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.


Introduction
Let X 1 t , X 2 t be the two-dimensional diffusion process defined by the stochastic differential equations dX i t f i X i t dt {v i X i t } 1/2 dW i t , 1.1 for i 1, 2, where v i • is nonnegative and W 1 t and W 2 t are independent standard Brownian motions.In this note, the problem of computing the probability that the process X 1 t , X 2 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 X 1 t , X 2 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 X 1 t in the interval a, b .Let Then, it is well known see Cox  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 X 1 t is a standard Brownian motion, so that f 1 x ≡ 0 and v 1 x ≡ 1, 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 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, define where D is a subset of R 2 for which the random variable T x 1 , x 2 is well defined.The moment generating function of T x 1 , x 2 , namely, M x 1 , x 2 ; s : E e −sT x 1 ,x 2 1.11 satisfies the Kolmogorov backward equation where M x i : ∂M/∂x i and M x i x i : ∂ 2 M/∂x 2 i .This partial differential equation is valid in the continuation region C : D c and is subject to the boundary condition In Section 2, the set C will be given by 1.14 and the function where T 1 is the random variable defined in 1.10 with D D 1 C c 1 , will be computed in important special cases, such as when X 1 t , X 2 t is a two-dimensional Wiener process.
In Section 3, we will choose We will calculate for important two-dimensional diffusion processes the probability where T 2 is the time taken by X 1 t , X 2 t to leave the set C 2 , starting from X i 0 Finally, a few remarks will be made in Section 4 to conclude.

First Hitting Place Probabilities when Starting between Two Circles
From the Kolmogorov backward equation 1.12 , we deduce that the function π x 1 , x 2 defined in 1.15 satisfies the partial differential equation in the set C 1 defined in 1.14 , and is subject to the boundary conditions

2.2
Because the two-dimensional process X 1 t , X 2 t is considered between two concentric circles, it seems natural to try to find a solution of the form where y : 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 C 1 is bounded, the solution to the problem 2.1 , 2.2 is unique.Therefore, if we can find a solution of the form π x 1 , x 2 q y , then we can state that it is indeed the solution we were looking for.

The Two-Dimensional Wiener Process
First, we take f i x i ≡ 0 and v i x i ≡ v 0 > 0. Then X 1 t , X 2 t is a two-dimensional Wiener process with zero infinitesimal means and infinitesimal variances both equal to v 0 .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 where c 1 and c 0 are constants.Therefore, The boundary condition 2.2 yields that , with v 01 / v 02 , 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 v 0 .

The Two-Dimensional Ornstein-Uhlenbeck Process
Next, we choose f i x i −αx i and v i x i ≡ v 0 for i 1, 2, where α is a positive parameter, so that X 1 t , X 2 t is a two-dimensional Ornstein-Uhlenbeck process with the same infinitesimal parameters.This time, 2.4 becomes v 0 yq y v 0 − αy q y 0, 2.9 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

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 X 1 t , X 2 t is a two-dimensional Bessel process.We assume that 0 < α < 2 and α / 1 ; then, the origin is a regular boundary for X 1 t and X 2 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 is q y c 1 y 1−α c 0 .

2.14
Finally, the solution that satisfies the boundary condition 2.2 is

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 v 0 1.
2 If the parameter α is greater than or equal to 2, the origin is an inaccessible boundary for X 1 t and X 2 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 x 1 , x 2 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.

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 x 1 , x 2 ; s N r, θ; s satisfies the Kolmogorov backward equation see 1.12 where r : 1/2 and θ : arctan with C 2 defined in 1.16 .The probability ν x 1 , x 2 defined in 1.17 satisfies the same partial differential equation as M x 1 , x 2 ; s in polar coordinates, with s 0. Furthermore, ν x 1 , x 2 ρ r, θ is such that

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.

The Two-Dimensional Wiener Process
When X 1 t , X 2 t is a two-dimensional Wiener process, with independent components and infinitesimal parameters 0 and v 0 , the partial differential equation that we must solve reduces to Looking for a solution of the form ρ r, θ F r G θ , we find that where c n is a constant; therefore, the separation constant must be given by for n 1, 2, . . .

3.13
We then consider the infinite series 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 Hence, the solution is for 0 ≤ θ ≤ θ 0 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.

The Two-Dimensional Ornstein-Uhlenbeck Process
When X 1 t and X 2 t are independent Ornstein-Uhlenbeck processes with infinitesimal parameters −αX i t and v 0 for i 1, 2, we must solve the partial differential equation

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 /θ 2 0 and G n θ c n sin nπθ θ 0 for n 1, 2, . . .

3.20
Next, the general solution of 3.19 can be written as

3.21
where k : −2α/v 0 and M •, •, • is a confluent hypergeometric function see Abramowitz and Stegun 10, p. 504 .We find at once that we must choose c 1 equal to zero.We then consider the infinite series ρ r, θ

3.22
Making use of the boundary condition ρ d, θ 1, we find that ρ r, θ

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

3.25
We assume that θ 0 is in the interval 0, π/2 .Writing that G θ H z , where z : sin θ, we find that this ordinary differential equation is transformed to The general solution of 3.26 can be written in the form

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 The condition G 0 0 implies that we must set c 2 equal to zero.Next, we must find the value s of the separation constant λ for which G θ 0 0; that is, Now, notice that 3.25 can be written in the form with P θ sin θ cos θ α−1 , S θ ≡ 0 and R θ ≡ P θ .If we assume that 0 < θ 00 < θ < θ 0 < π/2 in C 2 , then the problem of solving 3.25 together with the boundary conditions G θ 00 G θ 0 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 θ 00 0 and G θ 0 0 are satisfied.These eigenvalues constitute an increasing sequence λ 1 < λ 2 < • • • < 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 θ 00 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

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 x 1 , x 2 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 x 1 , x 2 , at least in the cases treated here.Similarly, in Section 3 we could have computed the probability that the process X 1 t , X 2 t will exit the continuation region through the radius θ 0, or through θ θ 0 .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, X 1 t and X 2 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 X 1 t , X 2 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 X 1 T , X 2 T .
and Miller 1, p. 230 , for instance that the moment generating function which is a Laplace transform p x : P X 1 τ x a | X 1 0 x .1.6Thefunction p x satisfies the ordinary differential equation see Cox and Miller 1, p. 231 .e., α 0 and f 2 x 2 1/2x 2 which corresponds to α 2 , with v i x i ≡ 2, we find that the partial differential equation that we must solve is Remark 3.2.There is at least another particular case of interest for which we can obtain an explicit expression when 0 < θ 00 < θ < θ 0 < π/2 .Indeed, if we choose f 1 x 1 −1/2x 1 i