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 equations𝑑𝑋𝑖(𝑡)=𝑓𝑖𝑋𝑖𝑣(𝑡)𝑑𝑡+𝑖𝑋𝑖(𝑡)1/2𝑑𝑊𝑖(𝑡),(1.1) for 𝑖=1,2, where 𝑣𝑖(⋅) 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 [ğ‘Ž,𝑏]. Let 𝜏(𝑥)∶=inf𝑡≥0∶𝑋1(𝑡)=ğ‘Žor𝑏∣𝑋1[](0)=ğ‘¥âˆˆğ‘Ž,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) 𝑒𝐿(𝑥;𝑠)∶=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)2ğ¿î…žî…ž(𝑥;𝑠)+𝑓1(𝑥)ğ¿î…ž(𝑥;𝑠)=𝑠𝐿(𝑥;𝑠),(1.4) and it is subject to the boundary conditions 𝐿(ğ‘Ž;𝑠)=𝐿(𝑏;𝑠)=1.(1.5)

Next, let 𝑝𝑋(𝑥)∶=𝑃1(𝜏(𝑥))=ğ‘Žâˆ£ğ‘‹1(0)=𝑥.(1.6) The function 𝑝(𝑥) satisfies the ordinary differential equation (see Cox and Miller [1, p. 231]) 𝑣1(𝑥)2ğ‘î…žî…ž(𝑥)+𝑓1(𝑥)ğ‘î…ž(𝑥)=0,(1.7) with 𝑝(ğ‘Ž)=1,𝑝(𝑏)=0.(1.8) It is therefore a simple matter to compute explicitly the probability 𝑝(𝑥) of hitting the point ğ‘Ž before 𝑏, starting from 𝑥∈[ğ‘Ž,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)=ğ‘âˆ’ğ‘¥ğ‘âˆ’ğ‘Žforğ‘Žâ‰¤ğ‘¥â‰¤ğ‘.(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𝑋=inf𝑡≥0∶1(𝑡),𝑋2(𝑡)∈𝐷∣𝑋𝑖(0)=𝑥𝑖,(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𝑒;𝑠∶=𝐸−𝑠𝑇(𝑥1,𝑥2)(1.11) satisfies the Kolmogorov backward equation2𝑖=1𝑣𝑖𝑥𝑖2𝑀𝑥𝑖𝑥𝑖+𝑓𝑖𝑥𝑖𝑀𝑥𝑖=𝑠𝑀,(1.12) where 𝑀𝑥𝑖∶=𝜕𝑀/𝜕𝑥𝑖 and 𝑀𝑥𝑖𝑥𝑖∶=𝜕2𝑀/𝜕𝑥2𝑖. This partial differential equation is valid in the continuation region 𝐶∶=𝐷𝑐 and is subject to the boundary condition 𝑀𝑥1,𝑥2𝑥;𝑠=1if1,𝑥2∈𝜕𝐷.(1.13)

In Section 2, the set 𝐶 will be given by𝐶1𝑥∶=1,𝑥2∈ℝ2∶𝑑21<𝑥21+𝑥22<𝑑22,(1.14) and the function𝜋𝑥1,𝑥2𝑋∶=𝑃21𝑇1𝑥1,𝑥2+𝑋22𝑇1𝑥1,𝑥2=𝑑22,(1.15) where 𝑇1 is the random variable defined in (1.10) with 𝐷=𝐷1=𝐶𝑐1, 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𝑥∶=1,𝑥2∈ℝ2𝑥∶0<21+𝑥221/2𝑥<𝑑,0<arctan2𝑥1<𝜃0.