A new integral equation for the first passage time density of the Ornstein-Uhlenbeck process

TL;DR

This paper introduces a novel Volterra integral equation for the first passage time density of the Ornstein-Uhlenbeck process, utilizing inverse Laplace transforms of parabolic cylinder functions, applicable to constant and time-dependent thresholds.

Contribution

It derives a new integral equation for the first passage time density using inverse Laplace transforms, extending the Fortet renewal equation and applicable to various thresholds.

Findings

The integral equation is valid for constant and time-dependent thresholds.

The kernel involves a parabolic cylinder function and is regular for q<=-1.

The Fortet renewal equation is a special case of the new integral equation.

Abstract

The Laplace transform of the first passage time density of the Ornstein--Uhlenbeck process for a constant threshold contains a ratio of two parabolic cylinder functions for which no analytical inversion formula is available. Recently derived inverse Laplace transforms for the product of two parabolic cylinder functions together with the convolution theorem of the Laplace transform then allow to derive a new Volterra integral equation for this first passage time density. The kernel of this integral equation contains a parabolic cylinder function and the Fortet renewal equation for the Ornstein-Uhlenbeck process emerges as a special case, namely when the order q of the parabolic cylinder function is set at 0. The integral equation is shown to hold both for constant as well as time dependent thresholds. Moreover, the kernel of the integral equation is regular for q<=-1.

Equations

$$f\left(x,t|x_0\right)=\frac{\partial }{\partial x}P\left\{X\left(t\right)⩽x|X\left(0\right)=x_0\right\},$$

$$f\left(x,t|x_0\right)=\frac{1}{\sqrt{{\sigma }^2\pi \theta \left(1-\exp \left(-2t/\theta \right)\right)}}\exp \left(-\frac{{\left(\left(x-\mu \theta \right)-\left(x_0-\mu \theta \right)\exp \left(-t/\theta \right)\right)}^2}{{\sigma }^2\theta \left(1-\exp \left(-2t/\theta \right)\right)}\right),$$

$$T_S=\underset{t⩾0}{\inf }\left\{t:X\left(t\right)>S|X\left(0\right)=x_0\right\}{x}_0<S,$$

$$g\left(S,t|x_0\right)=\frac{\partial }{\partial t}P\left\{T_S<t\right\}.$$

$$T_{S\left(t\right)}=\underset{t⩾0}{\inf }\left\{t:X\left(t\right)>S\left(t\right)|X\left(0\right)=x_0\right\}{x}_0<S\left(t\right),$$

$$g\left(S\left(t\right),t|x_0\right)=\frac{\partial }{\partial t}P\left\{T_{S\left(t\right)}<t\right\}.$$

$$g_{\lambda }\left(S|x_0\right)=\exp \left(\frac{{\left(x_0-\mu \theta \right)}^2-{\left(S-\mu \theta \right)}^2}{2{\sigma }^2\theta }\right)\frac{D_{-\lambda \theta }\left(\sqrt{\frac{2}{{\sigma }^2\theta }}\left(\mu \theta -x_0\right)\right)}{D_{-\lambda \theta }\left(\sqrt{\frac{2}{{\sigma }^2\theta }}\left(\mu \theta -S\right)\right)}{x}_0<S,$$

$$f\left(x,t|x_0\right)={\int }_0^{t}f\left(x,t|S,\tau \right)g\left(S,\tau |x_0\right)d\tau x⩾S,$$

$$f\left(x,t|x_0\right)={\int }_0^{t}f\left(x,t|S\left(\tau \right),\tau \right)g\left(S\left(\tau \right),\tau |x_0\right)d\tau x⩾S\left(t\right),$$

$${\int }_0^t\frac{1}{\sqrt{{\sigma }^2\pi \theta \left(1-\exp \left(-2\left(t-\tau \right)/\theta \right)\right)}}\exp \left(-\frac{{\left(\left(x-\mu \theta \right)-\left(S-\mu \theta \right)\exp \left(-\left(t-\tau \right)/\theta \right)\right)}^2}{{\sigma }^2\theta \left(1-\exp \left(-2\left(t-\tau \right)/\theta \right)\right)}\right)$$

