Similarity Solutions of Partial Differential Equations in Probability

Abstract

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.

1. Introduction

Let (𝑋₁(𝑡),𝑋₂(𝑡)) be the two-dimensional diffusion process defined by the stochastic differential equations𝑑𝑋_𝑖(𝑡)=𝑓_𝑖𝑋_𝑖(𝑡)𝑑𝑡+𝑣_𝑖𝑋_𝑖(𝑡)^1/2𝑑𝑊_𝑖(𝑡),(1.1) for 𝑖=1,2, where 𝑣_𝑖(⋅) is nonnegative and 𝑊₁(𝑡) and 𝑊₂(𝑡) are independent standard Brownian motions. In this note, the problem of computing the probability that the process (𝑋₁(𝑡),𝑋₂(𝑡)), 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 (𝑋₁(𝑡),𝑋₂(𝑡)) 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 𝑋₁(𝑡) in the interval [𝑎,𝑏]. Let 𝜏(𝑥)∶=inf𝑡≥0∶𝑋₁(𝑡)=𝑎or𝑏∣𝑋₁(0)=𝑥∈[𝑎,𝑏].(1.2)

Then, it is well known (see Cox and Miller [1, p. 230], for instance) that the moment generating function (which is a Laplace transform) 𝐿(𝑥;𝑠)∶=𝐸𝑒^{−𝑠𝜏(𝑥)}(1.3) of the first passage time 𝜏(𝑥), where 𝑠 is a nonnegative parameter, satisfies the Kolmogorov backward equation 𝑣₁(𝑥)2𝐿^(𝑥;𝑠)+𝑓₁(𝑥)𝐿^(𝑥;𝑠)=𝑠𝐿(𝑥;𝑠),(1.4) and it is subject to the boundary conditions 𝐿(𝑎;𝑠)=𝐿(𝑏;𝑠)=1.(1.5)

Next, let 𝑝(𝑥)∶=𝑃𝑋₁(𝜏(𝑥))=𝑎∣𝑋₁(0)=𝑥.(1.6) The function 𝑝(𝑥) satisfies the ordinary differential equation (see Cox and Miller [1, p. 231]) 𝑣₁(𝑥)2𝑝^(𝑥)+𝑓₁(𝑥)𝑝^(𝑥)=0,(1.7) with 𝑝(𝑎)=1,𝑝(𝑏)=0.(1.8) It is therefore a simple matter to compute explicitly the probability 𝑝(𝑥) of hitting the point 𝑎 before 𝑏, starting from 𝑥∈[𝑎,𝑏]. In particular, in the case when 𝑋₁(𝑡) is a standard Brownian motion, so that 𝑓₁(𝑥)≡0 and 𝑣₁(𝑥)≡1, we find at once that 𝑝(𝑥)=𝑏−𝑥𝑏−𝑎for𝑎≤𝑥≤𝑏.(1.9)

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 [2] and Spitzer [3], and also Wendel [4]. However, a rather small number of papers have been written on first hitting place problems; see, for instance, the papers by Yin and Wu [5] and by Yin et al. [6]. 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, define𝑇𝑥₁,𝑥₂=inf𝑡≥0∶𝑋₁(𝑡),𝑋₂(𝑡)∈𝐷∣𝑋_𝑖(0)=𝑥_𝑖,(1.10) where 𝐷 is a subset of ℝ² for which the random variable 𝑇(𝑥₁,𝑥₂) is well defined. The moment generating function of 𝑇(𝑥₁,𝑥₂), namely, 𝑀𝑥₁,𝑥₂;𝑠∶=𝐸𝑒^{−𝑠𝑇(𝑥1,𝑥2)}(1.11) satisfies the Kolmogorov backward equation²_𝑖=1𝑣^𝑖𝑥_𝑖2𝑀^{𝑥𝑖𝑥𝑖}+𝑓_𝑖𝑥_𝑖𝑀_𝑥𝑖=𝑠𝑀,(1.12) where 𝑀_𝑥𝑖∶=𝜕𝑀/𝜕𝑥_𝑖 and 𝑀_{𝑥𝑖𝑥𝑖}∶=𝜕²𝑀/𝜕𝑥^2𝑖. This partial differential equation is valid in the continuation region 𝐶∶=𝐷^𝑐 and is subject to the boundary condition 𝑀𝑥₁,𝑥₂;𝑠=1if𝑥₁,𝑥₂∈𝜕𝐷.(1.13)

