Friedrich Hubalek, Irene Klein, Josef Teichmann
A general proof of the Dybvig-Ingersoll-Ross Theorem on the monotonicity of long forward rates is presented. Some inconsistencies in the original proof of this theorem are discussed.
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The Dybvig-Ingersoll-Ross (DIR) theorem states that, in arbitrage-free term structure models, long-term yields and forward rates can never fall. We present a refined version of the DIR theorem, where we identify the reciprocal of the maturity date as the maximal order that long-term rates at earlier dates can dominate long-term rates at later dates. The viability assumption imposed on the market model is weaker than those appearing previously in the literature.
We consider a time inhomogeneous Cox-Ingersoll-Ross diffusion with positive jumps. We exploit a branching property to prove existence of a unique strong solution under a restrictive condition on the jump measure. We give Laplace transforms for the transition probabilities, with an interpretation in terms of limits of mixtures over Gamma laws.
In this paper, we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. Laplace transforms and limit theorems have been obtained, including law of large numbers, central limit theorems and large deviations.
Due to the importance of the Cox-Ingersoll-Ross process in different areas of finance, a broad spectrum of studies and investigations on this model have been carried out. In case of ambiguity, we characterize it by applying the $G$-expectation theory and the associated $G$-Brownian motion. In this paper, we provide the existence and uniqueness of the solution of the Cox-Ingersoll-Ross process in the presence of volatility uncertainty. In addition, some properties of the solution are indicated, such as the regularity and strong Markov property. Besides, we calculate some moments of the CIR proc…
In this paper the fractional Cox-Ingersoll-Ross process on $\mathbb{R}_+$ for $H<1/2$ is defined as a square of a pointwise limit of the processes $Y_{\varepsilon}$, satisfying the SDE of the form $d Y_{\varepsilon}(t)=( \frac{k}{ Y_{\varepsilon}(t)\mathbb{1}_{\{ Y_{\varepsilon}(t)>0\}}+\varepsilon}-a Y_{\varepsilon}(t))dt+σdB^H(t)$, as $\varepsilon\downarrow0$. Properties of such limit process are considered. SDE for both the limit process and the fractional Cox-Ingersoll-Ross process are obtained.
In this paper, the Harnack inequalities and super-Poincaré inequality for generalized CIR model are obtained. Since the noise is degenerate, the intrinsic metric has been introduced to construct the coupling by change of measure. By using isopermetric constant, the optimal estimate of the rate function in the super Poincaré inequality for the associated Dirichlet form is also obtained.
In this paper, we price the zero-coupon bond of the extended Cox-Ingersoll-Ross model by a Dyson type formula established in one of the authors' paper Jin, Peng and Schelllhorn (2016) using Malliavin calculus. This formula provides a fast convergent series to represent the bond price, and it depends on the given drift and volatility of the interest rate process but not the instantaneous forward rate used in Maghsoodi (1996). This expression can be also regarded as a new solution to a class of Reccati equations with time-dependent coefficients.
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…
We study the estimation of a stable Cox-Ingersoll-Ross model, which is a special subcritical continuous-state branching process with immigration. The process is characterized in terms of some stochastic equations. The exponential ergodicity and strong mixing property of the process and the heavy tail behavior of some related random sequences are studied. We also establish the convergence of some point processes and partial sums associated with the model. From those results, we derive the consistency and central limit theorems of the conditional least squares estimators and the weighted conditi…
In this paper, we establish a new connection between Cox-Ingersoll-Ross (CIR) and reflected Ornstein-Uhlenbeck (ROU) models driven by either a standard Wiener process or a fractional Brownian motion with $H>\frac{1}{2}$. We prove that, with probability 1, the square root of the CIR process converges uniformly on compacts to the ROU process as the mean reversion parameter tends to either $σ^2/4$ (in the standard case) or to $0$ (in the fractional case). This also allows to obtain a new representation of the reflection function of the ROU as the limit of integral functionals of the CIR processes…
