## Session 10: Probability and Statistics, Financial Mathematics Markov Skeleton Processes and Their Applications to Queueing Systems, Reliability Theory and Inventory Models

Zhenting Hou
School of Mathematics
Central South University
Changsha, Hunan, 410075, China
zthou@csu.edu.cn

Abstract : A stochastic process X(t) is called a Markov skeleton process(MSP) if it has the Markov property on a sequence of stopping times tn ­¥. The usual Markov process, semi-Markov process, deterministic Markov process and semi-regenerative process can be regarded as special cases of MSP. In this paper, first, backward and forward equations with which we can compute one-dimensional distribution is derived, and then formulas to compute finite-dimensional distribution and the existence and computation of limit distribution are also obtained. Based ourselves upon the above the results, we give a tentative study of queueing system, reliability system, and storage system. Transient distribution and formulas to compute limit distribution of the stochastic processes introduced for studying these system are presented in the latter half the paper.

## Dynamic Optimal Recovery to Credit Risk

Jianhui Huang
Department of Mathematics and Statistics
The University of Alberta
jhuang@stat.ualberta.ca

Abstract : This paper investigates the optimal recovery problem with respect to credit risk, which has not been well analyzed so far. In order to examine this problem, we develop a Backward Stochastic Differential Equation (BSDE) Representation for credit risk pricing in Reduced-form model under which Recovery of Market Value (RMV) scheme is presumed. Combining this BSDE Representation with the Stochastic Maximum Principle (SMP), we then derive conditions characterizing the optimal recovery policy for a general class of utility functions. In particular, we study two important cases where the utility is of the exponential form, power form respectively, and provide the optimal recovery policies as well as their market implications.

## A Study on the Test of Equal Mean Vector And Its Application to the Forensic Science

Congzhu Li* and Wang Jianwen

Department of Mathematics
North China University of Technology
Beijing, 100041, P.R. China
lcz@ncut.edu.cn

Abstract : Statistic with many good statistical characters is usually used in the test of equal mean vector, but its applying conditions are , which have limited its applying scope greatly, especially in the forensic science. In this paper, the condition is extended to and the background of application is provided. In the end, an example is given.

Key words: Mean Vector, Test, T2-statistic, Two-sample

## A Kind of Risk Models with Two Classes of Insurance Business

Zaiming Liu
School of Mathematics
Central South University
Changsha, 410075, P.R. China
Math_lzm@mail.csu.edu.cn

Abstract : In this paper we consider a risk model involving two classes of insurance business , one of which is delay. Let {N1(t)}, {N2(t)} be the claim number processes of two classes of insurance business and {Zi(1)}i = 1¥, {Zj(2)}j = 1¥ be the series of claim size random variables for two classes of insurance business. Let T1 be the delay variable, that is, nonnegative random variable. Then the surplus process is as following

 R(t) = u+c1t- N1(t) å i = 1 Zi(1) +{c2(t-T1)- N2(t-T1) å j = 1 Zj(2)}I{t > T1}
where u is the initial reserve, c1,c2 are the incomes of two classes of insurance business in the unit time, and IA is the index function. Furthermore, suppose that {N1(t)},{N2(t)}, {Zi(1)}i = 1¥, {Zj(2)}j = 1¥ are independent; {N1(t)} is a Poisson process with the parameter l1,{N2(t)} is a Erlang process with the parameters n,l2; and {Zi(1)}i = 1¥, {Zj(2)}j = 1¥ with the common distributions F1(z), F2(z) and EZ(i) = mi, Fi(0) = 0. And suppose that c1-l1m1 > 0, c2-l2m2 > 0.

In this paper, we give the integration equations about the ruin probability Y(u) , and the explicit expressions for Y(u) have been got.

Key Words: risk model, ruin probability, finite-horizon ruin probability.

## Ruin Probability in a Class of Stationary Renewal Model

Xuan Luo* and Wen-Fen Liu
Department of Mathematics
Information Engineering University
Zhengzhou Henan
xuanluo@yeah.net

Abstract : Considering a stationary risk process for which the claim inter-arrival distribution is Erlang(2), we get the ultimate ruin probability by using the anderson risk model. In particular, we also get the explicit solution of ultimate ruin probability in the case where the individual claim amount distribution is exponential. Finally, we calculate the ruin probability of a concrete example and analyse the variety of ruin probability affected by initial reserve and gross risk premium rate.

