We consider a family {Xtx,t≥0,x∈R} of one-dimensional homogeneous diffusion processes defined on a complete filtered probability space {Ω,F,{Ft}t≥0,P} by a stochastic differential equation (1) dXtx=b(Xtx)dt+a(Xtx)dWt,t≥0, with initial condition X0x=x∈R , where {Wt,t≥0} is a standard Ft -Wiener process. If the initial condition is not important, we will denote the process in question by X. Let the coefficients a,b of equation (1) be continuous and satisfy any conditions of the existence of a nonexplosive weak solution on  R . Assume also that a(x)≠0 for x∈R . We further introduce several objects related to the family {Xtx,t≥0,x∈R} .

The generator of a diffusion process X is defined for f∈C2(R) as Lf(x)=a(x)22f″(x)+b(x)f′(x). Define the functions φ(x0,x)=exp{−2∫x0xb(u)a(u)2du},Φ(x0,x)=∫x0xφ(x0,z)dz, x0,x∈R∪{−∞,+∞} . It is easy to see that, for a fixed x0∈R , the function Φ(x0,·) solves a second-order homogeneous differential equation LΦ(x0,·)=0 .

For x,y∈R , let τyx=inf{t≥0,Xtx=y} be the first moment of hitting the point y. For any (a,b)⊂R and x∈(a,b) , let τa,bx=inf{t≥0,Xtx∉(a,b)}=τax∧τbx be the first moment of exiting the interval (a,b) . (We use the convention inf∅=+∞ .)

For any t>0 0$]]> and y∈R , define the local time of the process Xx at the point y on the interval [0,t] by (2) Ltx(y)=a(y)2limε↓012ε∫0tI{|Xsx−y|≤ε}ds. (The factor a2(y) is included to agree with the general Meyer–Tanaka definition of the local time of a semimartingale [7].) The limit in (2) exists almost surely and defines a continuous nondecreasing process {Ltx(y),t≥0} for any x,y∈R . The local time on the whole interval [0,+∞) will be denoted by L∞x(y)=limt→+∞Ltx(y) .

In this article, we focus on the transient diffusion processes, that is, those converging to +∞ or −∞ as t→∞ . We use the explicit distribution of  L∞x to study integral functionals of the form J∞(f)=∫0∞f(Xsx)ds , which can be interpreted as continuous perpetuities in the framework of financial mathematics. We follow the approach of Salminen and Yor [8] to study integral functionals of a Wiener process with positive drift and generalize their results to homogeneous transient diffusion processes. Applying the results of [6], we establish criteria of convergence of almost sure finiteness of the functionals J∞(f) , calculate their moments and potentials, and bound their exponential moments.

In this section, we concentrate on the explicit distribution of L∞x(y) . According to the classical results (see, e.g., [4]), in the case where Φ(x,+∞)=−Φ(x,−∞)=+∞ for some (equivalently, for all) x∈R , the diffusion process X is recurrent, that is, P(lim supt→+∞Xtx=+∞,lim inft→+∞Xtx=−∞)=1. Therefore, L∞x(y)=+∞ for all x,y∈R a.s.

In what follows, we will consider only the case of a transient process X, where at least one of the integrals Φ(x,+∞) and Φ(x,−∞) is finite. We formulate the following statement concerning the distribution of L∞x(y) that can be easily deduced from the results of [3, 1].

Consider two examples where the parameters of the distribution of local time can be calculated explicitly. Example 1.

Let a(x)≡a≠0 and b(x)≡b be constant. Then φ(x,y)=e−2b(y−x)/a2 and Φ(x,y)=a22b(1−e−2b(y−x)/a2)forb≠0;Φ(x,y)=y−xforb=0. In this case, the diffusion process X is transient if and only if b≠0 ; moreover, −Φ(0,−∞)=+∞ and Φ(0,+∞)<+∞ for b>0 0$]]>, and −Φ(0,−∞)<+∞ and Φ(0,+∞)=+∞ for b<0 . The cases are symmetric; therefore, we consider only the case b>0 0$]]>.

Now ψx(0)=ba2 . Thus, for x≤y , the local time L∞x(y) is exponentially distributed with parameter ba2 . For x>y y$]]>, the local time is distributed as κξ , where ξ has an exponential distribution with parameter ba2 , and κ is a Bernoulli random variable independent of ξ and distributed as P(κ=1)=1−P(κ=0)=e−2b(x−y)/a2 . Using the properties of exponential distribution, we see that these cases can be combined: L∞x(y)=d(ξ−2(x−y)+)+ , where a+=a∨0 .

For a measurable function f:R→R such that f and f/a2 are locally integrable, define the integral functional J∞x(f)=∫0∞f(Xsx)ds. We will study the questions of finiteness and existence of moments of J∞x(f) . We start with the well-known occupation density formula (see, e.g., [4, 7]) (6) J∞x(f)=∫Rf(y)a(y)2L∞x(y)dy. If the process Xx is recurrent, then L∞x(y)=∞ a.s. for all y∈R , so J∞x(f) is undefined unless f is identically zero. Therefore, we will require that the process X is transient. We recall that this holds iff Φ(0,+∞) of Φ(0,−∞) is finite. Moreover, if Φ(0,+∞)<+∞ and Φ(0,−∞)=−∞ , then Xsx→+∞ a.s.; if Φ(0,+∞)=+∞ and Φ(0,−∞)>−∞ -\infty $]]>, then Xsx→−∞ a.s.; if Φ(0,+∞)<+∞ and Φ(0,−∞)>−∞ -\infty $]]>, then Xsx→+∞ on a set A+ of positive probability, and Xsx→−∞ on a set A−=Ω∖A+ of positive probability.

We start with a criterion of almost sure finiteness of J∞x(f) . It was obtained in [5] in the case where only one of the integrals Φ(0,+∞) and Φ(0,−∞) is finite; a complete analysis was made in [6]. Define I1(f)=∫0+∞|f(y)|a(y)2Φ(y,+∞)dy,I2(f)=∫−∞0|f(y)|a(y)2Φ(y,−∞)dy.

In what follows, we consider the case where Φ(0,+∞)<+∞ and Φ(0,−∞)=−∞ , the other cases being similar. The next result is a direct consequence of Proposition 1.

We further derive conditions for E[J∞x(f)] to be finite.

Now we continue with the moments of J∞(f) of higher order.

We conclude with the existence of potential and exponential moments. Some related results were obtained in [5].

The following result is an immediate corollary of Theorem 2.