$$\times g\left(S,\tau |x_0\right)d\tau =$$

$$\frac{1}{\sqrt{{\sigma }^2\pi \theta \left(1-\exp \left(-2t/\theta \right)\right)}}\exp \left(-\frac{{\left(\left(x-\mu \theta \right)-\left(x_0-\mu \theta \right)\exp \left(-t/\theta \right)\right)}^2}{{\sigma }^2\theta \left(1-\exp \left(-2t/\theta \right)\right)}\right)x⩾S,$$

$${\int }_0^t\frac{1}{\sqrt{{\sigma }^2\pi \theta \left(1-\exp \left(-2\left(t-\tau \right)/\theta \right)\right)}}\exp \left(-\frac{{\left(\left(x-\mu \theta \right)-\left(S\left(\tau \right)-\mu \theta \right)\exp \left(-\left(t-\tau \right)/\theta \right)\right)}^2}{{\sigma }^2\theta \left(1-\exp \left(-2\left(t-\tau \right)/\theta \right)\right)}\right)$$

$$\times g\left(S\left(\tau \right),\tau |x_0\right)d\tau =$$

$$\frac{1}{\sqrt{{\sigma }^2\pi \theta \left(1-\exp \left(-2t/\theta \right)\right)}}\exp \left(-\frac{{\left(\left(x-\mu \theta \right)-\left(x_0-\mu \theta \right)\exp \left(-t/\theta \right)\right)}^2}{{\sigma }^2\theta \left(1-\exp \left(-2t/\theta \right)\right)}\right)x⩾S\left(t\right).$$

$${\overline{f}}_1\left(\lambda \right)$$

$${\overline{f}}_2\left(\lambda \right)$$

$${\overline{f}}_1\left(\lambda \right){\overline{f}}_2\left(\lambda \right)={\int }_0^{\infty }\exp \left(-\lambda t\right)f_1\left(t\right)\ast f_2\left(t\right)dt,$$

$$f_1\left(t\right)\ast f_2\left(t\right)={\int }_0^t{f}_1\left(\tau \right)f_2\left(t-\tau \right)d\tau ,$$

$$\Gamma \left[\lambda +c\right]D_{-c-\lambda }\left(y\right)D_{q-\lambda }\left(z\right)-{\sqrt{\frac{\pi }{2}}|}_{y+z=0,c+q=1}=$$

$${\int }_0^{\infty }\exp \left(-\left(\lambda +c\right)t\right){\left(1-\exp \left(-2t\right)\right)}^{-\left(1+c+q\right)/2}\exp \left(-\frac{{\left(y+z\exp \left(-t\right)\right)}^2}{4\left(1-\exp \left(-2t\right)\right)}\right)$$

$$\times D_{c+q}\left(\frac{z+y\exp \left(-t\right)}{\sqrt{1-\exp \left(-2t\right)}}\right)dt$$

$$\left[\mathrm{Re}\left(\lambda +c\right)>0\text{ and }y+z>0\text{ or }y+z=0,c+q⩽1\right].$$

$$\overline{f}\left(\lambda \theta \right)=\frac{1}{\theta }L\left\{f\left(\frac{t}{\theta }\right)\right\}\theta >0,$$

$$\Gamma \left[\lambda \theta \right]D_{-\lambda \theta }\left(\sqrt{\frac{2}{{\sigma }^2\theta }}\left(\mu \theta -S\right)\right)D_{q-\lambda \theta }\left(\sqrt{\frac{2}{{\sigma }^2\theta }}\left(x-\mu \theta \right)\right)-{\sqrt{\frac{\pi }{2}}|}_{x=S,q=1}=$$

