Modern Stochastics: Theory and Applications

We consider a family $\{{X_{t}^{x}},t\ge 0,x\in \mathbb{R}\}$ of one-dimensional homogeneous diffusion processes defined on a complete filtered probability space $\{\varOmega ,\mathcal{F},\{\mathcal{F}_{t}\}_{t\ge 0},\mathsf{P}\}$ by a stochastic differential equation with initial condition ${X_{0}^{x}}=x\in \mathbb{R}$, where $\{W_{t},t\ge 0\}$ is a standard $\mathcal{F}_{t}$-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 $\mathbb{R}$. Assume also that $a(x)\ne 0$ for $x\in \mathbb{R}$. We further introduce several objects related to the family $\{{X_{t}^{x}},t\ge 0,x\in \mathbb{R}\}$.

The generator of a diffusion process X is defined for $f\in {C}^{2}(\mathbb{R})$ as Define the functions $x_{0},x\in \mathbb{R}\cup \{-\infty ,+\infty \}$. It is easy to see that, for a fixed $x_{0}\in \mathbb{R}$, the function $\varPhi (x_{0},\cdot )$ solves a second-order homogeneous differential equation $\mathcal{L}\varPhi (x_{0},\cdot )=0$.

For $x,y\in \mathbb{R}$, let ${\tau _{y}^{x}}=\inf \{t\ge 0,{X_{t}^{x}}=y\}$ be the first moment of hitting the point y. For any $(a,b)\subset \mathbb{R}$ and $x\in (a,b)$, let ${\tau _{a,b}^{x}}=\inf \{t\ge 0,{X_{t}^{x}}\notin (a,b)\}={\tau _{a}^{x}}\wedge {\tau _{b}^{x}}$ be the first moment of exiting the interval $(a,b)$. (We use the convention $\inf \varnothing =+\infty $.)

For any $t>0$ and $y\in \mathbb{R}$, define the local time of the process ${X}^{x}$ at the point y on the interval $[0,t]$ by (The factor ${a}^{2}(y)$ 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 $\{{L_{t}^{x}}(y),t\ge 0\}$ for any $x,y\in \mathbb{R}$. The local time on the whole interval $[0,+\infty )$ will be denoted by ${L_{\infty }^{x}}(y)=\lim _{t\to +\infty }{L_{t}^{x}}(y)$.

In this article, we focus on the transient diffusion processes, that is, those converging to $+\infty $ or $-\infty $ as $t\to \infty $. We use the explicit distribution of ${L_{\infty }^{x}}$ to study integral functionals of the form $J_{\infty }(f)={\int _{0}^{\infty }}f({X_{s}^{x}})ds$, 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 $J_{\infty }(f)$, calculate their moments and potentials, and bound their exponential moments.

In this section, we concentrate on the explicit distribution of ${L_{\infty }^{x}}(y)$. According to the classical results (see, e.g., [4]), in the case where $\varPhi (x,+\infty )=-\varPhi (x,-\infty )=+\infty $ for some (equivalently, for all) $x\in \mathbb{R}$, the diffusion process X is recurrent, that is, Therefore, ${L_{\infty }^{x}}(y)=+\infty $ for all $x,y\in \mathbb{R}$ a.s.

In what follows, we will consider only the case of a transient process X, where at least one of the integrals $\varPhi (x,+\infty )$ and $\varPhi (x,-\infty )$ is finite. We formulate the following statement concerning the distribution of ${L_{\infty }^{x}}(y)$ 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.

For a measurable function $f:\mathbb{R}\to \mathbb{R}$ such that f and $f/{a}^{2}$ are locally integrable, define the integral functional We will study the questions of finiteness and existence of moments of ${J_{\infty }^{x}}(f)$. We start with the well-known occupation density formula (see, e.g., [4, 7]) If the process ${X}^{x}$ is recurrent, then ${L_{\infty }^{x}}(y)=\infty $ a.s. for all $y\in \mathbb{R}$, so ${J_{\infty }^{x}}(f)$ is undefined unless f is identically zero. Therefore, we will require that the process X is transient. We recall that this holds iff $\varPhi (0,+\infty )$ of $\varPhi (0,-\infty )$ is finite. Moreover, if $\varPhi (0,+\infty )<+\infty $ and $\varPhi (0,-\infty )=-\infty $, then ${X_{s}^{x}}\to +\infty $ a.s.; if $\varPhi (0,+\infty )=+\infty $ and $\varPhi (0,-\infty )>-\infty $, then ${X_{s}^{x}}\to -\infty $ a.s.; if $\varPhi (0,+\infty )<+\infty $ and $\varPhi (0,-\infty )>-\infty $, then ${X_{s}^{x}}\to +\infty $ on a set $A_{+}$ of positive probability, and ${X_{s}^{x}}\to -\infty $ on a set $A_{-}=\varOmega \setminus A_{+}$ of positive probability.

We start with a criterion of almost sure finiteness of ${J_{\infty }^{x}}(f)$. It was obtained in [5] in the case where only one of the integrals $\varPhi (0,+\infty )$ and $\varPhi (0,-\infty )$ is finite; a complete analysis was made in [6]. Define

In what follows, we consider the case where $\varPhi (0,+\infty )<+\infty $ and $\varPhi (0,-\infty )=-\infty $, the other cases being similar. The next result is a direct consequence of Proposition 1.

We further derive conditions for $\mathsf{E}[{J_{\infty }^{x}}(f)]$ to be finite.

Now we continue with the moments of $J_{\infty }(f)$ 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.