## A Hyperfinite Model of Margin Trading

Siu-Ah Ng
School of Mathematical Sciences
University of Natal - Pietermaritzburg
3209 South Africa
ngs@nu.ac.za

Abstract : A binomial model of margin trading based on the hyperfinite timeline from nonstandard analysis is given, with application in computing the average margin call time and the average loss. Some strategies in margin trading are discussed. The combinatorial techniques behind the nonstandard method are explained, leading to a generalization of the Catalan numbers. Further use of the model in pricing barrier options is demonstrated.

## On the Martingale Property of Stochastic Exponentials

Bernard Wong*
School of Actuarial Studies,
University of New South Wales
Sydney, Australia
bernard.wong@unsw.edu.au

C.C. Heyde
Department of Statistics
Columbia University
New York, USA
chris@stat.columbia.edu.au

Abstract : We derive a necessary and sufficient condition for the martingale property of a continuous stochastic exponential. It is proved that the criteria for the true martingale property is related to whether a related process explodes. Applications of our theorem to problems arising in mathematical finance are also given.

## The Geometric Properties of the Brownian Sheet

Davar Khoshnevisan
Department of Mathematics
University of Utah
Salt Lake City, UT 84112, U.S.A. davar@math.utah.edu.

Yimin Xiao*
Department of Statistics and Probability
Michigan State University
East Lansing, MI 48823, U.S.A. xiao@stt.msu.edu

Abstract : An N-parameter Brownian sheet in Rd maps a non-random compact set F in RN+ to the random compact set B(F) in Rd. We prove the following results on the image-set B(F): (1) It has positive d-dimensional Lebesgue measure if and only if F has positive d/2-dimensional capacity. This generalizes greatly the earlier works of J. Hawkes (1977), J.-P.@ Kahane (1985a, 1995b) and D. Khoshnevisan (1999).

(2) If dimF > d/2, then with probability one, we can find a finite number of points z1,¼,zm Î Rd such that for any rotation matrix q that leaves F in RN+, one of the zi's is interior to B(qF). In particular, B(F) has interior-points a.s. This verifies a conjecture of T. S. Mountford  (1989). (3) If dimF < d/2, then B(F) is almost surely a Salem set.

The proofs of these results rely on two novel ideas: To prove (1), we introduce and analyze a family of bridged sheets. Items (2) and (3) are proved by developing a notion of ``sectorial local-non-determinism (LND).'' Both ideas may be of independent interest.

## Valuation of Asian Options

Hong-Kun Xu
School of Mathematical Sciences
University of KwaZulu-Natal
Westville Campus
Private Bag X54001, Durban 4000
South Africa
xuhk@ukzn.ac.za

Abstract : Assume that an underlying asset (stock) {S(t)}t ³ 0 follows a geometric Brownian motion {W(t)}t ³ 0. A European Asian option on the stock is a claim whose payoff depends on some sort of averages of the stock prices between the initiation and expiration time. The averages can be discrete (arithmetic or geometric) and continuous. We are going to talk about the risk-neutral valuation of and PDE approach to Asian options. Stochastic volatility in Asian options is possibly discussed. Some numerical methods will also be included.

## Optimal Utility With Some Additional Information

Zhongxing Ye
Department of Mathematics
Jiao Tong University
Shanghai 200030, P.R.China
Zxye@mail.sjtu.edu.cn

Abstract : We study the utility optimization problem with some additional information in continuous time setting.. First we consider three kinds of enlargement of the information filtration and obtain the martingale representation theorems for each case. As an application, we consider the utility optimization problem of the agents with inside or additional information. In the case of logarithmic utility, we obtain explicit formulas of optimal wealth and optimal strategy.

## Massive Data Analysis and Multi-Agent Fuse Trade System in Capital Market

Dongyun Yi
Department of Mathematics and System Science
National University of Defense Technology
Changsha 410073, Hunan, P.R. China
yidongyun@sohu.com

Abstract : With the aim of solving the intercrossing core problems emerge in these fields: new pattern of econometric model in capital market, massive data processing and multi-agent fusion trade system, this paper focuses on the techniques of dynamic and instantaneous massive data processing techniques, architecture design of multi-agent fusion system and robust control of fusion trade dealing. Integrating complex system evolution properties and real-time data interaction, a information perceptive intelligent platform has been studied, including three subsystems which are a primary security information auto-acquire from Web, a distribution background perceptive computing and a simulation trade test surroundings. The results show that a new breakthrough is expected in three fields simultaneously, and innovate production in the software technique and econometric theory of capital market would be available.

File translated from TEX by TTH, version 2.00.
On 08 Dec 2004, 11:36.