JPSJournal of Probability and Statistics1687-95381687-952XHindawi Publishing Corporation68942710.1155/2011/689427689427Research ArticleSimilarity Solutions of Partial Differential Equations in ProbabilityLefebvreMarioChowShein-chungDépartement de Mathématiques et de Génie IndustrielÉcole Polytechnique de MontréalC.P. 6079 Succursale Centre-VilleMontréal, QCH3C 3A7Canadapolymtl.ca2011982011201102052011060620112011Copyright © 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.

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(t)]dt+{vi[Xi(t)]}1/2dWi(t), 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 [a,b]. Let τ(x):=inf{t0:X1(t)=a  or  bX1(0)=x[a,b]}.

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)] 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), and it is subject to the boundary conditions L(a;s)=L(b;s)=1.

Next, let p(x):=P[X1(τ(x))=aX1(0)=x]. 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, with p(a)=1,p(b)=0. 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-afor  axb.

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, defineT(x1,x2)=inf{t0:(X1(t),X2(t))DXi(0)=xi}, where D is a subset of 2 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)] satisfies the Kolmogorov backward equationi=12{vi(xi)2Mxixi+fi(xi)Mxi}=sM, where Mxi:=M/xi and Mxixi:=2M/xi2. This partial differential equation is valid in the continuation region C:=Dc and is subject to the boundary condition M(x1,x2;s)=1if  (x1,x2)D.

In Section 2, the set C will be given byC1:={(x1,x2)R2:d12<x12+x22<d22}, and the functionπ(x1,x2):=P[X12(T1(x1,x2))+X22(T1(x1,x2))=d22], where T1 is the random variable defined in (1.10) with D=D1=C1c, 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<(x12+x22)1/2<d,0<arctan(x2x1)<θ0}. We will calculate for important two-dimensional diffusion processes the probabilityν(x1,x2):=P[X12(T2(x1,x2))+X22(T2(x1,x2))=d2], 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 equationi=12{vi(xi)2πxixi+fi(xi)πxi}=0 in the set C1 defined in (1.14), and is subject to the boundary conditionsπ(x1,x2)={1if  x12+x22=d22,0if  x12+x22=d12. 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), where y:=x12+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) becomesi=12{2vi(xi)xi2q(y)+[vi(xi)+2xifi(xi)]q(y)}=0.

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 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 yq(y)+q(y)=0. 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,

where c1 and c0 are constants. Therefore, π(x1,x2)=c1ln(x12+x22)+c0. The boundary condition (2.2) yields that π(x1,x2)=ln((x12+x22)/d12)ln(d22/d12)for  d12x12+x22d22.

Remark 2.2.

If we choose fi(xi)f00 or if vi(xi)v0i>0 for i=1,2, with v01v02, 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 v0yq(y)+(v0-αy)q(y)=0, the general solution of which can be expressed as q(y)=c1Ei(αyv0)+c0, where Ei(·) is the exponential integral function defined by Ei(z)=--ze-ttdtfor  z>0, in which the principal value of the integral is taken. It follows that π(x1,x2)=Ei(α(x12+x22)/v0)-Ei(αd12/v0)Ei(αd22/v0)-Ei(αd12/v0)for  d12x12+x22d22.

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 α1); 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. We find that, for α1, the function q(y) isq(y)=c1y1-α+c0. Finally, the solution that satisfies the boundary condition (2.2) is π(x1,x2)=(x12+x22)1-α-d12(1-α)d22(1-α)-d12(1-α)for  d12x12+x22d22.

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){x12r2Nrr-2x1x2r3Nrθ+x22r4Nθθ+x22r3Nr+2x1x2r4Nθ}+12v2(x2){x22r2Nrr+2x1x2r3Nrθ+x12r4Nθθ+x12r3Nr-2x1x2r4Nθ}+f1(x1){x1rNr-x2r2Nθ}+f2(x2){x2rNr+x1r2Nθ}, where r:=(x12+x22)1/2 and θ:=arctan(x2/x1). Let T(x1,x2):=inf{t0:r=d  or  θ=0  or  θ0  (>0)Xi(0)=xi,i=1,2}, that is, T(x1,x2)=inf{t0:(x1,x2)C2Xi(0)=xi,i=1,2}, 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)=1θ[0,θ0],ρ(r,0)=ρ(r,θ0)=0if  r<d.

