Low-dimensional Cox-Ingersoll-Ross process

The present paper investigates Cox-Ingersoll-Ross (CIR) processes of dimension less than 1, with a focus on obtaining an equation of a new type including local times for the square root of the CIR process. We utilize the fact that non-negative diffusion processes can be obtained by the transformation of time and scale of some reflected Brownian motion to derive this equation, which contains a term characterized by the local time of the corresponding reflected Brownian motion. Additionally, we establish a new connection between low-dimensional CIR processes and reflected Ornstein-Uhlenbeck (ROU) processes, providing a new representation of Skorokhod reflection functions.


Introduction 1.Background and motivation
The squared Bessel process as well as its generalization Cox-Ingersoll-Ross (CIR) process where x 0 ≥ 0, a > 0, b ∈ R, and their respective square roots are widely used in various fields, in particular physics (see e.g.[8,19] and the overview in [17, Section I]) and finance [11,12,13,18].One of the reasons for the popularity of these processes lies in the well-known fact (see e.g.[20, Chapter IV, Example 8.2]) that a > 0 in (1.2) implies that X(t) ≥ 0 for all t ≥ 0 with probability 1, which is a natural property for multiple real-life phenomena.Furthermore, if the Feller condition 2a ≥ σ 2 is satisfied, the paths of X in (1.2) are strictly positive a.s., which turns out to be very useful in multiple cases.For example, the well-known Heston model [18] utilizes Y := √ X as stochastic volatility and, under the Feller condition, Y has the dynamics of the form since it is evident that with probability 1 for all t ≥ 0. This equation can be used for e.g.simulation purposes (see, for example, [1,15,27]); moreover, the measure change procedure associated with the Heston model naturally involves the inverse volatility 1/Y which has far more transparent properties when X > 0 a.s.At the same time, empirical considerations indicate that the Feller condition 2a ≥ σ 2 can sometimes be too restrictive and models perform better when it is not satisfied.For instance, [22,Section 3.4] reports that the joint SPX-VIX fit of the Heston model turns out to be substantially better when the Feller condition is not demanded from the model parameters.Additionally, [2,Example 10.2.6] indicates that the Heston model with violated Feller condition can reproduce the upward VIX "smirk".In other words, there are cases when the process Y = √ X under relatively small values of a turns out to be more relevant for reflecting real-life phenomena despite the associated analytical challenges.Nevertheless, the majority of sources in the literature pay more attention to the case when the Feller condition is satisfied.Among notable exceptions, we mention [4,7,9,10] which discussed the SDEs of the type (1.3) when σ 2 4 < a < σ 2 2 .It is worth to note a more recent paper [25] which establishes a connection between Y = √ X and a reflected Ornstein-Uhlenbeck (ROU) process where L 0 is the corresponding Skorokhod reflection function, i.e. a continuous non-decreasing process that has points of growth exclusively when Y 0 (t) = 0 and such that Y 0 (t) ≥ 0. In particular, it is established that Additionally, [25,Theorem 2.4] provides a new representation of L 0 in terms of a limit of the CIR processes: with probability 1, for any positive sequence {ε n , n ≥ 1} such that ε n ↓ 0, n → ∞, and for all T > 0 sup where The representation of L 0 from [25] described above essentially concerns convergence of the CIR square roots as a → σ 2 4 + and does not cover what happens when a → σ 2 4 −.The reason is that analytic challenges associated to the process Y = √ X are especially acute when 0 < a < σ 2 4 , i.e. when the dimension (see e.g.[23]) k := 4a σ 2 of the process (1.2) is less than 1.Indeed, the integral in (1.4) is infinite after the first moment of hitting zero, the representation (1.3) does not hold and, furthermore, the process Y = √ X is not a semimartingale (see e.g.Example 1.2 and Appendix 1 in [24] or [16, p. 100]).In this regard, one must mention important contributions [5,6] which shed light on the behavior of Y = √ X when X is the squared Bessel process (1.1) of dimension k = a ∈ (0, 1).There, it is shown that Y satisfies the equation of the form where with ℓ being a jointly continuous in (t, y) normalized local time such that for any bounded measurable function

