Research Article | Open Access
Similarity Solutions of Partial Differential Equations in Probability
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 be the two-dimensional diffusion process defined by the stochastic differential equations for , where is nonnegative and and are independent standard Brownian motions. In this note, the problem of computing the probability that the process , starting between two concentric circles, will hit the larger circle first is solved for the most important particular cases. The process is also considered inside a circle centered at the origin, and, this time, the probability that will hit the boundary of the circle before either of two radii is treated. Again, the most important particular cases are analyzed.
Suppose that we consider only the process in the interval . Let
Then, it is well known (see Cox and Miller [1, p. 230], for instance) that the moment generating function (which is a Laplace transform) of the first passage time , where is a nonnegative parameter, satisfies the Kolmogorov backward equation and it is subject to the boundary conditions
Next, let The function satisfies the ordinary differential equation (see Cox and Miller [1, p. 231]) with It is therefore a simple matter to compute explicitly the probability of hitting the point before , starting from . In particular, in the case when is a standard Brownian motion, so that and , we find at once that
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  and Spitzer , and also Wendel . However, a rather small number of papers have been written on first hitting place problems; see, for instance, the papers by Yin and Wu  and by Yin et al. . Guilbault and Lefebvre (see [7, 8]) have considered problems related to the ones treated in the present note; however, in these problems, the processes were considered inside rectangles.
Now, define where is a subset of for which the random variable is well defined. The moment generating function of , namely, satisfies the Kolmogorov backward equation where and . This partial differential equation is valid in the continuation region and is subject to the boundary condition
In Section 2, the set will be given by and the function where is the random variable defined in (1.10) with , will be computed in important special cases, such as when is a two-dimensional Wiener process.
In Section 3, we will choose We will calculate for important two-dimensional diffusion processes the probability where is the time taken by to leave the set , starting from for .
Finally, a few remarks will be made in Section 4 to conclude.
2. First Hitting Place Probabilities when Starting between Two Circles
From the Kolmogorov backward equation (1.12), we deduce that the function defined in (1.15) satisfies the partial differential equation in the set defined in (1.14), and is subject to the boundary conditions Because the two-dimensional process is considered between two concentric circles, it seems natural to try to find a solution of the form where . Actually, this only works in a few, but very important, special cases, some of which will be presented below. The partial differential equation (2.1) becomes
Remark 2.1. Because the region is bounded, the solution to the problem (2.1), (2.2) is unique. Therefore, if we can find a solution of the form , then we can state that it is indeed the solution we were looking for.
2.1. The Two-Dimensional Wiener Process
First, we take and . Then is a two-dimensional Wiener process with zero infinitesimal means and infinitesimal variances both equal to . Equation(2.4) can be rewritten as Notice that this is a first-order linear ordinary differential equation for . It is a simple matter to find that
where and are constants. Therefore, The boundary condition (2.2) yields that
Remark 2.2. If we choose or if for , with , then the particular case of the method of similarity solutions that we have used above fails. Notice also that the solution does not depend on the parameter .
2.2. The Two-Dimensional Ornstein-Uhlenbeck Process
Next, we choose and for , where is a positive parameter, so that is a two-dimensional Ornstein-Uhlenbeck process with the same infinitesimal parameters. This time, (2.4) becomes the general solution of which can be expressed as where Ei is the exponential integral function defined by in which the principal value of the integral is taken. It follows that
2.3. The Two-Dimensional Bessel Process
The last particular case that we consider is the one when and for . Again, is a positive parameter, so that is a two-dimensional Bessel process. We assume that (and ); then, the origin is a regular boundary for and (see Karlin and Taylor [9, p. 238-239]).
Remarks 2.3. (1) When , the ordinary differential equation (2.14) reduces to the one obtained in Section 2.1 with the two-dimensional Wiener process, if .
(2) If the parameter is greater than or equal to 2, the origin is an inaccessible boundary for and ; that is, it cannot be reached in finite time. Therefore, in this case the continuation region could be the region between the two concentric circles, but inside the first quadrant (for instance).
In the next section, the problem of computing explicitly the function defined in (1.17) for important two-dimensional diffusion processes in angular sectors will be treated. This time, we will work in polar coordinates and make use of the method of separation of variables, which can be viewed as a special case of the method of similarity solutions. The solutions will be expressed as generalized Fourier series and will therefore be more involved than the simple solutions obtained in this section.
3. First Hitting Place Probabilities when Starting in Angular Sectors
We consider the two-dimensional processes defined by the stochastic differential equations (1.1) inside the circle of radius centered at the origin. In polar coordinates, the function satisfies the Kolmogorov backward equation (see (1.12)) where and . Let that is, with defined in (1.16). The probability defined in (1.17) satisfies the same partial differential equation as in polar coordinates, with . Furthermore, is such that
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.
