Tail distribution of the integrated Jacobi diffusion process

In this paper, we study the distribution of the integrated Jacobi diffusion processes with Brownian noise and fractional Brownian noise. Based on techniques of Malliavin calculus, we develop a unified method to obtain explicit estimates for the tail distribution of these integrated diffusions.


Introduction
We consider Jacobi diffusion process that is defined as the solution of the scalar stochastic differential equation where the initial conditions X 0 ∈ (0, 1), a, b, c are positive constants and W t is a standard Brownian motion. It is known that this diffusion plays an important role in various fields. In population biology, it is called a Wright-Fisher diffusion with migration, see e.g. [10]. In the finance context, the Jacobi process was first used by De Jong et al. [8] to model the exchange rates in a target zone and by Delbaen and Shirakawa [3] to model interest rates. Since then, Jacobi process has been became one of popular interest rate models in finance. The main advantage of the model is that it admits lower and upper boundaries for the interest rate, hence preventing negative interest rates. Many different properties of Jacobi model can be found in the literature. Among others, we mention the works by Gouriéroux and Jasiak [6] for a multidimensional version and several applications, Ackerer et al. [2] for a new stochastic volatility model, etc. From financial point of view, the integrated diffusion process of the form is one of fundamental objects that needs studying, see e.g. [11] and references therein. In particular, the information about the distribution function will be very useful for applications. The reader can consult the seminal paper [4] for the results related to Cox-Ingersoll-Ross interest rate model, also see [1,7] for other models. However, for Some fundamental properties of the fractional model (3), including the existence and uniqueness of solutions, were already discussed in [5]. However, the distribution of Y H t has not been discussed yet. Motivated by the above observations, the aim of this paper is to study the distribution of the integrated diffusions (2) and (4). Because of the complexity of stochastic calculus with respect to fBm, it is almost impossible to compute the distribution of Y H t explicitly. On the other hand, we would like to develop a unified method for Y t and Y H t . Hence, we will focus on providing the estimates for the tail distribution of Y t and Y H t . The main tools of the present paper are the techniques of Malliavin calculus (stochastic calculus of variations) which have been successfully used to investigate many financial models, see e.g. Chapter 6 in [12]. By using the flexible transforms, we are able to bound Malliavin derivatives of Y t and Y H t , and hence, we obtain explicit estimates for the tail distributions. More specifically, the main contributions of this paper are as follows • In Theorem 3.1, we provide two explicit estimates (13) and (14) for the tail distribution of Y t . The first one is an inequality of Gaussian type and the second one is a Bernstein-type inequality. • Analogously, in Theorem 3.2, we also obtain two explicit estimates (19) and (20) for the tail distribution of Y H t . The rest of the paper is organized as follows. In Section 2, we recall some elements of Malliavin calculus and a general estimate for the tail distribution of Malliavin differentiable random variables. The main results of the paper are stated and proved in Section 3. The conclusion is given in Section 4.

Preliminaries
Let us recall some elements of stochastic calculus of variations (for more details see [12]). We suppose that Brownian motion (W t ) t∈[0,T ] is defined on a complete probability space ( Let S denote the dense subset of L 2 (Ω, F, P ) consisting of smooth random variables of the form If F has the form (5), we define its Malliavin derivative as the process DF := {D t F, t ∈ [0, T ]} given by We shall denote by D 1,2 the closure of S with respect to the norm A random variable F is said Malliavin differentiable if it belongs to D 1,2 . The next lemma comes from Corollary 4.7.4 in [14].

Lemma 2.1
Let Z be a centered random variable in D 1,2 . Assume there exists a non-random constant β such that Then, the following estimate for tail probabilities holds 3. The main results

Jacobi process with Brownian noise
In this subsection, we provide two explicit estimate for the tail distribution of Y t defined by (2). We always assume that the parameters a, b and c satisfy This assumption ensures the equation (1) has a unique solution belonging to (0, 1) and the boundaries {0} and {1} are inaccessible, see e.g. Chapter 4 in [9]. In order to be able apply Lemma 2.1, we have to prove the Malliavin differentiability of the solution to the equation (1). (1) is Malliavin differentiable and its derivative is given by

Proof
We first investigate the Malliavin differentiability of stochastic process x t := arcsin(2X t − 1) ∈ (− π 2 , π 2 ). By Itô differential formula, x t solves the following equation Let us compute the directional where x ε t solves the following equation where ε ∈ [0, 1) and for the simplicity, we put g( We have By using Taylor expansion, the above equation becomes for some random variable ξ s lying between 0 and 1. The solution to (10) is given by We now observe that As a consequence, by the dominated convergence theorem, we obtain where the limit holds in L 2 (Ω). So, by the classical results of Sugita [13], we can conclude that x t ∈ D 1,2 , and its derivative is given by We now apply the chain rule of Malliavin derivatives to X t = 1+sin xt 2 and we obtain This completes the proof.