Main results
In our paper, we consider a more general case of the CIR process (1.2) with b ∈ R and 0 < a < σ 2 4 (we call such a process a low-dimensional CIR) and study the properties of Y = √ X.More precisely, we represent Y as a transformation of a reflected Brownian motion W and use the properties of the local time L W of the latter to study the local time L Y of Y .Afterwards, we use the connection between L W and L Y to get a representation in the spirit of (1.6): namely, we prove that Y is a strong solution of the equation where with ℓ being an explicitly given normalized transformation of a local time L Y of the process Y : where ℓ(t, 0) := lim y→0+ ℓ(t, y) is defined by continuity.Finally, we close the gap of [25] mentioned above and obtain a representation of the Skorokhod reflection function for the ROU process in terms of CIR processes of dimension k = 4a σ 2 < 1.It is worth noting that our approach is simpler than the one in [5,6] and is based on the following machinery: we notice that Itô's formula applied to X(t) + ε followed by moving ε ↓ 0 implies that Y = √ X satisfies the equation of the form (1.9) with L represented as an a.s.-limit and {ε n , n ≥ 0} being some sequence converging to zero.After that, we utilize the fact that the CIR process X is a regular diffusion and hence can be obtained from some reflected Brownian motion W by a transformation of time and scale (see e.g.[29, Chapter V, Section 7]).We find the explicit shape of this transformation, use it to establish the connection between the local times of W and Y .Finally, we exploit this link to show that the limit (1.11) is equal to (1.10).The technique described above seems to be more transparent than the one employed in [5,6] and additionally allows to get a clear intuition behind the process ℓ in (1.8).

Structure of the paper
The paper is organized as follows.In Section 2, we present some preliminary calculations and discuss the representation (1.9)- (1.11).Section 3 is devoted to the case 0 < a < σ 2 4 and contains Theorem 3.5 that can be regarded as the main result of the paper.In Section 4, we discuss the results and compare them with the behavior of the limit in (1.11) when a ≥ σ 2 4 .In Section 5, we establish a new connection between CIR processes of dimension k = 4a σ 2 < 1 and ROU processes and obtain a new representation of Skorokhod reflection function.

Preliminary calculations
Let a, σ > 0, b ≥ 0, W = {W (t), t ≥ 0} be a Brownian motion, and let us consider the continuous modification of a standard CIR process (1.2).Note that, by [21, Chapter IV, Example 8.2], the paths of X are non-negative with probability 1 provided that a > 0 and hence one can define the squareroot process Y = {Y (t), t ≥ 0} := { X(t), t ≥ 0}.In order to analyze the dynamics of Y , take ε > 0 and observe that, by Itô's formula, (2.1) Fix an arbitrary T > 0 and note that the left-hand side of (2.1) converges to Y (t) uniformly on [0, T ] with probability 1 as ε ↓ 0. It is also evident that sup Next, by [28, Chapter XI], E ∞ 0 ½ {X(s)=0} ds = 0 and hence, by the Burkholder- Davis-Gundy inequality, for any T > 0 This implies that for each In particular, (2.1) as well as convergences (2.2) and (2.3) imply that the left-hand side of ds converges uniformly on compacts in probability as ε ↓ 0. Therefore, there exists a ucp-limit and the process Y = √ X satisfies the SDE of the form where Remark 2.1.Ucp convergence in (2.4) implies that for an arbitrary T > 0 there exists a sequence {ε n , n ≥ 1} (depending on T ) such that, for any t ∈ [0, T ], with probability 1 as n → ∞.Later on, we will see that the a.s.convergence (2.5) holds for an arbitrary sequence {ε n , n ≥ 1} such that ε n ↓ 0 as n → ∞.
Moreover, it will be shown that the set of full probability where (2.5) holds can be chosen independently of a particular sequence {ε n , n ≥ 1}.
Remark 2.2.Note that the process L defined by (2.4) is continuous a.s.since 3 Stochastic representation of L when 0 < a < σ 2

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Our strategy for the analysis of L will be as follows.Since the CIR process X in (1.2) is a non-negative regular diffusion, it can be represented (see e.g. [29, Chapter V, Section 7]) in the form for some change of time τ and change of scale S of a reflected Brownian motion W = { W (t), t ≥ 0}.Then, we re-write the integral in the limit (2.4) in terms of the local time L W = L W (t, x) of W and exploit Hölder continuity of the latter to find an explicit representation of L in terms of L W .