In Section 2, the set 𝐶 will be given by𝐶₁∶=𝑥₁,𝑥₂∈ℝ²∶𝑑²¹<𝑥²¹+𝑥²²<𝑑²²,(1.14) and the function𝜋𝑥₁,𝑥₂∶=𝑃𝑋²¹𝑇₁𝑥₁,𝑥₂+𝑋²²𝑇₁𝑥₁,𝑥₂=𝑑²²,(1.15) where 𝑇₁ is the random variable defined in (1.10) with 𝐷=𝐷₁=𝐶^𝑐1, will be computed in important special cases, such as when (𝑋₁(𝑡),𝑋₂(𝑡)) is a two-dimensional Wiener process.

In Section 3, we will choose𝐶₂∶=𝑥₁,𝑥₂∈ℝ²∶0<𝑥²¹+𝑥²²^1/2<𝑑,0<arctan𝑥²𝑥₁<𝜃⁰.(1.16) We will calculate for important two-dimensional diffusion processes the probability𝜈𝑥₁,𝑥₂∶=𝑃𝑋²¹𝑇₂𝑥₁,𝑥₂+𝑋²²𝑇₂𝑥₁,𝑥₂=𝑑²,(1.17) where 𝑇₂ is the time taken by (𝑋₁(𝑡),𝑋₂(𝑡)) to leave the set 𝐶₂, starting from 𝑋_𝑖(0)=𝑥_𝑖 for 𝑖=1,2.

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 𝜋(𝑥₁,𝑥₂) defined in (1.15) satisfies the partial differential equation²_𝑖=1𝑣^𝑖𝑥_𝑖2𝜋^{𝑥𝑖𝑥𝑖}+𝑓_𝑖𝑥_𝑖𝜋_𝑥𝑖=0(2.1) in the set 𝐶₁ defined in (1.14), and is subject to the boundary conditions𝜋𝑥₁,𝑥₂=⎧⎪⎨⎪⎩1if𝑥²¹0if𝑥₂₁+𝑥²²=𝑑²²,+𝑥²²=𝑑²¹.(2.2) Because the two-dimensional process (𝑋₁(𝑡),𝑋₂(𝑡)) is considered between two concentric circles, it seems natural to try to find a solution of the form 𝜋𝑥₁,𝑥₂=𝑞(𝑦),(2.3) where 𝑦∶=𝑥²¹+𝑥²². 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²_𝑖=12𝑣_𝑖𝑥_𝑖𝑥^2𝑖𝑞^(𝑦)+𝑣_𝑖𝑥_𝑖+2𝑥_𝑖𝑓_𝑖𝑥_𝑖𝑞^(𝑦)=0.(2.4)

Remark 2.1. Because the region 𝐶₁ is bounded, the solution to the problem (2.1), (2.2) is unique. Therefore, if we can find a solution of the form 𝜋(𝑥₁,𝑥₂)=𝑞(𝑦), then we can state that it is indeed the solution we were looking for.

2.1. The Two-Dimensional Wiener Process

First, we take 𝑓_𝑖(𝑥_𝑖)≡0 and 𝑣_𝑖(𝑥_𝑖)≡𝑣₀>0. Then (𝑋₁(𝑡),𝑋₂(𝑡)) is a two-dimensional Wiener process with zero infinitesimal means and infinitesimal variances both equal to 𝑣₀. Equation(2.4) can be rewritten as 𝑦𝑞^(𝑦)+𝑞^(𝑦)=0.(2.5) Notice that this is a first-order linear ordinary differential equation for ℎ(𝑦)∶=𝑞^(𝑦). It is a simple matter to find that 𝑞(𝑦)=𝑐₁ln(𝑦)+𝑐₀,(2.6)

where 𝑐₁ and 𝑐₀ are constants. Therefore, 𝜋𝑥₁,𝑥₂=𝑐₁ln𝑥²¹+𝑥²²+𝑐₀.(2.7) The boundary condition (2.2) yields that 𝜋𝑥₁,𝑥₂=ln𝑥²¹+𝑥²²/𝑑²¹ln𝑑²²/𝑑²¹for𝑑²¹≤𝑥²¹+𝑥²²≤𝑑²².(2.8)

Remark 2.2. If we choose 𝑓_𝑖(𝑥_𝑖)≡𝑓₀≠0 or if 𝑣_𝑖(𝑥_𝑖)≡𝑣_0𝑖>0 for 𝑖=1,2, with 𝑣₀₁≠𝑣₀₂, 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 𝑣₀.

