From: Nils Christian Framstad <ncf@math.uio.no>
Subject: Re: A Brownian motion question
Date: 30 Apr 1999 13:41:35 +0200
Newsgroups: sci.math.research
Keywords: Stochastic differential equation

fhchen@my-dejanews.com writes:

> Hi,
> 
> Does anyone know how to get an explicit expression for the following
> value function V(x,t)?
>
> Consider the Brownian motion X(t) with drift mu and diffusion
> coefficient sigma. That is, dX=(mu)dt+(sigma)dW, where W is a
> standard Wiener process. X(0)=0. Given a "retirement" value R, find
> V(X,t)=E_X [\int_0^\tau {exp(-rt) mu dt}+exp(-r(tau))R] where
> tau=inf{t:X(t)=at+b}, b<0.
>
> Can anyone help me, or does anybody know of a good reference for
> this type of problems? Thanks.


First a general comment: Problems of this kind are treated in most
textbooks on stochastic calculus and applications to boundary value
problems and/or optimal control, for example Bernt Øksendal:
Stochastic Differential Equations on Springer Universitext.

(Disclaimer: I am not the author's cousin, but I am his student.)


The tricky part is that most of these references use the Dynkin
formula, which requires the stopping time to be integrable. Yours
isn't - it is even infinite with positive probability. You may go back
to Dynkin's original formula in his book "Markov Processes" (vol II
page 132, theorem 5.1 if I remember correctly), which is a corollary
to a similar theorem with "discounting" and possibly infinite stopping
time, but this book is not easy to read. 


To solve your particular case: Presuming you have constant
coefficients with nonzero sigma, we can get rid of some of them:

X(t) = mu t + sigma W(t) 

and X(t) = at+b iff W(t)= ((a-mu)t + b) /sigma  =: -pt + q

and tau=inf{t; W(t) + pt = q}


Furthermore, solving the integral we get 
V= mu + (R-mu)/r  * E[exp(-r tau)]


Then I apply to Borodin and Salminen: Handbook of Brownian Motion
(Birkhäuser 1996), formula 2.0.1 p223: 

E[exp (-r tau)] = exp (pq - |q|*sqrt(2r+p^2) )


The rest is now trivial.


nc

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