$${\int }_0^{\infty }\exp \left(-\lambda t\right)\{\frac{1}{\theta }{\left(1-\exp \left(-2t/\theta \right)\right)}^{-\left(1+q\right)/2}\exp \left(-\frac{{\left(\left(\mu \theta -S\right)+\left(x-\mu \theta \right)\exp \left(-t/\theta \right)\right)}^2}{2{\sigma }^2\theta \left(1-\exp \left(-2t/\theta \right)\right)}\right)$$

$$\times D_q\left(\frac{\sqrt{2}\left(\left(x-\mu \theta \right)+\left(\mu \theta -S\right)\exp \left(-t/\theta \right)\right)}{\sqrt{{\sigma }^2\theta \left(1-\exp \left(-2t/\theta \right)\right)}}\right)\}dt$$

$$\left[\mathrm{Re}\lambda >0\text{ and }x>S\text{ or }x=S,q⩽1\right].$$

$${\overline{f}}_1\left(\lambda \right){\overline{f}}_2\left(\lambda \right)=\Gamma \left[\lambda \theta \right]\exp \left(\frac{{\left(x_0-\mu \theta \right)}^2-{\left(S-\mu \theta \right)}^2}{2{\sigma }^2\theta }\right)D_{-\lambda \theta }\left(\sqrt{\frac{2}{{\sigma }^2\theta }}\left(\mu \theta -x_0\right)\right)$$

$$\times D_{q-\lambda \theta }\left(\sqrt{\frac{2}{{\sigma }^2\theta }}\left(x-\mu \theta \right)\right)-{\sqrt{\frac{\pi }{2}}|}_{x=S,q=1}{g}_{\lambda }\left(S|x_0\right)x_0<S.$$

$${\int }_0^{\infty }\exp \left(-\lambda t\right)\{\exp \left(\frac{{\left(x_0-\mu \theta \right)}^2-{\left(S-\mu \theta \right)}^2}{2{\sigma }^2\theta }\right)\frac{1}{\theta }{\left(1-\exp \left(-2t/\theta \right)\right)}^{-\left(1+q\right)/2}$$

$$\times \exp \left(-\frac{{\left(\left(\mu \theta -x_0\right)+\left(x-\mu \theta \right)\exp \left(-t/\theta \right)\right)}^2}{2{\sigma }^2\theta \left(1-\exp \left(-2t/\theta \right)\right)}\right)$$

$$\times D_q\left(\frac{\sqrt{2}\left(\left(x-\mu \theta \right)+\left(\mu \theta -x_0\right)\exp \left(-t/\theta \right)\right)}{\sqrt{{\sigma }^2\theta \left(1-\exp \left(-2t/\theta \right)\right)}}\right)\}dt$$

$$-\left\{{\sqrt{\frac{\pi }{2}}|}_{x=S,q=1}g\left(S,t|x_0\right)\right\}$$

$$\left[\mathrm{Re}\lambda >0\text{ and }x>S\text{ or }x=S,q⩽1\right].$$

$${\int }_0^t{\left(1-\exp \left(-2\left(t-\tau \right)/\theta \right)\right)}^{-\left(1+q\right)/2}\exp \left(-\frac{{\left(\left(x-\mu \theta \right)-\left(S-\mu \theta \right)\exp \left(-\left(t-\tau \right)/\theta \right)\right)}^2}{2{\sigma }^2\theta \left(1-\exp \left(-2\left(t-\tau \right)/\theta \right)\right)}\right)$$

$$\times D_q\left(\frac{\sqrt{2}\left(\left(x-\mu \theta \right)-\left(S-\mu \theta \right)\exp \left(-\left(t-\tau \right)/\theta \right)\right)}{\sqrt{{\sigma }^2\theta \left(1-\exp \left(-2\left(t-\tau \right)/\theta \right)\right)}}\right)g\left(S,\tau |x_0\right)d\tau =$$