3.1. The Two-Dimensional Wiener Process
When is a two-dimensional Wiener process, with independent components and infinitesimal parameters 0 and , the partial differential equation that we must solve reduces to Looking for a solution of the form , we find that so that we obtain the ordinary differential equations where is the separation constant. The ordinary differential equation (3.7) is subject to the boundary conditions whereas It is well known that the function must be of the form where is a constant; therefore, the separation constant must be given by
Next, the solution of (3.8) (which is an Euler-Cauchy equation), with , that is such that = 0 is We then consider the infinite series where is a constant. This series, as a function of , is a Fourier series. The condition implies that Hence, the solution is for and .
Remark 3.1. If the infinitesimal mean of is not equal to zero, we cannot separate the variables in the partial differential equation satisfied by the function . So, as in Section 2, the cases for which the technique we have used will work are actually rather few. Fortunately, it does work in the most important cases for applications.
3.2. The Two-Dimensional Ornstein-Uhlenbeck Process
When and are independent Ornstein-Uhlenbeck processes with infinitesimal parameters and for , we must solve the partial differential equation Writing , we obtain the ordinary differential equations The boundary conditions are the same as in Section 3.1. Therefore, we find that we still have and
Next, the general solution of (3.19) can be written as where and is a confluent hypergeometric function (see Abramowitz and Stegun [10, p. 504]). We find at once that we must choose equal to zero. We then consider the infinite series Making use of the boundary condition = 1, we find that for and .
3.3. The Two-Dimensional Bessel Process
Finally, with and for , we obtain the partial differential equation It follows that we must solve the ordinary differential equation We assume that is in the interval . Writing that , where , we find that this ordinary differential equation is transformed to The general solution of (3.26) can be written in the form where and is a hypergeometric function (see Abramowitz and Stegun [10, p. 556]). Hence, we have where .
The condition implies that we must set equal to zero. Next, we must find the value(s) of the separation constant for which = 0; that is,
Now, notice that (3.25) can be written in the form with , and . If we assume that in , then the problem of solving (3.25) together with the boundary conditions = 0 is a regular Sturm-Liouville problem. It follows that we can state (see Edwards Jr. and Penney [11, p. 519], for instance) that there exist an infinite number of eigenvalues for which the conditions = 0 and = 0 are satisfied. These eigenvalues constitute an increasing sequence of real numbers with . Moreover, we can also state (see Butkov [12, p. 337–340]) that the eigenfunctions corresponding to the eigenvalues are orthogonal to each other with respect to the weight function . However, computing these eigenvalues explicitly is another problem.
Here, we consider the case where , so that we do not have a regular Sturm-Liouville problem. However, one can check graphically, using a computer software, that there exist an infinite number of positive constants for which where for We thus have, apart from an arbitrary constant, for .
Finally, we must solve the ordinary differential equation subject to = 0. This is an Euler-Cauchy differential equation; for all positive eigenvalues , we can write that where . Since there are an infinite number of such eigenvalues, we can consider the infinite series (a generalized Fourier series) Making use of the boundary condition , we can write that the constant is given by (see Butkov [12, p. 339])
Remark 3.2. There is at least another particular case of interest for which we can obtain an explicit expression (when ). Indeed, if we choose (i.e., ) and (which corresponds to ), with , we find that the partial differential equation that we must solve is
This equation is separable; the two ordinary differential equations that result from the separation of variables are
Writing with , (3.38) becomes which we can solve to find Moreover, (3.39) is again an Euler-Cauchy equation; the solution that satisfies the boundary condition = 0 is (for positive eigenvalues ) Hence, proceeding as above, we can obtain the function , expressed as a generalized Fourier series, in this case too.
4. Concluding Remarks
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 2 and 3. Furthermore, we have arbitrarily chosen in Section 2 to compute the probability of hitting the larger circle first. It would be a simple matter to obtain the probability of hitting the smaller circle first instead. Actually, because the continuation region is bounded, the probability of hitting the smaller circle first should simply be , at least in the cases treated here. Similarly, in Section 3 we could have computed the probability that the process will exit the continuation region through the radius = 0, or through .
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, and , had the same infinitesimal parameters; it would be interesting to try to find the solutions to the first hitting place problems in the general cases.
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 will hit a given part of the boundary of the stopping region first. Another problem would be to try to obtain the distribution of .
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Copyright © 2011 Mario Lefebvre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.