CIR process as the transformation of a reflected Brownian motion
In order to implement our approach, we first need to represent the CIR process as a transformation of a reflected Brownian motion.For a given set of parameters a, b, σ of the SDE (1.2), define a scale function S: and observe that, since S is strictly increasing and S(∞) = ∞, there exists its inverse S −1 .Define also a speed measure Proposition 3.1.Let X be the unique strong solution to the CIR equation Then there exists a reflected Brownian motion W starting at S(x 0 ) such that Before moving to the proof of Proposition 3.1, let us make some remarks regarding its formulation.Remark 3.2.
Remark 3.3.The transformation (3.3) is invertible.Indeed, it is straightforward to check that, with probability 1, where ϕ = τ −1 can be expressed as the inverse of the mapping t → Proof of Proposition 3.1.We will split the proof into two steps.First, we will follow [29, Chapter V, Section 7, §48] to prove that, for some given reflected Brownian motion W , the process X(t) := S −1 W (τ t ) is the weak solution to the SDE (1.2).Then we will utilize the invertability of transformation (3.3) outlined in Remark 3.3 to establish the existence of a reflected Brownian motion W together with the required representation for the given CIR process X.
Step 1.Let Z = { Z(t), t ≥ 0} be a standard Brownian motion starting at Z(0) = S(x 0 ).Consider a reflected Brownian motion where Z(t) := t 0 sign Z(s)d Z(s) is a Brownian motion and L Z (t) is the local time of L Z at zero.Put V (t) := S −1 W (t) and observe that, by the extension of Itô's formula in [29, Lemma IV.45.9], Hence, by Itô's formula, for any infinitely differentiable function with compact support h, is a local martingale, where is the generator of (1.2).Recall that and observe that, by (3.6), for any infinitely differentiable function with compact support h, simple change of variables yields that Ah(X(s))ds.
Since C τt (h), t ≥ 0, is a local martingale by the optional stopping theorem, is also a local martingale and therefore, by [29, V.19-V.20],X is the weak solution to (1.2).
Step 2. Let now X be the unique strong solution to (1.2).By Remark 3.3 and Step 1, the process where ϕ is defined as the inverse of the mapping t → t 0 1 ρ(X(s)) ds, is a reflected Brownian motion for which X admits the representation (3.3).

Characterization of L in terms of L W
Having the representation (3.3) at our disposal, we are now ready to characterize the process L from (2.4) in terms of the local time L W of the corresponding reflected Brownian motion.
Let X be the unique strong solution to the SDE (1.2) with 0 < a < σ 2

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and W be the reflected Brownian motion such that Denote L W = {L W (t, x), t ≥ 0, x ≥ 0} the jointly continuous modification of the local time of W so that for any bounded measurable f , with probability 1.First of all, let us express the local time be the square root of the CIR process X.Then, for any bounded measurable f , where, with probability 1, with ϕ being defined by (3.5).
Proof.For any bounded measurable f , we can write The final result is obtained by recalling that Define a normalized local time of the process Y as follows.Set and ℓ(t, 0) := lim y→0+ ℓ(t, y).
Note that ℓ(t, y) is continuous in (t, y) because However, we want to stress that ℓ(t, y) is a function of the local time L Y of the process Y without mentioning the auxiliary Brownian motion W .
Theorem 3.5.Let X be the CIR process satisfying (1.2) and W be the reflected Brownian motion such that X(t) = S −1 ( W (τ t )), t ≥ 0.Then, with probability 1, the process Y = √ X satisfies the SDE of the form where with probability 1 for any t ≥ 0.Moreover, since L W (t, •) is Hölder continuous of order up to 1 2 a.s.(see e.g.calculations in [29, Section IV.44]), for any δ ∈ 0, 1  2 and any fixed t > 0 there exists a random variable C > 0 such that, with probability 1, (3.12) Hence, on the one hand, by (3.11).On the other hand, take δ ∈ 0, k 2(2−k) and observe that (3.12) where and C is a (random) constant that varies from line to line.Now we are ready to proceed to the proof of Theorem 3.5.
Proof of Theorem 3.5.In Section 2, we obtained the representation (3.9) with L being a ucp-limit of the form ds.
Hence, one is left to prove that this limit exists in the sense of a.s.convergence and check that the last equality in (3.10) holds.
Let k := 4a σ 2 ∈ (0, 1) denote the dimension of the CIR process, i.e. we have to study the a.s.-limit of the form ds.
Observe that and, similarly, Let us study separately the asymptotics of as ε ↓ 0. First, observe that for any y ≥ 0 and note that by Remark 3.7, Thus, by the dominated convergence theorem, with probability 1, Next, observe that, with probability 1, On the other hand, take an arbitrary δ ∈ 0, k 2(2−k) , denote δ ′ := (2 − k)δ and observe that (3.12) yields where β := 2b σ 2 .Hence, by substituting z = y/ √ ε in (3.14), we can write with probability 1 as ε ↓ 0. Summarizing (3.13) and (3.15), we obtain that, with probability 1, lim Finally, integration by parts yields dy and the right-hand side of the last equation is equal to dy. and Summarizing all of the above and recalling that k = 4a σ 2 , we finally obtain that with probability 1 which ends the proof.
Remark 3.8.Theorem 3.5 implies that the limit exists a.s.for any sequence {ε n , n ≥ 1} such that ε n ↓ 0 and does not depend on the particular choice of the sequence.Moreover, the proof of Theorem 3.5 yields that the existence of the limit (3.16) is ensured for all ω such that L W (ω; t, •) is Hölder continuous.In other words, the set of full probability where (3.16) holds can be chosen independently of a particular sequence {ε n , n ≥ 1}, as anticipated in Remark 2.1.