$${\left(1-\exp \left(-2t/\theta \right)\right)}^{-\left(1+q\right)/2}\exp \left(-\frac{{\left(\left(x-\mu \theta \right)-\left(x_0-\mu \theta \right)\exp \left(-t/\theta \right)\right)}^2}{2{\sigma }^2\theta \left(1-\exp \left(-2t/\theta \right)\right)}\right)$$

$$\times D_q\left(\frac{\sqrt{2}\left(\left(x-\mu \theta \right)-\left(x_0-\mu \theta \right)\exp \left(-t/\theta \right)\right)}{\sqrt{{\sigma }^2\theta \left(1-\exp \left(-2t/\theta \right)\right)}}\right)-\theta {\sqrt{\frac{\pi }{2}}|}_{x=S,q=1}g\left(S,t|x_0\right)$$

$$\left[x>S\text{ or }x=S,q⩽1\right].$$

Full text

1. Introduction

First passage time probability density functions (pdf) play a prominent role in various fields such as neurophysiology (see [2] and the overview in [3]), psychology [4], climate studies [5], biochemistry [6] and finance [7]. However, closed form expressions for the first passage time pdf can only be obtained in a very limited number of cases, see [3].

Hence, applications rely on numerical procedures that are based on a variety of methods. For instance, [8] discusses three such methods for the Ornstein–Uhlenbeck process, namely an eigenfunction expansion, an approach that relies on the link between the Ornstein–Uhlenbeck process and the three dimensional Bessel bridge, and an integral representation consisting of cosine transforms. An alternative and frequently employed integral representation is the renewal equation of Fortet [9]. This approach allows to infer the first passage time pdf when the transition pdf is known (see[10], [11] and [3] for more detail). Numerical applications based on this method are complicated by the fact that the kernel of this Volterra equation of the first kind is weakly singular, see [3]. However, this problem can be circumvented, for instance, by transforming the Fortet renewal equation into a Volterra equation of the second kind, see [12], [13], [14].

This paper derives a new Volterra integral equation for the first passage time pdf of the Ornstein–Uhlenbeck process of which the properties may be attractive for numerical applications. We start from the inverse Laplace transforms for products of two parabolic cylinder functions that were recently derived in [15] and [16]. As the Laplace transform of the first passage time pdf for a constant threshold contains a ratio of two parabolic cylinders (see [17], [18], [19]), the convolution theorem of the Laplace transform then can be used to derive a new integral equation for this first passage time pdf, namely one in which the kernel contains a parabolic cylinder function. This integral equation is a Volterra equation of the first kind, with the exception of a very specific, but useful, combination of parameters for which it is of the second kind. The kernel of this integral equation is regular for $q\leqslant-1$, where $q$ denotes the order of the parabolic cylinder function. The Fortet renewal equation for the Ornstein–Uhlenbeck process is a special case of the new integral equation as it emerges for $q=0$. As the Fortet renewal equation holds both for constant as well as for time depending thresholds, we verify whether this property also holds for the new integral equation for values of $q$ that differ from [math]. This indeed is the case and is verified numerically by using the fact that the first passage time pdf of the Ornstein–Uhlenbeck process can be obtained in analytical form for specific exponential time dependent thresholds as noted in [20] and implemented in [21].

The remainder of this paper is organized as follows. Section $2$ presents the notation and definitions. Section $3$ uses the convolution theorem of the Laplace transform to derive an integral equation for the first passage time pdf of the Ornstein–Uhlenbeck process of which the kernel contains a parabolic cylinder function of order $q$. For $q=0$, the integral equation simplifies into the Fortet renewal equation for the Ornstein–Uhlenbeck process. It is also shown that the integral equation holds for constant as well as time dependent thresholds and that the kernel is regular for $q\leqslant-1$. Section $4$ illustrates two approaches through which the only known closed form expression for the first passage time pdf of the Ornstein–Uhlenbeck process under a constant boundary can easily be obtained from the results in this paper.

