| As the science and technology is developing, it requires us model and discuss certain process precisely so as to evalute or study its properties and developing inclination. The theory of diffusion process is one of the most important model which is relevant in various biological, physical, chemical and mathematical finance applications. But for the particular diffussion process ?Brownian motion, being the most simple stochastic process with conticuous time parameter and continuous state space, It has been attented by more and more bachelors and experts as its use in science, technology, economy wider and wider; On the other hand, the study on the Brownian motion has been the base of modern probability as it is filtering into many branches of probability theory and mathematical analysis. Therefore, the research on diffusion process and Brownian motion is of great importance in theory and in practice.In this paper, we'll show that the moments of the first hitting time to a general region D for the homogeneous and inhomogeneous diffusion processes are the solutions of certain partial differential equations with suitable boundary and initial conditions. Then, we get some identities about Bessel functions and the moments of the first hitting time, the last exit time from a ball for the nth iterated Brownian motion.Let X = {Xt}t>0 be a diffusion process in Rd, and D be a relative compact smooth region. Denoting by TD = inf{t > 0, Xt D}, the firsthitting time of X to the region D. Let B = (Bt)t>o be a standard Brownian motion in Rd, BT={x,x € Rd, \x\ < r}, the ball centered at 0 with radius r. BT = {x, x R1, |x| = r} be the surface of the ball. Px, the conditional probability starting from point or, and the corresponding numerical evalutionsis Ex. Denoting by Tr = Br (Bs)ds the sojourn time of Brownian motionin the ball Br and Lr = sup{t > 0, Bt Br} the last exit time of Brownianmotion B = (Bt) t >0 from the ball Br.There are three parts in this paper. In Section 1, the moments of the first hitting times to the rigion D for the homogeneous and inhomogeneous diffusion process are shown to be the solutions of certain partial differential equations with suitable boundary and initial conditions. The corresponding moments for Brownian motion on a general region are discussed. In particular, we get the first three moments of Tr and Lr for the ball respectively. The results are as follows:? Homogeneous diffusion process denoting by L the second order elipse differential operatorMain result 1: For x ?D,t > 0, then P(t,x) = Px(rD > t) is the solution of the following partial differential equation :Main result 2 : For x D, f(x) = EXTD is the solution of the follow-ing partial differential equation :Main result 3:the following PDE: is the solution of Inhomogeneous diffusion process denoting by A + ^ the second order elipse differential operatorMain result 4 : Fors D, t > s > 0, u(s, x, t) = PS> x(TD -s>t) is the solution of thefollowing partial differential equations :Main result 5: x D, s > 0, then u(s, x) = ES,X(TD - s) is thesolution of the PDE as follows:Main result 6: For x D, s > 0, n N, then u(s,x)=ES,x( TD - s)n satisfies the following equation:As is well known, Bessel function plays an important role in the study of the functional properties of Brownian motion. In Section 2, the writer computed the three moments of the first hitting time and the sojourn times of Brownian motion on a ball respectively. Then we get some identities for Bessel functions by means of probability methods which is diflBcult to obtain by the method of pure mathematics.In Section 3, the moments of the hitting time and the last exit time for the iterated Brownian motion are also discussed.Let {Bj(i),t > 0}, i=l,2,---n be indispending Brownian motion starting from point 0. If X?t) = Bi(t) , XW(t) = B^B^t)]), - - -, X(t) = Bi(\Xtn-V(t)\)t then we call {X^(t),t > 0} the nth iterated Brownian motion. For some r > 0, define T$n\ a^ be the first hi... |
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