BOUNDARY CROSSING PROBLEMS AND FUNCTIONAL TRANSFORMATIONS FOR ORNSTEIN-UHLENBECK PROCESSES

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BOUNDARY CROSSING PROBLEMS AND

FUNCTIONAL TRANSFORMATIONS FOR

ORNSTEIN-UHLENBECK PROCESSES

Aria Ahari1∗

, Larbi Alili1†

, Massimiliano Tamborrino1‡

1Department of Statistics, University of Warwick, Coventry, CV4 7AL, United Kingdom.

Abstract

We are interested in the law of the ﬁrst passage time of an Ornstein-Uhlenbeck process

to time-varying thresholds. We show that this problem is connected to the laws of the ﬁrst

passage time of the process to members of a two-parameter family of functional transforma-

tions of a time-varying boundary. For speciﬁc values of the parameters, these transformations

appear in a realisation of a standard Ornstein-Uhlenbeck bridge. We provide three diﬀerent

proofs of this connection. The ﬁrst one is based on a similar result for Brownian motion,

the second uses a generalisation of the so-called Gauss-Markov processes and the third relies

on the Lie group symmetry method. We investigate the properties of these transformations

and study the algebraic and analytical properties of an involution operator which is used in

constructing them. We also show that these transformations map the space of solutions of

Sturm-Liouville equations into the space of solutions of the associated nonlinear ordinary dif-

ferential equations. Lastly, we interpret our results through the method of images and give

new examples of curves with explicit ﬁrst passage time densities.

Keywords: First passage times; Lie algebras; Sturm-Liouville equations; Orstein-Uhlenbeck;

Fokker Planck equation; Ornstein-Uhlenbck bridge; Brownian motion.

2020 Mathematics Subject Classiﬁcation: Primary 35K05, 60J50, 60J60.

1 Introduction

Let U:= (Ut)t≥0be a one-dimensional Ornstein-Uhlenbeck (OU for short) process deﬁned on

a ﬁltered probability space (Ω,(F)t≥0,F,P) as the unique solution to the following stochastic

diﬀerential equation (SDE)

dUt=−kUtdt +dBt, U0= 0,(1)

where (Bt)t≥0is a standard Brownian motion (BM) starting at 0 and k∈Ris a constant. The

OU process is a Gauss-Markov process with transition density function given by

pt(x, y) := ∂

∂y P(Ut≤y|U0=x) = ekt

pr(t)ϕ yekt −x

pr(t)!, x, y ∈R,(2)

where ϕ(z) = e−z2

2/√2π,z∈R, is the probability density function of the standard normal

distribution and

r(t) = (e2kt −1)/2k, t ≥0,

s(t) = ln (2kt + 1)/2k, t ≤ζ(k),

∗aria.ahari@warwick.ac.uk

†l.alili@warwick.ac.uk

‡massimiliano.tamborrino@warwick.ac.uk

arXiv:2210.01658v3 [math.PR] 25 Mar 2024

where

ζ(k)=(−1

2kif k < 0;

+∞otherwise.

It is well known by the Dambis, Dubins-Schwarz theorem (see, e.g., Theorem V.1.6 in [34]), that

the OU process can be written in terms of a time changed BM (Wt)t≥0as

Ut=e−ktWr(t), t ≥0.(3)

Let f∈ C([0,∞),R) be such that f(0) ̸= 0, with C(I, K) denoting the space of continuous

functions from Iinto Kfor some intervals Iand K⊆R. We are interested in the ﬁrst passage

time (FPT) of the OU to fgiven by

k= inf{t > 0; Ut=f(t)},

with inf{∅} =∞. The main goal of this paper is to derive, through diﬀerent methods, an explicit

analytical expression linking the distribution of Tf

kto that of TSα,β

k. Here, the two-parameter

family of curves {Sα,β

kf;α̸= 0, β ∈R}is deﬁned by

Sα,β

kf(t) = 1 + αβr(t)

α2kα2r(t)

1 + αβr(t)+ 11/2

e−kt fsα2r(t)

1 + αβr(t), t < ζk,α,β,(4)

where

ζk,α,β =









s−1

αβ if αβ < 0, k ≥0;

s−1

αβ+2kα2if 0 <2k

αβ+2kα2<1, k < 0;

+∞otherwise.