2. Notation and preliminaries

Let $\left\{X\left(t\right),t\geqslant 0\right\}$ with $X\left(0\right)=x_{0}$ be an Ornstein–Uhlenbeck process with drift $\left(-x/\theta+\mu\right)$, $\theta>0$, and infinitesimal variance $\sigma^{2}$. Its transition pdf is defined as

[TABLE]

which is given by

[TABLE]

see Equation (5.29) in [3].

The first passage time at the constant threshold $S$ with $S>x_{0}$ is the random variable that is defined by

[TABLE]

and the first passage time pdf is defined as

[TABLE]

For the time dependent threshold $S\left(t\right)$ with $S\left(t\right)>x_{0}$, the following definitions apply

[TABLE]

and

[TABLE]

The Laplace transform of the first passage time pdf for the constant threshold $S$ with $x_{0}<S$ is denoted by $g_{\lambda}\left(S\left|x_{0}\right.\right)$ and for the Ornstein–Uhlenbeck process is given by

[TABLE]

see Equation (5.37) in [3], where $D_{\nu}\left(z\right)$ denotes the parabolic cylinder function of order $\nu$ and argument $z$, see [22].

The Fortet renewal equation [9] is a Volterra integral equation of the first kind that connects the transition pdf and the first passage time pdf for constant $S$ as follows

[TABLE]

and for time dependent $S\left(t\right)$ via

[TABLE]

see [3]. Specializing the Fortet equations (3) and (4) for the Ornstein–Uhlenbeck process, i.e. when using the transition density (1), gives

[TABLE]

and

[TABLE]

The Laplace transforms of the original functions $f_{1}\left(t\right)$ and $f_{2}\left(t\right)$ are denoted by

[TABLE]

where $\operatorname{Re}\lambda>0$. The convolution theorem of the Laplace transform then states

[TABLE]

where $f_{1}\left(t\right)\ast f_{2}\left(t\right)$ is the convolution of $f_{1}\left(t\right)$ and $f_{2}\left(t\right)$ that is to be obtained from

[TABLE]

see [23].

3. The integral equation and its properties

The first passage time pdf $g\left(S,t\left|x_{0}\right.\right)$ and its Laplace transform $g_{\lambda}\left(S\left|x_{0}\right.\right)$ will represent $f_{1}\left(t\right)$ and $\overline{f}_{1}\left(\lambda\right)$, respectively, within the convolution. Thus, $\overline{f}_{1}\left(\lambda\right)$ contains a ratio of parabolic cylinder functions, see (2). The initial function $f_{2}\left(t\right)$ then will need to be chosen in such manner that the inversion formula for $\overline{f}_{1}\left(\lambda\right)\overline{f}_{2}\left(\lambda\right)$ is known.

The paper starts from the inverse Laplace transform for the product of two parabolic cylinder functions with different arguments and orders that was recently derived in [16] and that generalized the results in [15]. The inverse Laplace transform in Equation (2.1) in [16] first will be extended via the result in Equation (4.7) in [15]. This gives the following inverse Laplace transform in which attention is restricted to real arguments and orders of the parabolic cylinder functions

[TABLE]

The term $\left.\sqrt{\pi/2}\right|_{{}_{y+z=0,c+q=1}}$ in the inverse Laplace transform (9) equals $\sqrt{\pi/2}$ when $y+z=0$ and $c+q=1$, and equals [math] otherwise. This additional term finds its origin in the use of the differentiation property of the Laplace transform in the derivation of the inverse Laplace transform in Equation (4.7) in [15]. This extra term turns out to be quite useful as it later allows to obtain the only known closed form expression for the first passage time pdf under a constant threshold in a very simple manner, see Section $4$. For $y+z=0$ and $c+q>1$, the term $f\left(0\right)$ within the application of the differentiation property of the Laplace transform in [15] becomes infinite such that the inverse Laplace transform (9) requires the restriction $c+q\leqslant 1$ when $y+z=0$, but no such restriction on $c+q$ is to be imposed when $y+z>0$.

