The generalization of a class of impulse stochastic control models of a geometric Brownian motion

Recently, international academic circles advanced a class of new stochastic control models of a geometric Brownian motion which is an important kind of impulse control models whose cost structure is different from the others before, and it has a broad applying background and important theoretical significance in financial control and management of investment.This paper generalizes substantially the above stochastic control models under quite extensive conditions and describes the models more exactly under more normal theoretical system of stochastic process.By establishing a set of proper variational equations and proving the existence of its solution, and applying the means of stochastic analysis, this paper proves that the generalized stochastic control models have optimal controls.Meanwhile, we also analyze the structure of optimal controls carefully.Besides, we study the solution function of variational equations in a relatively deep-going way, which constitutes the value function of control models to some extent.Because the analysis methods of this paper are greatly different from those of original reference, this paper possesses considerable originality to some extent.In addition, this paper gives the strict proof to the part of original reference which is not fairly well-knit in analyses, and makes analyses and discussions of the model have the exactitude of mathematical sense.

MULTI-DIMENSIONAL GEOMETRIC BROWNIANMOTIONS, ONSAGER-MACHLUP FUNCTIONS, AND APPLICATIONS TO MATHEMATICAL FINANCE

The solutions of the following bilinear stochastic differential equation are studied [GRAPHICS] where A(t)(k), B-t are (deterministic) continuous matrix-valued functions of t and w(1) (t),..., w(m) (t) are m independent standard Brownian motions. Conditions are given such that the solution is positive if the initial condition is positive. The equation the most probable path must satisfy is also derived and applied to a mathematical finance problem.

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.

Evolution-Based Uncertainty Analysis for Incline Hoist

The effect of uncertainty and its evolution with time on the incline hoist reliability are investigated in this paper.The performance of incline hoist is changed over time and gradually degraded.The degradation will influence the safe usage and reliability of incline hoist.Degradation process can be described by stochastic process.The degradation process of incline hoist is modeled in geometric Brownian motions(GBM),and the drift rate and diffusion rate of this process can reflect the failure extent and fluctuation of the system.Evolution-based uncertainty analysis(EBUA)method is proposed to describe the dynamic reliability of the incline hoist,and the system of incline hoist can be designed with the specified reliability value at the given time.

Integro-Differential Equations for a Jump-Diffusion Risk Process with Dependence between Claim Sizes and Claim Intervals

The classical Poisson risk model in ruin theory assumed that the interarrival times between two successive claims are mutually independent, and the claim sizes and claim intervals are also mutually independent. In this paper, we modify the classical Poisson risk model to describe the surplus process of an insurance portfolio. We consider a jump-diffusion risk process compounded by a geometric Brownian motion, and assume that the claim sizes and claim intervals are dependent. Using the properties of conditional expectation, we establish integro-differential equations for the Gerber-Shiu function and the ultimate ruin probability.

A Useful Extension of It 's Formula with Applications to Optimal Stopping

Given a continuous semimartingale M = （Mt）t≥〉0 and a d-dimensional continuous process of locally bounded variation V = （V^1,……, V^d）, the multidimensional Ito Formula states that f（Mt, Vt） - f（M0, V0） = ∫[0, t] Dx0f（Ms, Vs）dMs＋∑i=1^d∫[0, t] Dxi F（Ms, Vs）dVs^i＋1/2∫[0, t] Dx0^2 f（Ms, Vs）d 〈M〉s if f（x0,……,xd） is of C^2-type with respect to x0 and of C^1-type with respect to the other arguments This formula is very useful when solving various optimal stopping problems based on Brownian motion. However, in such application the function f typically fails to satisfy the stated conditions in that its first partial derivative with respect to x0 is only absolutely continuous. We prove that the formula remains true for such functions and demonstrate its use with two examples from Mathematical Finance.

Computations of VαR for European Options on Dividend-paying Securities

Options are a kind of financial derivative tools having the characteristic of nonlinear curvature. Compared with other derivatives such as futures, the market risk of options is much more difficult to measure. Options are based on certain underlying securities, when the security is the one having dividend-paying, the price of the security would immediately drop according to the arbitrage theorem, which results in the fact that we couldn＇t assume the price of the security follow the geometric Brownian motion when we compute the value or VaR for options. Therefore, based on the assumption that we reinvest all dividends in the purchase of additional shares, this paper derives the values of European options on dividend-paying securities at some instant time, and the utmost losses of options during the period of holding options, i.e. the computational formulations of VαR, consequently, we can implement the real-time monitoring of market risk of European options on dividend-paying securities.