Alpha-CIR Model with Branching Processes in Sovereign Interest Rate Modelling

Abstract

We introduce a class of interest rate models, called the α-CIR model, which gives a natural extension of the standard CIR model by adopting the α-stable Lévy process and preserving the branching property. This model allows to describe in a unified and parsimonious way several recent observations on the sovereign bond market such as the persistency of low interest rate together with the presence of large jumps at local extent. We emphasize on a general integral representation of the model by using random fields, with which we establish the link to the CBI processes and the affine models. Finally we analyze the jump behaviors and in particular the large jumps, and we provide numerical illustrations.

Asymptotic properties of maximum likelihood estimator for the growth rate of a stable CIR process based on continuous time observations

“…✷ 3 Joint Laplace transform of Y t and t 0 Y s ds Using Theorem 4.10 in Keller-Ressel [22] we derive a formula for the joint Laplace transform of Y t and t 0 Y s ds, where t ∈ R + . We note that this form of the joint Laplace transform in question is a consequence of Theorem 5.3 in Filipović [13] and a special case of Proposition 3.3 in Jiao et al [19] as well.…”

“…where the last event has probability 0, implying P For (vi), see Proposition 3.7 in Jiao et al [19].…”

“…The next proposition is about the existence and uniqueness of a strong solution of the SDE (1.1) stating also that Y is a CBI process with explicitly given branching and immigration mechanisms and we also collect some other useful properties of Y based on Dawson and Li [10], Fu and Li [15], Li [25] and Jiao et al [19].…”

Asymptotic properties of maximum likelihood estimator for the growth rate for a jump-type CIR process based on continuous time observations

“…Here we use the fact that Y t , t 0 Y s ds t∈[0,∞) is a 2-dimensional CBI process following also from Keller-Ressel [26,Theorem 4.10] or Filipović et al [17,paragraph before Theorem 4.3]. For completeness, we note that Keller-Ressel [26, Theorem 4.10] derived a formula for the joint Laplace transform of a regular affine process and its integrated process containing the solutions of Riccati-type differential equations, and Jiao et al [24,Proposition 4.3] derived a formula for that of a general CBI process and its integrated process. We point out that our proof of technique of Theorem 3.1 is different from those of Keller-Ressel [26, Theorem 4.10] and Jiao et al [24,Proposition 4.3], and we make the solutions of Riccati-type differential equations explicit in case of (Y t ) t∈[0,∞) .…”

“…For completeness, we note that Keller-Ressel [26, Theorem 4.10] derived a formula for the joint Laplace transform of a regular affine process and its integrated process containing the solutions of Riccati-type differential equations, and Jiao et al [24,Proposition 4.3] derived a formula for that of a general CBI process and its integrated process. We point out that our proof of technique of Theorem 3.1 is different from those of Keller-Ressel [26, Theorem 4.10] and Jiao et al [24,Proposition 4.3], and we make the solutions of Riccati-type differential equations explicit in case of (Y t ) t∈[0,∞) . Section 4 is devoted to study the existence and uniqueness of the MLE b T of b based on observations (Y t ) t∈[0,T ] with T ∈ (0, ∞).…”