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. 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, so that we obtain the ordinary differential equationsG(θ)=λG(θ),  r2F(r)+rF(r)+λF(r)=0, where λ is the separation constant. The ordinary differential equation (3.7) is subject to the boundary conditionsG(0)=G(θ0)=0, whereas F(0)=0. It is well known that the function G(θ) must be of the form Gn(θ)=cnsin(nπθθ0)for  n=1,2,, where cn is a constant; therefore, the separation constant must be given by λ=λn=-(nπ)2θ02for  n=1,2,

Next, the solution of (3.8) (which is an Euler-Cauchy equation), with λ=-(πn)2/θ02, that is such that F(0) = 0 is Fn(r)=const.  rnπ/θ0. We then consider the infinite series ρ(r,θ)=n=1ansin(nπθθ0)rnπ/θ0, where an is a constant. This series, as a function of θ, is a Fourier series.   The condition ρ(d,θ)=1 implies that an=2θ00θ0d-nπ/θ0sin(nπθθ0)dθ=2d-nπ/θ0(-1)n+1+1nπ. Hence, the solution is ρ(r,θ)=2n=1(rd)nπ/θ01+(-1)n+1nπsin(nπθθ0), for 0θθ0 and 0rd.

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. Writing ρ(r,θ)=F(r)G(θ), we obtain the ordinary differential equations G(θ)=λG(θ),r2F(r)+rF(r)-2αv0r3F(r)+λF(r)=0. The boundary conditions are the same as in Section 3.1. Therefore, we find that we still have λ=λn=-(πn)2/θ02 and Gn(θ)=cnsin(nπθθ0)for  n=1,2,

Next, the general solution of (3.19) can be written as F(r)=c1r-2-λM(-12-λ,1--λ,-12kr2)+c2r2-λM(12-λ,1+-λ,-12kr2),           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=1ansin(nπθθ0)r2nπ/θ0M(nπ2θ0,1+nπθ0,-12kr2). Making use of the boundary condition ρ(d,θ) = 1, we find that ρ(r,θ)=n=11+(-1)n+1nπsin(nπθθ0)(rd)2nπ/θ0M(nπ/2θ0,1+nπ/θ0,-(1/2)kr2)M(nπ/2θ0,1+nπ/θ0,-(1/2)kd2), for 0θθ0 and 0rd.

3.3. The Two-Dimensional Bessel Process

Finally, with fi(xi)=(α-1)/2xi  (0<α<2,α1) and vi(xi)1 for i=1,2, we obtain the partial differential equation 12{ρrr+1rρr+1r2ρθθ}+α-12{2rρr+1r2(cosθsinθ-sinθcosθ)ρθ}=0. It follows that we must solve the ordinary differential equationG(θ)+(α-1)(cosθsinθ-sinθcosθ)G(θ)+λG(θ)=0. 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)1-2z2zH(z)+λH(z)=0. The general solution of (3.26) can be written in the form H(z)=z2-αc1F(12-(γ2+λ)1/22,12+(γ2+λ)1/22;32-γ2;z2)+c2F(γ2-(γ2+λ)1/22,γ2+(γ2+λ)1/22;12+γ2;z2), 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;sin2θ)+c2F(α-12-δ,α-12+δ;α2;sin2θ),   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.

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, 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 limnλn=. 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+δn;2-α2;sin2θ0)=0, 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+δn;2-α2;sin2θ) for 0θθ0.

Finally, we must solve the ordinary differential equation r2F(r)+(2α-1)rF(r)=λnF(r), 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, 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)=n=1anrknGn(θ). Making use of the boundary condition ρ(d,θ)=1, we can write that the constant an is given by (see Butkov [12, p. 339]) an=d-kn0θ0R(θ)Gn(θ)dθ0θ0R(θ)Gn2(θ)  dθ.

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(1cosθsinθ)ρθ=0. This equation is separable; the two ordinary differential equations that result from the separation of variables are G(θ)+(12cosθsinθ)G(θ)+λG(θ)=0,r2F(r)+rF(r)=λF(r).

Writing G(θ)=H(z) with z:=sinθ, (3.38) becomes(1-z2)H(z)-zH(z)+12zH(z)+λH(z)=0, which we can solve to find H(z)=z1/2c1F(14-λ2,14+λ2;54;z2)+c2F(-λ2,λ2;34;z2). 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.  rλn. 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)).

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