(1.16) We will calculate for important two-dimensional diffusion processes the probability𝜈𝑥1,𝑥2𝑋∶=𝑃21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 𝑋𝑖(0)=𝑥𝑖 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 equation2𝑖=1𝑣𝑖𝑥𝑖2𝜋𝑥𝑖𝑥𝑖+𝑓𝑖𝑥𝑖𝜋𝑥𝑖=0(2.1) in the set 𝐶1 defined in (1.14), and is subject to the boundary conditions𝜋𝑥1,𝑥2=⎧⎪⎨⎪⎩1if𝑥21+𝑥22=𝑑22,0if𝑥21+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=ğ‘ž(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) becomes2𝑖=12𝑣𝑖𝑥𝑖𝑥2ğ‘–ğ‘žî…žî…žî€ºğ‘£(𝑦)+𝑖𝑥𝑖+2ğ‘¥ğ‘–ğ‘“ğ‘–î€·ğ‘¥ğ‘–ğ‘žî€¸î€»î…žî€¾(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)=ğ‘ž(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 𝑓𝑖(𝑥𝑖)≡0 and 𝑣𝑖(𝑥𝑖)≡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 ğ‘¦ğ‘žî…žî…ž(𝑦)+ğ‘žî…ž(𝑦)=0.(2.5) Notice that this is a first-order linear ordinary differential equation for ℎ(𝑦)∶=ğ‘žî…ž(𝑦). It is a simple matter to find that ğ‘ž(𝑦)=𝑐1ln(𝑦)+𝑐0,(2.6)

where 𝑐1 and 𝑐0 are constants. Therefore, 𝜋𝑥1,𝑥2=𝑐1𝑥ln21+𝑥22+𝑐0.(2.7) The boundary condition (2.2) yields that 𝜋𝑥1,𝑥2=𝑥ln21+𝑥22/𝑑21𝑑ln22/𝑑21for𝑑21≤𝑥21+𝑥22≤𝑑22.(2.8)

Remark 2.2. If we choose 𝑓𝑖(𝑥𝑖)≡𝑓0≠0 or if 𝑣𝑖(𝑥𝑖)≡𝑣0𝑖>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 𝑓𝑖(𝑥𝑖)=−𝛼𝑥𝑖 and 𝑣𝑖(𝑥𝑖)≡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 𝑣0ğ‘¦ğ‘žî…žî…žî€·ğ‘£(𝑦)+0î€¸ğ‘žâˆ’ğ›¼ğ‘¦î…ž(𝑦)=0,(2.9) the general solution of which can be expressed as ğ‘ž(𝑦)=𝑐1Ei𝛼𝑦𝑣0+𝑐0,(2.10) where Ei(⋅) is the exponential integral function defined by Ei(𝑧)=âˆ’âˆžâˆ’ğ‘§ğ‘’âˆ’ğ‘¡ğ‘¡ğ‘‘ğ‘¡for𝑧>0,(2.11) in which the principal value of the integral is taken. It follows that 𝜋𝑥1,𝑥2=𝛼𝑥Ei21+𝑥22/𝑣0−Ei𝛼𝑑21/𝑣0Ei𝛼𝑑22/𝑣0−Ei𝛼𝑑21/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 𝑓𝑖(𝑥𝑖)=(𝛼−1)/2𝑥𝑖 and 𝑣𝑖(𝑥𝑖)≡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 ğ‘¦ğ‘žî…žî…ž(𝑦)+ğ›¼ğ‘žî…ž(𝑦)=0.