2.2. The Two-Dimensional Ornstein-Uhlenbeck Process

Next, we choose 𝑓_𝑖(𝑥_𝑖)=−𝛼𝑥_𝑖 and 𝑣_𝑖(𝑥_𝑖)≡𝑣₀ for 𝑖=1,2, where 𝛼 is a positive parameter, so that (𝑋₁(𝑡),𝑋₂(𝑡)) is a two-dimensional Ornstein-Uhlenbeck process with the same infinitesimal parameters. This time, (2.4) becomes 𝑣₀𝑦𝑞^(𝑦)+𝑣₀−𝛼𝑦𝑞^(𝑦)=0,(2.9) the general solution of which can be expressed as 𝑞(𝑦)=𝑐₁Ei𝛼𝑦𝑣₀+𝑐⁰,(2.10) where Ei(⋅) is the exponential integral function defined by Ei(𝑧)=−^∞−𝑧𝑒^−𝑡𝑡𝑑𝑡for𝑧>0,(2.11) in which the principal value of the integral is taken. It follows that 𝜋𝑥₁,𝑥₂=Ei𝛼𝑥²¹+𝑥²²/𝑣₀−Ei𝛼𝑑²¹/𝑣₀Ei𝛼𝑑²²/𝑣₀−Ei𝛼𝑑²¹/𝑣₀for𝑑²¹≤𝑥²¹+𝑥²²≤𝑑²².(2.12)

2.3. The Two-Dimensional Bessel Process

The last particular case that we consider is the one when 𝑓_𝑖(𝑥_𝑖)=(𝛼−1)/2𝑥_𝑖 and 𝑣_𝑖(𝑥_𝑖)≡1 for 𝑖=1,2. Again, 𝛼 is a positive parameter, so that (𝑋₁(𝑡),𝑋₂(𝑡)) is a two-dimensional Bessel process. We assume that 0<𝛼<2 (and 𝛼≠1); then, the origin is a regular boundary for 𝑋₁(𝑡) and 𝑋₂(𝑡) (see Karlin and Taylor [9, p. 238-239]).

Equation (2.4) takes the form 𝑦𝑞^(𝑦)+𝛼𝑞^(𝑦)=0.(2.13) We find that, for 𝛼≠1, the function 𝑞(𝑦) is𝑞(𝑦)=𝑐₁𝑦^1−𝛼+𝑐₀.(2.14) Finally, the solution that satisfies the boundary condition (2.2) is 𝜋𝑥₁,𝑥₂=𝑥²¹+𝑥²²^1−𝛼−𝑑^12(1−𝛼)𝑑^22(1−𝛼)−𝑑^12(1−𝛼)for𝑑²¹≤𝑥²¹+𝑥²²≤𝑑²².(2.15)