By doing so, we generalize the results obtained for a BM in [3], which can be immediately

recovered from ours by letting k→0, i.e. Tf

0, Sα,β

0and ζ0,α,β. To simplify the notation and for

consistency with [3], we drop the subscript 0 when referring to the BM.

To the best of our knowledge, explicit results for the FPT of the OU to fonly exist for

constants [1, 36] or hyperbolic type boundaries [13, 15]. Further results for the boundary crossing

problem of Gauss-Markov processes to moving boundaries have been obtained in, e.g., [14, 17,

18, 31]. We also refer to [8, 39] for applications of Lie symmetries to FPT problems. Our main

result, stated in Theorem 3.1, allows to map those results for the law of the FPT of Tf

kto that

of TSα,β

kfor the OU process. Such problems are of great interest, as the OU process has been

used in many applications to model objects such as interest rates in ﬁnance or the evolution of

the neuronal membrane voltages in neuroscience, see e.g. [1] and citations therein. In this paper,

we focus on the OU without drift, as the results for the OU process with drift can be directly

obtained from our results after some transformations, as discussed in Remark 3.2.

The paper is organised as follows. In Subsection 2.1, we introduce some notations and provide

the key results for the Sα,β operator for the BM. The OU functional setting is presented in

Subsection 2.2. In Subsection 2.3, we recall the diﬀerent constructions of OU bridges and their

properties. In particular, the process (S1,−1/r(T)

kUt,0< t ≤T) has the same law as an OU bridge

of length Tfrom 0 to 0, for some T > 0. Section 3 is devoted to the statement of Theorem 3.1,

which contains the main result of the paper, and two examples of its application. In Section 4,

we discuss the properties of the Sα,β

kftransformation with its connection to a certain nonlinear

diﬀerential equation (Lemma 4.1), while in Section 5, we prove Theorem 3.1 in three diﬀerent

ways. In the ﬁrst proof, we use the relationship between the FPT of an OU and that of a BM.

In the second one, we use a generalisation of the Gauss-Markov processes introduced in Section

3.2 of [3] and ﬁnd an analogue version of that proof in our case. In the third one, we use the

Lie algebra to ﬁnd the symmetries of the Fokker-Planck equation or the Kolmogorov forward

diﬀerential equation

∂h

∂t =1

∂2h

∂x2+kx∂h

∂x +kh. (5)

Then, we use these symmetries to construct the function hα,β

kof equation (30), derive our

transformation Sα,β

kfand relate the FPT distribution of Tf

kto that of TSα,β f

k, in Section 5.3. In

Section 6, we discuss the asymptotic distribution of TSα,β

kand the transience of the transformed

curves Sα,β

kf. We provide the analogue of the Kolmogorov–Erd¨os–Petrovski transience test [19]

in the OU case and show its connection to the asymptotic behaviour of the FPT. Lastly, in

Section 7, we use the method of images to obtain new classes of boundaries yielding explicit FPT

distributions and use our Sα,β

kftransformation (4) to produce new examples. A limitation of this

method is that it only works for boundaries with certain properties given in Lemma 7.1.

As we were ﬁnalising the paper, we discovered that the Sα,β

ktransformation, a variant of

the boundary crossing identity (18) in Theorem 3.1 and the Lie symmetries (31) had previously

appeared in [28] (our (4) can be obtained by setting A=k2and B= 0 in equation (39) therein),

using the Lie approach. When comparing our results, we found misprints in one of their Lie

symmetries and boundary crossing identity, as discussed in Section 5.3.