(2.13) We find that, for 𝛼≠1, the function ğ‘ž(𝑦) isğ‘ž(𝑦)=𝑐1𝑦1−𝛼+𝑐0.(2.14) Finally, the solution that satisfies the boundary condition (2.2) is 𝜋𝑥1,𝑥2=𝑥21+𝑥221−𝛼−𝑑12(1−𝛼)𝑑22(1−𝛼)−𝑑12(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)) 1𝑠𝑁=2𝑣1𝑥1𝑥21𝑟2𝑁𝑟𝑟𝑥−21𝑥2𝑟3𝑁𝑟𝜃+𝑥22𝑟4𝑁𝜃𝜃+𝑥22𝑟3𝑁𝑟𝑥+21𝑥2𝑟4𝑁𝜃+12𝑣2𝑥2𝑥22𝑟2𝑁𝑟𝑟𝑥+21𝑥2𝑟3𝑁𝑟𝜃+𝑥21𝑟4𝑁𝜃𝜃+𝑥21𝑟3𝑁𝑟𝑥−21𝑥2𝑟4𝑁𝜃+𝑓1𝑥1𝑥1𝑟𝑁𝑟−𝑥2𝑟2𝑁𝜃+𝑓2𝑥2𝑥2𝑟𝑁𝑟+𝑥1𝑟2𝑁𝜃,(3.1) where 𝑟∶=(𝑥21+𝑥22)1/2 and 𝜃∶=arctan(𝑥2/𝑥1). Let 𝑇𝑥1,𝑥2∶=inf𝑡≥0∶𝑟=𝑑or𝜃=0or𝜃0(>0)∣𝑋𝑖(0)=𝑥𝑖,𝑖=1,2,(3.2) that is, 𝑇𝑥1,𝑥2𝑥=inf𝑡≥0∶1,𝑥2∉𝐶2∣𝑋𝑖(0)=𝑥𝑖,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(𝑑,𝜃)=1∀𝜃∈0,𝜃0,𝜌(𝑟,0)=𝜌𝑟,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 𝜌𝑟𝑟+1𝑟𝜌𝑟+1𝑟2𝜌𝜃𝜃=0.(3.5) Looking for a solution of the form 𝜌(𝑟,𝜃)=𝐹(𝑟)𝐺(𝜃), we find that ğ¹î…žî…ž1(𝑟)𝐺(𝜃)+ğ‘Ÿğ¹î…ž1(𝑟)𝐺(𝜃)+𝑟2𝐹(𝑟)ğºî…žî…ž(𝜃)=0,(3.6) so that we obtain the ordinary differential equationsğºî…žî…žğ‘Ÿ(𝜃)=𝜆𝐺(𝜃),(3.7)2ğ¹î…žî…ž(𝑟)+ğ‘Ÿğ¹î…ž(𝑟)+𝜆𝐹(𝑟)=0,(3.8) where 𝜆 is the separation constant. The ordinary differential equation (3.7) is subject to the boundary conditions𝐺𝜃(0)=𝐺0=0,(3.9) whereas 𝐹(0)=0.(3.10) It is well known that the function 𝐺(𝜃) must be of the form 𝐺𝑛(𝜃)=𝑐𝑛sin𝑛𝜋𝜃𝜃0for𝑛=1,2,…,(3.11) where 𝑐𝑛 is a constant; therefore, the separation constant must be given by 𝜆=𝜆𝑛=−(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 𝐹𝑛(𝑟)=const.𝑟𝑛𝜋/𝜃0.(3.13) We then consider the infinite series 𝜌(𝑟,𝜃)=âˆžî“ğ‘›=1ğ‘Žğ‘›î‚µsin𝑛𝜋𝜃𝜃0𝑟𝑛𝜋/𝜃0,(3.14) where ğ‘Žğ‘› is a constant. This series, as a function of 𝜃, is a Fourier series. The condition 𝜌(𝑑,𝜃)=1 implies that ğ‘Žğ‘›=2𝜃0𝜃00𝑑−𝑛𝜋/𝜃0sin𝑛𝜋𝜃𝜃0𝑑𝜃=2𝑑−𝑛𝜋/𝜃0(−1)𝑛+1+1.𝑛𝜋(3.15) Hence, the solution is 𝜌(𝑟,𝜃)=2âˆžî“ğ‘›=1𝑟𝑑𝑛𝜋/𝜃01+(−1)𝑛+1𝑛𝜋sin𝑛𝜋𝜃𝜃0,(3.16) for 0≤𝜃≤𝜃0 and 0≤𝑟≤𝑑.