Remarks 2.3. (1) When 𝛼=1, the ordinary differential equation (2.14) reduces to the one obtained in Section 2.1 with the two-dimensional Wiener process, if 𝑣₀=1.
(2) If the parameter 𝛼 is greater than or equal to 2, the origin is an inaccessible boundary for 𝑋₁(𝑡) and 𝑋₂(𝑡); 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 𝜈(𝑥₁,𝑥₂) 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 𝑑 centered at the origin. In polar coordinates, the function 𝑀(𝑥₁,𝑥₂;𝑠)=𝑁(𝑟,𝜃;𝑠) satisfies the Kolmogorov backward equation (see (1.12)) 𝑠𝑁=12𝑣¹𝑥₁𝑥𝑟²¹²𝑁_𝑟𝑟−2𝑥¹𝑥₂𝑟³𝑁_𝑟𝜃+𝑥²²𝑟⁴𝑁_𝜃𝜃+𝑥²²𝑟³𝑁_𝑟+2𝑥¹𝑥₂𝑟⁴𝑁_𝜃+12𝑣²𝑥₂𝑥𝑟²²²𝑁_𝑟𝑟+2𝑥¹𝑥₂𝑟³𝑁_𝑟𝜃+𝑥²¹𝑟⁴𝑁_𝜃𝜃+𝑥²¹𝑟³𝑁_𝑟−2𝑥¹𝑥₂𝑟⁴𝑁_𝜃+𝑓₁𝑥₁𝑥𝑟¹𝑁^𝑟−𝑥²𝑟²𝑁_𝜃+𝑓²𝑥₂𝑥²𝑟𝑁^𝑟+𝑥¹𝑟²𝑁_𝜃,(3.1) where 𝑟∶=(𝑥²¹+𝑥²²)^1/2 and 𝜃∶=arctan(𝑥₂/𝑥₁). Let 𝑇𝑥₁,𝑥₂∶=inf𝑡≥0∶𝑟=𝑑or𝜃=0or𝜃₀(>0)∣𝑋_𝑖(0)=𝑥_𝑖,𝑖=1,2,(3.2) that is, 𝑇𝑥₁,𝑥₂=inf𝑡≥0∶𝑥₁,𝑥₂∉𝐶₂∣𝑋_𝑖(0)=𝑥_𝑖,𝑖=1,2,(3.3) with 𝐶₂ defined in (1.16). The probability 𝜈(𝑥₁,𝑥₂) defined in (1.17) satisfies the same partial differential equation as 𝑀(𝑥₁,𝑥₂;𝑠)in polar coordinates, with 𝑠=0. Furthermore, 𝜈(𝑥₁,𝑥₂)=𝜌(𝑟,𝜃) is such that 𝜌(𝑟𝜌,(0𝑑),=𝜃)𝜌=𝑟1,𝜃₀∀𝜃∈0,𝜃₀,=0if𝑟<𝑑.(3.4)

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 (𝑋₁(𝑡),𝑋₂(𝑡)) is a two-dimensional Wiener process, with independent components and infinitesimal parameters 0 and 𝑣₀, the partial differential equation that we must solve reduces to 𝜌_𝑟𝑟+1𝑟𝜌^𝑟+1𝑟²𝜌_𝜃𝜃=0.(3.5) Looking for a solution of the form 𝜌(𝑟,𝜃)=𝐹(𝑟)𝐺(𝜃), we find that 𝐹^(𝑟)𝐺(𝜃)+1𝑟𝐹^(𝑟)𝐺(𝜃)+1𝑟²𝐹(𝑟)𝐺^(𝜃)=0,(3.6) so that we obtain the ordinary differential equations𝑟𝐺^(𝜃)=𝜆𝐺(𝜃),(3.7)₂𝐹^(𝑟)+𝑟𝐹^(𝑟)+𝜆𝐹(𝑟)=0,(3.8) where 𝜆 is the separation constant. The ordinary differential equation (3.7) is subject to the boundary conditions𝐺(0)=𝐺𝜃₀=0,(3.9) whereas 𝐹(0)=0.(3.10) It is well known that the function 𝐺(𝜃) must be of the form 𝐺_𝑛(𝜃)=𝑐_𝑛sin𝑛𝜋𝜃𝜃₀for𝑛=1,2,…,(3.11) where 𝑐_𝑛 is a constant; therefore, the separation constant must be given by 𝜆=𝜆_𝑛=−(𝑛𝜋)²𝜃²⁰for𝑛=1,2,…(3.12)