Discussion of the results
It is evident that the nature of the limit in (2.4) heavily depends on the relation between parameters a and σ.Therefore, in order to put our findings from Section 3 into context, let us provide some relevant results from [25] on the behavior of Y when a ≥ σ 2 4 .
4.1 Square root of the CIR process when a ≥ σ 2

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Case I: a > σ 2 4 .Observe that, if ds < ∞ a.s., (4.1) then the limit (2.4) is equal to ds by monotone convergence.This is clearly the case for a ≥ σ 2 2 : indeed a ≥ σ 2 2 implies that X (and hence Y ) has strictly positive paths a.s.(see e.g.[3] or [14]) and therefore (4.For a more detailed discussion of this phenomenon, we refer the reader to [7] whereas a comprehensive overview of SDEs of the type (4.3) can be found in [10].
However, the limit L from (2.4) has a simple interpretation in terms of Skorokhod reflections (see e.g. the seminal works [30,31]) as summarized in the following theorem.2) The processes Y := √ X and L defined by (2.4) is the (unique) solution to Skorokhod problem with L being the corresponding Skorokhod reflection function, i.e. a continuous non-decreasing process starting at 0 with points of growth occurring only at zeros of Y and such that Y (t) ≥ 0.
Remark 4.4.Item 2) of Theorem 4.3 states that, when a = σ 2 4 , the square root process Y = √ X coincides with a reflected Ornstein-Uhlenbeck (ROU) process.More details on the latter can be found in e.g.[32].

Comparison to the low-dimensional case
As we have seen in Section 3, the case 0 < a < σ 2 4 is arguably the most challenging one and leads to the most involved value of the limit (2.4).First of all, note that (4.1) does not hold due to Theorem 4.3 together with the comparison theorem for solutions of SDEs (see e.g.[20]).Next, the limit L in (2.4) cannot be non-decreasing in t as it happens when a ≥ σ 2 4 .Indeed, consider τ ≥ 0 such that X(τ ) > 0.Then, by a.s.continuity of X, there exists a neighborhood τ − < τ < τ + such that X is bounded away from zero on (τ − , τ + ).Denote now −δ := a − σ 2 4 , δ > 0.Then, with probability 1, for all τ On the other hand, L is not strictly decreasing on the entire [0, T ]: if it is strictly decreasing (and, since L(0) = 0, non-positive), then Y ≤ U , where U is the standard Ornstein-Uhlenbeck process defined by However, it is not possible since Y cannot take negative values.

Connection to Skorokhod reflections
Finally, let us present the connection of low-dimensional CIR processes with Skorokhod problems.For δ > − σ 2 4 , consider a family of CIR processes {X δ } with a = a(δ) = σ 2 4 + δ and defined by As described above in Sections 3-4, the process Y δ := √ X δ satisfies the SDE of the form where the term L δ depends on the parameter δ as follows: ds and the integral is well-defined and finite with probability 1; -if δ = 0, L 0 is the Shorokhod reflection function, i.e. a continuous non-decreasing process with points of growth occurring only at zeros of Y 0 and such that Y 0 ≥ 0, which is a symmetric local time of Y 0 at 0; in particular, Y 0 is a reflected Ornstein-Uhlenbeck process.
The dynamics of Y δ with δ ≥ 0 described above allowed [25] to obtain the following alternative representation to the Skorokhod reflection function L 0 .

Acknowledgements
The present research is carried out within the frame and support of the ToppForsk project nr.274410 of the Research Council of Norway with title STORM: Stochastics for Time-Space Risk Models.The first author is supported by The Swedish Foundation for Strategic Research, grant Nr.UKR22-0017.

t 0 1 ρ
(X(s)) ds.For more details on transformations of this type, we refer the reader to [21, Chapter IV, §7].