Remark 3.1. If the infinitesimal mean of 𝑋𝑖(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 −𝛼𝑋𝑖(𝑡) and 𝑣0 for 𝑖=1,2, we must solve the partial differential equation 12𝑣0𝜌𝑟𝑟+1𝑟𝜌𝑟+1𝑟2𝜌𝜃𝜃−𝛼𝑟𝜌𝑟=0.(3.17) Writing 𝜌(𝑟,𝜃)=𝐹(𝑟)𝐺(𝜃), we obtain the ordinary differential equations ğºî…žî…žğ‘Ÿ(𝜃)=𝜆𝐺(𝜃),(3.18)2ğ¹î…žî…ž(𝑟)+ğ‘Ÿğ¹î…žğ›¼(𝑟)−2𝑣0𝑟3ğ¹î…ž(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=𝜆𝑛=−(𝜋𝑛)2/𝜃20 and 𝐺𝑛(𝜃)=𝑐𝑛sin𝑛𝜋𝜃𝜃0forn=1,2,…(3.20)

Next, the general solution of (3.19) can be written as 𝐹(𝑟)=𝑐1𝑟√−2−𝜆𝑀−12√√−𝜆,1−1−𝜆,−2𝑘𝑟2+𝑐2𝑟2√−𝜆𝑀12√√−𝜆,1+1−𝜆,−2𝑘𝑟2,(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)=âˆžî“ğ‘›=1ğ‘Žğ‘›î‚µsin𝑛𝜋𝜃𝜃0𝑟2𝑛𝜋/𝜃0𝑀𝑛𝜋2𝜃0,1+𝑛𝜋𝜃01,−2𝑘𝑟2.(3.22) Making use of the boundary condition 𝜌(𝑑,𝜃) = 1, we find that 𝜌(𝑟,𝜃)=âˆžî“ğ‘›=11+(−1)𝑛+1𝑛𝜋sin𝑛𝜋𝜃𝜃0𝑟𝑑2𝑛𝜋/𝜃0𝑀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 𝑓𝑖(𝑥𝑖)=(𝛼−1)/2𝑥𝑖(0<𝛼<2,𝛼≠1) and 𝑣𝑖(𝑥𝑖)≡1 for 𝑖=1,2, we obtain the partial differential equation 12𝜌𝑟𝑟+1𝑟𝜌𝑟+1𝑟2𝜌𝜃𝜃+𝛼−122𝑟𝜌𝑟+1𝑟2cos𝜃−sin𝜃sin𝜃𝜌cos𝜃𝜃=0.(3.24) It follows that we must solve the ordinary differential equationğºî…žî…žî‚€(𝜃)+(𝛼−1)cos𝜃−sin𝜃sin𝜃𝐺cosğœƒî…ž(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−𝑧2î€¸ğ»î…žî…ž(𝑧)âˆ’ğ‘§ğ»î…ž(𝑧)+(𝛼−1)1−2𝑧2ğ‘§ğ»î…ž(𝑧)+𝜆𝐻(𝑧)=0.(3.26) The general solution of (3.26) can be written in the form 𝐻(𝑧)=𝑧2−𝛼𝑐1𝐹12−𝛾2+𝜆1/22,12+𝛾2+𝜆1/22;32−𝛾2;𝑧2+𝑐2𝐹𝛾2−𝛾2+𝜆1/22,𝛾2+𝛾2+𝜆1/22;12+𝛾2;𝑧2,(3.27) where 𝛾∶=𝛼−1 and 𝐹(ğ‘Ž,𝑏;𝑐;𝑧) is a hypergeometric function (see Abramowitz and Stegun [10, p. 556]). Hence, we have 𝐺(𝜃)=(sin𝜃)2−𝛼𝑐1𝐹121−𝛿,2𝛼+𝛿;2−2;sin2𝜃+𝑐2𝐹𝛼−12−𝛿,𝛼−12𝛼+𝛿;2;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121−𝛿,2𝛼+𝛿;2−2;sin2𝜃0=0.(3.29)