Next, the solution of (3.8) (which is an Euler-Cauchy equation), with 𝜆=−(𝜋𝑛)²/𝜃²⁰, that is such that 𝐹(0) = 0 is 𝐹_𝑛(𝑟)=const.𝑟^{𝑛𝜋/𝜃0}.(3.13) We then consider the infinite series 𝜌(𝑟,𝜃)=^∞_𝑛=1𝑎_𝑛sin𝑛𝜋𝜃𝜃₀𝑟^{𝑛𝜋/𝜃0},(3.14) where 𝑎_𝑛 is a constant. This series, as a function of 𝜃, is a Fourier series. The condition 𝜌(𝑑,𝜃)=1 implies that 𝑎_𝑛=2𝜃₀^𝜃0₀𝑑^{−𝑛𝜋/𝜃0}sin𝑛𝜋𝜃𝜃₀𝑑𝜃=2𝑑^{−𝑛𝜋/𝜃0}(−1)^𝑛+1+1𝑛𝜋.(3.15) Hence, the solution is 𝜌(𝑟,𝜃)=2^∞_𝑛=1𝑑𝑟^{𝑛𝜋/𝜃0}1+(−1)^𝑛+1𝑛𝜋sin𝑛𝜃𝜋₀𝜃,(3.16) for 0≤𝜃≤𝜃₀ and 0≤𝑟≤𝑑.

Remark 3.1. If the infinitesimal mean of 𝑋_𝑖(𝑡) is not equal to zero, we cannot separate the variables in the partial differential equation satisfied by the function 𝜌(𝑟,𝜃). 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 𝑋₁(𝑡) and 𝑋₂(𝑡) are independent Ornstein-Uhlenbeck processes with infinitesimal parameters −𝛼𝑋_𝑖(𝑡) and 𝑣₀ for 𝑖=1,2, we must solve the partial differential equation 12𝑣⁰𝜌^𝑟𝑟+1𝑟𝜌^𝑟+1𝑟²𝜌_𝜃𝜃−𝛼𝑟𝜌^𝑟=0.(3.17) Writing 𝜌(𝑟,𝜃)=𝐹(𝑟)𝐺(𝜃), we obtain the ordinary differential equations 𝑟𝐺^(𝜃)=𝜆𝐺(𝜃),(3.18)₂𝐹^(𝑟)+𝑟𝐹^(𝑟)−2𝛼𝑣₀𝑟³𝐹^(𝑟)+𝜆𝐹(𝑟)=0.(3.19) The boundary conditions are the same as in Section 3.1. Therefore, we find that we still have 𝜆=𝜆_𝑛=−(𝜋𝑛)²/𝜃²⁰ and 𝐺_𝑛(𝜃)=𝑐_𝑛sin𝑛𝜋𝜃𝜃₀forn=1,2,…(3.20)

Next, the general solution of (3.19) can be written as 𝐹(𝑟)=𝑐₁𝑟^{√−2−𝜆}𝑀−12√−𝜆,1−√−𝜆,−12𝑘𝑟²+𝑐²𝑟^2√−𝜆𝑀12√−𝜆,1+√−𝜆,−12𝑘𝑟²,(3.21) where 𝑘∶=−2𝛼/𝑣₀ and 𝑀(⋅,⋅,⋅) is a confluent hypergeometric function (see Abramowitz and Stegun [10, p. 504]). We find at once that we must choose 𝑐₁ equal to zero. We then consider the infinite series 𝜌(𝑟,𝜃)=^∞_𝑛=1𝑎_𝑛sin𝑛𝜋𝜃𝜃₀𝑟^{2𝑛𝜋/𝜃0}𝑀𝑛𝜋2𝜃₀,1+𝑛𝜋𝜃₀,−12𝑘𝑟².(3.22) Making use of the boundary condition 𝜌(𝑑,𝜃) = 1, we find that 𝜌(𝑟,𝜃)=^∞_𝑛=11+(−1)^𝑛+1𝑛𝜋sin𝑛𝜃𝜋₀𝜃𝑑𝑟^{2𝑛𝜋/𝜃0}𝑀𝑛𝜋/2𝜃₀,1+𝑛𝜋/𝜃₀,−(1/2)𝑘𝑟²𝑀𝑛𝜋/2𝜃₀,1+𝑛𝜋/𝜃₀,−(1/2)𝑘𝑑²,(3.23) for 0≤𝜃≤𝜃₀ and 0≤𝑟≤𝑑.