The time scaling property of the Laplace transform states

[TABLE]

see Equation (29.2.13) in [24]. Using the latter property, $c=0$, $y=\sqrt{\dfrac{2}{\sigma^{2}\theta}}\left(\mu\theta-S\right)$ and $z=\sqrt{\dfrac{2}{\sigma^{2}\theta}}\left(x-\mu\theta\right)$ allows to rewrite (9) as follows

[TABLE]

The two terms on the left hand side of the Laplace transform (11) together specify the Laplace transform $\overline{f}_{2}\left(\lambda\right)$ that will be used within the convolution. The term within curly brackets on the right hand side of the Laplace transform (11) then represents the original function $f_{2}\left(t\right)$.

The product $\overline{f}_{1}\left(\lambda\right)\overline{f}_{2}\left(\lambda\right)$, given the above choices for $\overline{f}_{1}\left(\lambda\right)$ and $\overline{f}_{2}\left(\lambda\right)$, then emerges as

[TABLE]

Employing the inverse Laplace transform (9) and using the above values for $c$, $y$ and $z$ allow to express the inversion formula for $\overline{f}_{1}\left(\lambda\right)\overline{f}_{2}\left(\lambda\right)$ as

[TABLE]

Hence, $f_{1}\left(t\right)\ast f_{2}\left(t\right)$ is to be obtained from the two terms within curly brackets in the inversion formula (13). The convolution theorem (8) then gives – after some straightforward simplifications – the following integral equation for the first passage time pdf of the Ornstein–Uhlenbeck process for the constant threshold $S$

[TABLE]

The kernel of the Volterra integral equation (14) is a combination of terms that contain exponential functions as well as a parabolic cylinder function of order $q$. Note that no restriction on $q$ is required for $x>S$, whereas $q\leqslant 1$ is to be imposed when $x=S$. This integral equation, notwithstanding the fact that it relies on an argument for constant $S$, actually also holds for the time dependent threshold $S\left(t\right)$. In order to illustrate this, we first specialize the integral equation (14) in terms of the Fortet equation (5). Using $q=0$ simplifies the parabolic cylinder function into an exponential function via the following identity

[TABLE]

see Equation (46:4:1) in [22]. Plugging the latter property into the integral equation (14) and simplifying then indeed gives the Fortet equation for the Ornstein–Uhlenbeck process with constant threshold $S$ that is presented in expression (5). The renewal equation (5) extends also to the time dependent threshold $S\left(t\right)$ resulting in the renewal equation (6). Or, the specialization of the integral equation (14) for $q=0$ likewise extends to the time dependent threshold $S\left(t\right)$. The integral equations (5) and (6) then raise the question whether or not this extension of applicability from constant to time dependent threshold also holds for values of $q$ that differ from [math]. The answer is affirmative and this will be illustrated for the case of a time dependent threshold for which the first passage time pdf is known in closed form. As noted in [20], a closed form expression for the first passage time pdf of the Ornstein–Uhlenbeck process can be obtained for thresholds that depend on time in a specific exponential manner. In fact, [20] uses the Doob transformation [25] to express the problem for the Ornstein–Uhlenbeck process in terms of the Wiener process with a constant or affine threshold for which closed form expressions exist. The resulting transition and first passage time pdf are illustrated in [21] for the threshold $S\left(t\right)_{\exp}$ with

[TABLE]

see Equation (2.2) in [21], where $d_{1}$ and $d_{2}$ are arbitrary constants. The first passage time pdf for the latter time dependent threshold is

[TABLE]

see Equation (2.13) in [21]. The applicability of the integral equation (14) for both constant as well as time dependent thresholds then can be tested by inserting the time dependent threshold (15) and its first passage time pdf (16) into the integral equation (14). The resulting integral cannot be evaluated analytically. However, numerical evaluation reveals that the integral equals the term on the right hand side for all values of $q$. Hence, the integral equation (14) can indeed be generalized towards the time dependent threshold $S\left(t\right)$. This gives the following integral equation for the first passage time pdf of the Ornstein–Uhlenbeck process with time dependent threshold $S\left(t\right)$

[TABLE]

No restrictions apply to the order $q$ in the integral equation (17) except when $x=S$ $\left(t\right)$ in which case the restriction $q\leqslant 1$ is to be imposed. The integral equation (17) is a Volterra equation of the first kind for $x>S\left(t\right)$ or $q\neq 1$, and is of the second kind when $x=S\left(t\right)$ and $q=1$.