Now, notice that (3.25) can be written in the form 𝑑𝑑𝑑𝜃𝑃(𝜃)𝑑𝜃𝐺(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 𝜆𝑛 for which the conditions 𝐺(𝜃00) = 0 and 𝐺(𝜃0) = 0 are satisfied. These eigenvalues constitute an increasing sequence 𝜆1<𝜆2<⋯<𝜆𝑛<⋯ of real numbers with limğ‘›â†’âˆžğœ†ğ‘›=∞. Moreover, we can also state (see Butkov [12, p. 337–340]) that the eigenfunctions 𝐺𝑛(𝜃) corresponding to the eigenvalues 𝜆𝑛 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 𝜆𝑛 for which 𝐹12−𝛿𝑛,12+𝛿𝑛𝛼;2−2;sin2𝜃0=0,(3.31) where 𝛿𝑛∶=(1/2)[(𝛼−1)2+𝜆𝑛]1/2 for 𝑛=1,2,… We thus have, apart from an arbitrary constant, 𝐺𝑛(𝜃)∶=(sin𝜃)2−𝛼𝐹12−𝛿𝑛,12+𝛿𝑛𝛼;2−2;sin2𝜃(3.32) for 0≤𝜃≤𝜃0.

Finally, we must solve the ordinary differential equation 𝑟2ğ¹î…žî…ž(𝑟)+(2𝛼−1)ğ‘Ÿğ¹î…ž(𝑟)=𝜆𝑛𝐹(𝑟),(3.33) subject to 𝐹(0) = 0. This is an Euler-Cauchy differential equation; for all positive eigenvalues 𝜆𝑛, we can write that 𝐹𝑛(𝑟)=const.𝑟𝑘𝑛,(3.34) where 𝑘𝑛∶=(1−𝛼)+[(𝛼−1)2+𝜆𝑛]1/2. Since there are an infinite number of such eigenvalues, we can consider the infinite series (a generalized Fourier series) 𝜌(𝑟,𝜃)=âˆžî“ğ‘›=1ğ‘Žğ‘›ğ‘Ÿğ‘˜ğ‘›ğºğ‘›(𝜃).(3.35) Making use of the boundary condition 𝜌(𝑑,𝜃)=1, we can write that the constant ğ‘Žğ‘› is given by (see Butkov [12, p. 339]) ğ‘Žğ‘›=𝑑−𝑘𝑛∫𝜃00𝑅(𝜃)𝐺𝑛(𝜃)𝑑𝜃∫𝜃00𝑅(𝜃)𝐺2𝑛(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 𝑣𝑖(𝑥𝑖)≡2, we find that the partial differential equation that we must solve is 2𝜌𝑟𝑟+1𝑟𝜌𝑟+1𝑟2𝜌𝜃𝜃+1𝑟21𝜌cos𝜃sin𝜃𝜃=0.(3.37) This equation is separable; the two ordinary differential equations that result from the separation of variables are ğºî…žî…žî‚€1(𝜃)+𝐺2cos𝜃sinğœƒî…žğ‘Ÿ(𝜃)+𝜆𝐺(𝜃)=0,(3.38)2ğ¹î…žî…ž(𝑟)+𝑟𝐹(𝑟)=𝜆𝐹(𝑟).(3.39)
Writing 𝐺(𝜃)=𝐻(𝑧) with 𝑧∶=sin𝜃, (3.38) becomes1−𝑧2î€¸ğ»î…žî…ž(𝑧)âˆ’ğ‘§ğ»î…ž1(𝑧)+𝐻2ğ‘§î…ž(𝑧)+𝜆𝐻(𝑧)=0,(3.40) which we can solve to find 𝐻(𝑧)=𝑧1/2𝑐1𝐹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 𝜆𝑛) 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)).