2 Notation and preliminaries

We ﬁrst introduce some functional spaces, transformations and related results for the BM, as

in [3], in Subsection 2.1, and then deﬁne the corresponding functional objects for the OU in

Subsection 2.2. We end this section by providing diﬀerent representations of OU bridges and

highlighting their connection to our functional transformations.

2.1 Brownian motion setting

We start by introducing a nonlinear operator τdeﬁned on the space of functions whose reciprocals

are square integrable in some (possibly inﬁnite) interval of R+= [0,∞) by

τf (t) = Zt

f−2(z)dz,

and use it to deﬁne

A∞=[

a>0[

b>0

A(a, b) and A(a, b) = ±f∈ C([0, a],R+) : τf (a) = b,

where a, b ∈R+. In [3], the authors derived the following relationship between the laws of the

FPTs of Tfand TSα,β f,

P(TSα,β f∈dt) = α3(1 + αβt)−5

2e−αβ

2(1+αβt)(Sα,β f(t))2Sα,β(P(Tf∈dt)), t < ζ0,α,β,(6)

where Sα,β is the two-parameter family of transformations Sα,β :A∞→A∞given by

Sα,βf(t) = 1 + αβt

αfα2t

1 + αβt, α ̸= 0, β ∈R.(7)

Equation (6) extends a previous result by the same authors for a relationship between the laws

of the FPT of Tfand TS1,β f, with the one-parameter family of transformations {S1,β f, β ∈R}

obtained using the construction of Brownian bridges, see [2]. The connection to Brownian bridges

naturally appears as (S1,−1/T (B)t,0≤t<T) is a Brownian bridge of length Tfrom 0 to 0, see

page 64 of [11]. Besides deriving (6) in [3], the authors showed also that the transformation Sα,β

can be obtained as

Sα,β = Σ ◦Πα,−β◦Σ (8)

where ◦denotes the composition operator. Here, Σ : A∞→A∞is the involution operator, i.e.,

Σ◦Σ = Id, speciﬁed by

Σf(t) = 1

f(ρ◦τf (t)),

where ρis the inversion operator acting on the space of continuous monotone functions i.e.,

ρf ◦f(t) = t. For α∈R∗=R\{0}, β ∈R, Πα,β :A∞→A∞is the family of nonlinear operators

given by

Πα,βf(t) = f(t)(α+βτ f(t)).(9)

As explained in the beginning of Section 2 of [3] and Appendix 8 of [34], the operators (9) are

closely related to the Sturm-Liouville equation

ϕ′′ =µϕ, (10)

where µdenotes a positive Radon measure on R+and ϕ′′ is the second derivative in the sense of

distributions. In fact, if ϕsolves (10), then the vectorial space {Πα,β ϕ;α̸= 0, β ∈R}is the set of

other solutions to the same equation. Moreover, all positive solutions are convex and described

by the set {Πα,β φ;α > 0, β ≥0}, where φis the unique, positive, decreasing solution such that

φ(0) = 1.

Moreover, it was noted in [3] that Sα,β ◦Sα′,β′=Sαα′,αβ′+β

α′, for all couples (α, β) and (α′, β′)∈

R∗×R, and that (S1,β)β≥0is a semi-group, while (S1,β)β∈Rand (Seα,0)α∈Rare groups.

2.2 Ornstein-Uhlenbeck setting

We shall now deﬁne the operators of interest on the space of functions

Ak,∞=[

a>0[

b>0

Ak(a, b),(11)

where a, b ∈R+, and

Ak(a, b) = ±f∈ C([0, a],R+) : τ f sgn(k)ssgn(k)a=b

1−{k<0}2kb,(12)

with sgn(k) being the sign of k. Note that Ak,∞is the set of continuous functions which are of

constant sign on some nonempty interval [0, l], l > 0. Let the isomorphic nonlinear operators Λk

and Σkbe deﬁned, on Ak,∞, by

Λkf(t) = eks(t)f(s(t)) (13)

and

Σk= Λk−1◦Σ◦Λk,(14)

respectively. Note that the inverse of Λkis Λ−1

kf(t) = e−ktf(r(t)). Here, Λkmaps the curves

from the OU setting to the corresponding curves in the BM setting, with limits limk→0Λk=Id,

and limk→0Σk= Σ. From (14), we can immediately see that Σkcan also be represented as

