Parameter sensitivity of CIR process

We study the diﬀerentiability of the CIR process with respect to its parameters. We give a stochastic representation of these derivatives in termes of the paths of V .


Introduction
The CIR process is defined as the unique solution of the following stochastic differential equation: where a, σ, v ≥ 0 and b ∈ R (see [5] for the existence and uniqueness of the SDE). This paper is to investigate the sensitivity of the solution of (1.1), when parameters change. The knowledge of the derivatives of the CIR process with respect to its parameters is important if we want to measure the impact of these parameters in the evolution of asset prices modeled, or whose variance is modeled by this process. For technical reasons, we will rather consider the square root of V v , denoted X v . Throughout this paper, we assume that Under this assumption, we have for any T, v > 0, P (∀t ∈ [0, T ] : V v t > 0) = 1. The process X v is the unique solution of the following stockastic differential equation We will also assume b > 0. Note that the following results remain valid for b ≤ 0

Differentiability
We start by studying the differentiability of X with respect to the initial position v, often called 'flow' derivative of the SDE. We consider here the L p -differentiability of the function v −→ X v , i.e the existence of a processẊ v so that σ and its derivative, denoted byẊ v , is given aṡ In particular,V v is a solution of the SDE : Remark 2.1. The hypothesis 4a > 3σ 2 is used only in order to ensure that the random variable sup s≤t X s has moments of order p, for some p geq1. It is therefore not an optimal condition, however we need the condition known as 'Feller condition' (2a ≥ σ 2 )) because without this assumption, V will reach 0 with a nonzero probability, which will cause problems for the differentiability of its root.
Differentiability with respect to σ, a and b: Here we study the differentiability of X with respect to the parameter sigma. We propose a generalization of the result of Benhamou et al (cf. [1]) show that σ −→ X is C 2 in neighborhood of 0. We will show that this function is C 1 in We obtain a similar result for the differentiability of a and b.
Then the function a −→ X a is L p -differentiable at a 0 , for any 1 ≤ p ≤ 2a 0 σ 2 − 1 and its derivative, denote byẊ a , is given byẊ A Proof of Proposition 2.1 For v > 0, X v is given by It follows that R 0 (t) can be written as As 4a 2 > 3σ 2 > σ 2 , then for any t > 0, The processẊ v is solution of Now consider the process It follows from (A.1) and (A.2) that R 1 is solution of On the other hand, we have It follows that R 1 (t) can be written as The proof is based on the following lemma (cf. [3], Lemma 2.3.2.) Lemma A.1. Assume 2a > σ 2 . Then for any t > 0, It follows from this lemma that

B Proof of Proposition 2.3
The proof is similar to the case σ = 0, which can be found in [1]. Denote X the unique solution of the SDE : Set R 0 (t) = X t − X t , then noting that the dynamics of R 0 is given as we deduce that R 0 can be written as where U is given as Applying the Itô formula to the product (U t ) −1 W t , gives Applying Lemma A.1 gives We have Ẋ σ p ≤ C. Furthermore, we can easily check thatẊ σ is solution to the SDE : In particular, R 1 is solution to the SDE : On the other hand, we can easily check that It follows that R 1 can be written as Finally, using A.1, we get

C Proof of Proposition 2.4
DenoteX the unique solution of The process R 1 is solution of NOte that In follows that R 1 can be written as Thus, using Lemme A.1,

D Proof of Proposition 2.6
As V t = X 2 t , V is differentiable with respect to the parameters a, b and σ under the assumptions of Propositions 2.3, 2.4, 2.5. The derivativesV σ is given as solution of the SDE : One can see that the process Z t :=V σ (t) − 2 σ V t is solution of the SDE : On the other hand, applying Itô formula to the process ZV α , for α ∈ R * ,gives It follows that, for α = − 1 2 , the process Y = ZV − 1 2 , Y has finite variation and is given as solution of We can easily solve this equation, we get