The kernel in the integral equation (17) is regular for a wide range of values of the order $q$. First, we examine the behaviour of the kernel when evaluating the limit $\tau\rightarrow t$ for $x>S\left(t\right)$. The kernel then goes to zero for all values of $q$. Indeed, the parabolic cylinder function moves to [math] given $\underset{z\rightarrow+\infty}{\lim}\left[D_{\nu}\left(z\right)\right]=0\,$, see Equation (46:7:1) in [22]. Also, the limit of the second term on the left hand side of (17) vanishes for $\tau\rightarrow t$. The first term grows larger than $1$ for $q>-1$, but as the second and third term go to [math] at a faster rate, the kernel vanishes for $\tau\rightarrow t$ and $x>S\left(t\right)$ irrespective of the level of $q$. Second, the argument in the second term goes to [math] for $\tau\rightarrow t$ when $x=S\left(t\right)$. Also the argument in the parabolic cylinder function then equals [math] which gives

[TABLE]

see Equation (46:7:1) in [22]. The latter expression is finite except for $q=1,3,5,7,...$ However, the first term in the kernel becomes infinite for $q>-1$. Taken together, the kernel of the integral equation (17) is regular for $q\leqslant-1$ and $x=S\left(t\right)$ and is weakly singular for $q>-1$ and $x=S\left(t\right).$

Finally, it is to be noted that the kernel of the integral equation (17) can be expressed in terms of other special functions and polynomials for specific values of $q$. For positive integer values of $q$, the parabolic cylinder function simplifies into the Hermite polynomials via the relation

[TABLE]

see Equation (46:4:1) in [22]. The parabolic cylinder function reduces into the complementary error function erfc$\left(z\right)$ for negative integers via

[TABLE]

see p. 326 in [26]. The kernel can also be specified in terms of the modified Bessel function of the second kind given the property

[TABLE]

see Equation (46:4:4) in [22].

4. Retrieving the first passage time pdf for $\boldsymbol{S=\mu\theta}$

The first passage time pdf for the Ornstein–Uhlenbeck process with a constant threshold at $S=\mu\theta$ possesses the following well known closed form expression

[TABLE]

see [18] and [20], who obtained this result via Doob’s transformation [25]. However, the results from the previous section can be used to obtain the first passage time pdf (19) in two ways that do not rely on the latter transformation.

First, the integral equation (17) simplifies into the following Volterra equation of the second kind for $x=S$ and $q=1$

[TABLE]

of which the derivation used the property

[TABLE]

see Equation (46:4:1) in [22]. Using $x=\mu\theta$, i.e. $S=\mu\theta$, causes the integral in the integral equation (20) to vanish such that indeed the first passage time pdf (19) emerges.

Second, plugging $S=\mu\theta$ into the Laplace transform of the first passage time pdf (2) and using the property (18) gives

[TABLE]

The inversion formula for the Laplace transform (22) can be obtained from the inverse Laplace transform (9) when using $y=0$, applying the time scaling property (10) and setting $c$ and $q$ at $1$ and [math], respectively. Subsequently, the properties (18) and (21) are to be used and further simplifications rely on the Legendre duplication formula for the gamma function

[TABLE]

see Equation (6.1.18) in [24]. These simplifications then yield the following inverse Laplace transform

[TABLE]

Specializing the inverse Laplace transform (23) for $z=\sqrt{\dfrac{2}{\sigma^{2}\theta}}\left(\mu\theta-x_{0}\right),$ i.e. for $z>0$, and using the resulting inversion formula for the Laplace transform (22) then gives the first passage time pdf (19).