Σkf(t) = e−kρ◦τ f (r(t))−kt

f(ρ◦τf (r(t))).(15)

Inspired by (8) in the BM setting, for α∈R∗and β∈R, deﬁne Sα,β

k:Ak,∞→Ak,∞by

Sα,β

k= Σk◦Πα,−β◦Σk.(16)

Alternatively, Sα,β

kcan be also obtained as Λ−1

k◦Sα,β ◦Λk, as shown in the proof of Lemma 4.1.

This result provides us with an intuitive interpretation of this family of transformations. First,

we map the boundary ffor the OU process to its corresponding boundary for the standard BM

using the Λktransformation. Then, we use Sα,β on this curve for the standard BM, and ﬁnally

we revert it back to the OU problem via Λ−1

k, as illustrated in the following ﬁgure.

OU f(t)1+αβr(t)

αq2kα2r(t)

1+αβr(t)+ 1 e−kt fsα2r(t)

1+αβr(t)

BM √2kt + 1f(s(t)) 1+αβt

αq2kα2t

1+αβt + 1 fsα2t

1+αβt 

Sα,β

Λk

Sα,β

Λ−1

Figure 1: Flow chart of Sα,β

2.3 Ornstein-Uhlenbeck bridges

As mentioned in the introduction, Sα,β

kfor speciﬁc values of αand βrelates to a representa-

tion of the standard OU bridge. We shall now deﬁne this process and then recall its diﬀerent

representations, using the detailed analysis of both Wiener and OU bridges provided in [5, 11].

Deﬁnition 2.1. An OU bridge Ubr ={Ubr

t:t∈[0, T ]}from a to b, of length T, is characterised

by the following properties:

(i) Ubr

0=aand Ubr

T=b(each with probability 1).

(ii) Ubr is a Gaussian process.

(iii) E[Ubr

t] = asinh (k(T−t))

sinh (kt)+bsinh (kt)

sinh (kT ).

(iv) Cov(Ubr

s, Ubr

t) = sinh (ks) sinh (k(T−t))

ksinh (kT ),0≤s≤t<T.

(v) The paths of Ubr are almost surely continuous.

In what follows, we present three diﬀerent representations of OU bridges. First, consider the

following linear SDE

dUbr

t=−kcoth (k(T−t))Ubr

t+kb

sinh (k(T−t))dt +dBt,0≤t < T, (17)

with initial condition Ubr

0=a. This has a unique strong solution given by

Uir

t:= (asinh (k(T−t))

sinh (kT )+bsinh (kt)

sinh (kT )+Rt

sinh (k(T−t))

sinh (k(T−s)) dBsif 0 ≤t < T,

bif t=T.

This is referred to as the integral representation (ir) of the OU bridge. The following anticipative

(av) and the space-time (st) versions can be obtained by using the diﬀerent representations of

the Wiener bridge,

Uav

t=asinh (k(T−t))

sinh (kT )+bsinh (kt)

sinh (kT )+Ut−sinh (kt)

sinh (kT )UT,0≤t < T,

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BOUNDARYCROSSINGPROBLEMSANDFUNCTIONALTRANSFORMATIONSFORORNSTEIN-UHLENBECKPROCESSESAriaAhari1∗,LarbiAlili1†,MassimilianoTamborrino1‡1DepartmentofStatistics,UniversityofWarwick,Coventry,CV47AL,UnitedKingdom.AbstractWeareinterestedinthelawofthefirstpassagetimeofanOrnstein-Uhlenbeckprocesstotime-varying...

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BOUNDARY CROSSING PROBLEMS AND FUNCTIONAL TRANSFORMATIONS FOR ORNSTEIN-UHLENBECK PROCESSES

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