3.3. The Two-Dimensional Bessel Process

Finally, with 𝑓_𝑖(𝑥_𝑖)=(𝛼−1)/2𝑥_𝑖(0<𝛼<2,𝛼≠1) and 𝑣_𝑖(𝑥_𝑖)≡1 for 𝑖=1,2, we obtain the partial differential equation 12𝜌^𝑟𝑟+1𝑟𝜌^𝑟+1𝑟²𝜌_𝜃𝜃+𝛼−122𝑟𝜌^𝑟+1𝑟²csoins𝜃𝜃−csoins𝜃𝜃𝜌^𝜃=0.(3.24) It follows that we must solve the ordinary differential equation𝐺^(𝜃)+(𝛼−1)cos𝜃sin𝜃−csoins𝜃𝜃𝐺^(𝜃)+𝜆𝐺(𝜃)=0.(3.25) We assume that 𝜃₀ is in the interval (0,𝜋/2). Writing that 𝐺(𝜃)=𝐻(𝑧), where 𝑧∶=sin𝜃, we find that this ordinary differential equation is transformed to1−𝑧²𝐻^(𝑧)−𝑧𝐻^(𝑧)+(𝛼−1)1−𝑧2𝑧²𝐻^(𝑧)+𝜆𝐻(𝑧)=0.(3.26) The general solution of (3.26) can be written in the form 𝐻(𝑧)=𝑧^2−𝛼𝑐₁𝐹12−𝛾²+𝜆^1/22,12+𝛾²+𝜆^1/2+𝑐₂2;32−𝛾2;𝑧²𝐹𝛾2−𝛾²+𝜆^1/22,𝛾2+𝛾²+𝜆^1/22;12+𝛾2;𝑧²,(3.27) where 𝛾∶=𝛼−1 and 𝐹(𝑎,𝑏;𝑐;𝑧) is a hypergeometric function (see Abramowitz and Stegun [10, p. 556]). Hence, we have 𝐺(𝜃)=(sin𝜃)^2−𝛼𝑐₁𝐹12−𝛿,12+𝛿;2−𝛼2;sin²+𝑐₂𝜃𝐹𝛼−12−𝛿,𝛼−21+𝛿;𝛼2;sin²𝜃,(3.28) where 𝛿∶=(1/2)(𝛾²+𝜆)^1/2.

The condition 𝐺(0)=0 implies that we must set 𝑐₂ equal to zero. Next, we must find the value(s) of the separation constant 𝜆 for which 𝐺(𝜃₀) = 0; that is, sin𝜃₀^2−𝛼𝐹12−𝛿,12+𝛿;2−𝛼2;sin²𝜃₀=0.(3.29)

Now, notice that (3.25) can be written in the form 𝑑𝑑𝜃𝑃(𝜃)𝑑𝑑𝜃𝐺(𝜃)−𝑆(𝜃)𝐺(𝜃)+𝜆𝑅(𝜃)𝐺(𝜃)=0,(3.30) with 𝑃(𝜃)=(sin𝜃cos𝜃)^𝛼−1, 𝑆(𝜃)≡0 and 𝑅(𝜃)≡𝑃(𝜃). If we assume that 0<𝜃₀₀<𝜃<𝜃₀<𝜋/2 in 𝐶₂, then the problem of solving (3.25) together with the boundary conditions 𝐺(𝜃₀₀)=𝐺(𝜃₀) = 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 𝜆_𝑛 for which the conditions 𝐺(𝜃₀₀) = 0 and 𝐺(𝜃₀) = 0 are satisfied. These eigenvalues constitute an increasing sequence 𝜆₁<𝜆₂<⋯<𝜆_𝑛<⋯ of real numbers with lim_𝑛→∞𝜆_𝑛=∞. Moreover, we can also state (see Butkov [12, p. 337–340]) that the eigenfunctions 𝐺_𝑛(𝜃) corresponding to the eigenvalues 𝜆_𝑛 are orthogonal to each other with respect to the weight function 𝑅(𝜃). However, computing these eigenvalues explicitly is another problem.

Here, we consider the case where 𝜃₀₀=0, 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 𝜆_𝑛 for which 𝐹12−𝛿^𝑛,12+𝛿^𝑛;2−𝛼2;sin²𝜃₀=0,(3.31) where 𝛿_𝑛∶=(1/2)[(𝛼−1)²+𝜆_𝑛]^1/2 for 𝑛=1,2,… We thus have, apart from an arbitrary constant, 𝐺_𝑛(𝜃)∶=(sin𝜃)^2−𝛼𝐹12−𝛿^𝑛,12+𝛿^𝑛;2−𝛼2;sin²𝜃(3.32) for 0≤𝜃≤𝜃₀.

