respectively.     

Last exit decompositions and regularity at the boundary of transition probabilities

Summary The purpose of this paper is to give a probabilistic approach to studying the regularity at the boundary of the transition probabilities of certain hypoelliptic diffusions with boundary conditions. The main tools are last exit decompositions of Brownian motion, the Malliavin calculus, the theory of excursions, and the calculus of variations on Brownian excursions.     

A Set-indexed Fractional Brownian Motion

We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations with the Lévy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no “really nice” set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, behavior of the set-indexed fractional Brownian motion along increasing paths is analysed.