Two-dimensional diffusion processes are considered between concentric circles and in angular sectors. The aim of the paper is to compute the probability that the process will hit a given part of the boundary of the stopping region first. The appropriate partial differential equations are solved explicitly by using the method of similarity solutions and the method of separation of variables. Some solutions are expressed as generalized Fourier series.

Let

Suppose that we consider only the process

Then, it is well known (see Cox and Miller [

Next, let

Many papers have been devoted to first passage time problems for diffusion processes, either in one or many dimensions; see, in particular, the classic papers by Doob [

Now, define

In Section

In Section

Finally, a few remarks will be made in Section

From the Kolmogorov backward equation (

Because the region

First, we take

where

If we choose

Next, we choose

The last particular case that we consider is the one when

Equation (

(1) When

(2) If the parameter

In the next section, the problem of computing explicitly the function

We consider the two-dimensional processes defined by the stochastic differential equations (

As in the previous section, we will obtain explicit (and exact) solutions to the first hitting place problem set up above for the most important particular cases.

When

Next, the solution of (

If the infinitesimal mean of

When

Next, the general solution of (

Finally, with

The condition

Now, notice that (

Here, we consider the case where

Finally, we must solve the ordinary differential equation

There is at least another particular case of interest for which we can obtain an explicit expression (when

Writing

We have considered, in this note, the problem of computing first hitting place probabilities for important two-dimensional diffusion processes starting between two concentric circles or in an angular sector. We have obtained explicit (and exact) solutions to a number of problems in Sections

Now, there are other important two-dimensional diffusion processes for which the techniques used in this note do not work. In particular, there is the two-dimensional Wiener process with nonzero infinitesimal means and also the geometric Brownian motion. Moreover, we have always assumed, except in the last remark above, that the two diffusion processes,

Next, we could also try to find explicit solutions to first hitting place problems, but in three or more dimensions. It should at least be possible to solve some special problems.

Finally, we have computed the probability that the process