TAIL DISTRIBUTION OF THE INTEGRATED JACOBI DIFFUSION PROCESS
We also need the following lemma to bound the Malliavin derivative of (X t ) t∈[0,T ] .
where the constant M is defined by

Proof
The proof is straightforward, so we omit it. Here we only note that the function h(x) : We now are in a position to state the first main results of this paper.
be the integrated Jacobi diffusion process defined by (2). We have I. For each t ∈ [0, T ], the expected value of Y t is given by and Proof The proof of the part I is straightforward. Indeed, we have Let us now prove the part II. By the property of Malliavin derivatives we have Moreover, by the formula (8) and Lemma 3.1, we deduce In order to prove (13) we observe that Thus the condition (6) of Lemma 2.1 are fulfilled. Hence, we can use the estimate (7) and we obtain So (13) is verified. It only remains prove (14). We and use the chain rule for Malliavin derivatives to get This, together with (15), gives us By the Hölder inequality we deduce We therefore obtain which points out that the random variable Z also fulfills the condition (6) of Lemma 2.1. Hence, the estimate (7) and the fact that E[

TAIL DISTRIBUTION OF THE INTEGRATED JACOBI DIFFUSION PROCESS
So we obtain (14) . This completes the proof.
Remark 3.1. Clearly, when fixed t > 0, we have Hence, the bound (13) is better than (14) when x is large sufficiently. However, when t → 0 + , we have µ t ≃ X 0 t and Hence, the bound (14) will give us a better estimate if t ≃ 0 and x ≃ µ t . That is why we provided both bounds (13) and (14) as in Theorem 3.1.

Jacobi process with fractional Brownian noise
In this subsection, we investigate the tail distribution of Y H t defined by (4). Following the results obtained in [5], we always assume that This assumption ensures that the equation (3) has a unique solution and the fractional stochastic integral s is well defined as a pathwise Riemann-Stieltjes integral (see [15] for a detailed presentation of this integral). We recall that fBm W H admits the so-called Volterra representation (see e.g. [12] pp.

277-279)
where the kernel K H is given by  [5] provides the following which is similar to Proposition 3.1.

has a unique solution in
Moreover, this solution belongs to (0, 1) and is Malliavin differentiable with We also have a similar estimate to that obtained in Lemma 3.1.

Lemma 3.2
For all x ∈ (0, 1), we have where the constant M H is defined by Stat., Optim. Inf. Comput. Vol. 8, September 2020 We now note that, unlike the case of Brownian noises, the expected value of Y H t is not easy to compute explicitly. This is due to the fact that the expectation of fractional stochastic integrals is non zero. In the next proposition, we gives an estimate for E[Y H t ]. Proposition 3. 3 We have, for 0 ≤ t ≤ T, where, for α H := H(2H − 1), the functions A s , B s are defined by As a consequence, x H s ds.

Proof
We have We therefore obtain

Inserting this relation into (18) yields
By Comparison theorem for differential equations we have The above equation is a linear ordinary differential equation and its solution is given by So we can finish the proof.

TAIL DISTRIBUTION OF THE INTEGRATED JACOBI DIFFUSION PROCESS
The next statement is the second main result of the present paper.
T ] be the integrated fractional Jacobi diffusion process defined by (4). For each t ∈ (0, T ], the tail distribution of Y H t satisfies and Proof From the derivative formula (16) and Lemma 3.2 we deduce Since the function v → K(v, r) is non-decreasing, this implies and hence, We therefore obtain Thus the random variable Z H also fulfills the condition (6), and hence, the estimate (7) gives us So the claim (20) is proved. The proof of Theorem is complete.

Conclusion
In this paper, we used the techniques of Malliavin calculus to investigate the tail distribution of the integrated Jacobi diffusion with Brownian noise and fractional Brownian noise. Our contribution is that we are able to develop a unified method to obtain explicit estimates for the tail distributions. Our work provides one more fundamental property of Jacobi models. In this sense, we partly enrich the knowledge of Jacobi models.