Finally, we must solve the ordinary differential equation 𝑟²𝐹^(𝑟)+(2𝛼−1)𝑟𝐹^(𝑟)=𝜆_𝑛𝐹(𝑟),(3.33) subject to 𝐹(0) = 0. This is an Euler-Cauchy differential equation; for all positive eigenvalues 𝜆_𝑛, we can write that 𝐹_𝑛(𝑟)=const.𝑟^𝑘𝑛,(3.34) where 𝑘_𝑛∶=(1−𝛼)+[(𝛼−1)²+𝜆_𝑛]^1/2. Since there are an infinite number of such eigenvalues, we can consider the infinite series (a generalized Fourier series) 𝜌(𝑟,𝜃)=^∞_𝑛=1𝑎_𝑛𝑟^𝑘𝑛𝐺_𝑛(𝜃).(3.35) Making use of the boundary condition 𝜌(𝑑,𝜃)=1, we can write that the constant 𝑎_𝑛 is given by (see Butkov [12, p. 339]) 𝑎_𝑛=𝑑^−𝑘𝑛∫^𝜃0₀𝑅(𝜃)𝐺_𝑛(𝜃)𝑑𝜃∫^𝜃0₀𝑅(𝜃)𝐺^2𝑛(𝜃)𝑑𝜃.(3.36)

Remark 3.2. There is at least another particular case of interest for which we can obtain an explicit expression (when 0<𝜃₀₀<𝜃<𝜃₀<𝜋/2). Indeed, if we choose 𝑓₁(𝑥₁)=−1/2𝑥₁ (i.e., 𝛼=0) and 𝑓₂(𝑥₂)=1/2𝑥₂ (which corresponds to 𝛼=2), with 𝑣_𝑖(𝑥_𝑖)≡2, we find that the partial differential equation that we must solve is 2𝜌_𝑟𝑟+1𝑟𝜌^𝑟+1𝑟²𝜌_𝜃𝜃+1𝑟²cos𝜃1sin𝜃𝜌^𝜃=0.(3.37) This equation is separable; the two ordinary differential equations that result from the separation of variables are 𝐺^(𝜃)+12cos𝜃sin𝜃𝐺^𝑟(𝜃)+𝜆𝐺(𝜃)=0,(3.38)₂𝐹^(𝑟)+𝑟𝐹(𝑟)=𝜆𝐹(𝑟).(3.39)
Writing 𝐺(𝜃)=𝐻(𝑧) with 𝑧∶=sin𝜃, (3.38) becomes1−𝑧²𝐻^(𝑧)−𝑧𝐻^(𝑧)+21𝑧𝐻^(𝑧)+𝜆𝐻(𝑧)=0,(3.40) which we can solve to find 𝐻(𝑧)=𝑧^1/2𝑐₁𝐹14−√𝜆2,14+√𝜆2;54;𝑧²+𝑐²𝐹−√𝜆2,√𝜆2;34;𝑧².(3.41) Moreover, (3.39) is again an Euler-Cauchy equation; the solution that satisfies the boundary condition 𝐹(0) = 0 is (for positive eigenvalues 𝜆_𝑛) 𝐹(𝑟)=const.𝑟^√𝜆𝑛.(3.42) Hence, proceeding as above, we can obtain the function 𝜌(𝑟,𝜃), 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 𝜋(𝑥₁,𝑥₂) 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 1−𝜋(𝑥₁,𝑥₂), at least in the cases treated here. Similarly, in Section 3 we could have computed the probability that the process (𝑋₁(𝑡),𝑋₂(𝑡)) will exit the continuation region through the radius 𝜃 = 0, or through 𝜃=𝜃₀.

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, 𝑋₁(𝑡) and 𝑋₂(𝑡), 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 (𝑋₁(𝑡),𝑋₂(𝑡)) will hit a given part of the boundary of the stopping region first. Another problem would be to try to obtain the distribution of (𝑋₁(𝑇),𝑋₂(𝑇)).