We consider a stochastic differential equation of the form $dr_t = (a - b r_t) dt + σ\sqrt{r_t}dW_t$, where $a$, $b$ and $σ$ are positive constants. The solution corresponds to the Cox-Ingersoll-Ross process. We study the estimation of an unknown drift parameter $(a,b)$ by continuous observations of a sample path $\{r_t,t\in[0,T]\}$. First, we prove the strong consistency of the maximum likelihood estimator. Since this estimator is well-defined only in the case $2a>σ^2$, we propose another estimator that is defined and strongly consistent for all positive $a$, $b$, $σ$. The quality of the esti…
The Doss-Sussmann (DS) approach is used for uniform simulation of the Cox-Ingersoll-Ross (CIR) process. The DS formalism allows to express trajectories of the CIR process through solutions of some ordinary differential equation (ODE) depending on realizations of a Wiener process involved. By simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving the ODE, we uniformly approximate the trajectories of the CIR process. In this respect special attention is payed to simulation of trajectories near zero. From a conceptual point of view th…
Driven by Brownian motion Cox-Ingersoll-Ross and squared Bessel processes: interaction and phase transition
本文研究了由布朗运动驱动的CIR过程与Bessel过程的异同，证明平方Bessel过程是CIR过程的相变并探讨了两者的随机稳定性差异。
This work is denoted to studying the tail behavior of Cox-Ingersoll-Ross (CIR) processes with regime-switching. One essential difference shown in this work between CIR process with regime-switching and without regime-switching is that the stationary distribution for CIR process with regime-switching could be heavy-tailed. Our results provide a theoretical evidence of the existence of regime-switching for interest rates model based on its heavy-tailed empirical evidence. In this work, we first provide sharp criteria to justify the existence of stationary distribution for the CIR process with re…
Large deviations for Cox-Ingersoll-Ross processes with state-dependent fast switching
本文研究带小噪声和状态依赖快速切换的CIR过程的大偏差，借助Hamilton-Jacobi方程得到平均原理刻画的极限方程，证明了作用积分型速率函数的大偏差原理，创新在于奇异情形下比较原理的建立，采用非线性半群方法完成证明。
We consider the fractional Cox-Ingersoll-Ross process satisfying the stochastic differential equation (SDE) $dX_t = aX_t\,dt + σ\sqrt{X_t}\,dB^H_t$ driven by a fractional Brownian motion (fBm) with Hurst parameter exceeding $\frac{2}{3}$. The integral $\int_0^t\sqrt{X_s}dB^H_s$ is considered as a pathwise integral and is equal to the limit of Riemann-Stieltjes integral sums. It is shown that the fractional Cox-Ingersoll-Ross process is a square of the fractional Ornstein-Uhlenbeck process until the first zero hitting. Based on that, we consider the square of the fractional Ornstein-Uhlenbeck p…
In this paper, we consider the Cox--Ingersoll--Ross (CIR) process in the regime where the process does not hit zero. We construct additive and multiplicative discrete approximation schemes for the price of asset that is modeled by the CIR process and geometric CIR process. In order to construct these schemes, we take the Euler approximations of the CIR process itself but replace the increments of the Wiener process with iid bounded vanishing symmetric random variables. We introduce a "truncated" CIR process and apply it to prove the weak convergence of asset prices. We establish the fact that …
In this paper we define the fractional Cox-Ingersoll-Ross process as $X_t:=Y_t^2\mathbf{1}_{\{t<\inf\{s>0:Y_s=0\}\}}$, where the process $Y=\{Y_t,t\ge0\}$ satisfies the SDE of the form $dY_t=\frac{1}{2}(\frac{k}{Y_t}-aY_t)dt+\fracσ{2}dB_t^H$, $\{B^H_t,t\ge0\}$ is a fractional Brownian motion with an arbitrary Hurst parameter $H\in(0,1)$. We prove that $X_t$ satisfies the stochastic differential equation of the form $dX_t=(k-aX_t)dt+σ\sqrt{X_t}\circ dB_t^H$, where the integral with respect to fractional Brownian motion is considered as the pathwise Stratonovich integral. We also show that for $…
Weak convergence rate for the Cox-Ingersoll-Ross process
研究Cox-Ingersoll-Ross过程在严格为正区域中漂移隐式离散化方案的弱收敛速率，采用PDE方法结合连续时间SDE表示，在特定条件下证明弱收敛速率为$\mathcal{O}(1/N)$，证明方法简单且有望扩展。
We consider a non-stationary Cox-Ingersoll-Ross process. We establish a sharp large deviation principle for the maximum likelihood estimator of its drift parameter.
We derive tail asymptotics for the running maximum of the Cox-Ingersoll-Ross process. The main result is proved by the saddle point method, where the tail estimate uses a new monotonicity property of the Kummer function. This auxiliary result is established by a computer algebra assisted proof. Moreover, we analyse the coefficients of the eigenfunction expansion of the running maximum distribution asymptotically.
Entropies of Cox-Ingersoll-Ross and Bessel processes as functions of time and of related parameters
本文研究 CIR 过程与平方 Bessel 过程各类熵的长时间渐近行为，基于非中心卡方分布推导熵存在条件，并证明 CIR 熵收敛到平稳分布、平方 Bessel 熵的发散或有限性及两过程熵的收敛关系。
In this paper, we define a generalised fractional Cox-Ingersoll-Ross process as a square of singular stochastic differential equation with respect to fractional Brownian motion with Hurst parameter H in (0,1) and continuous drift function. Firstly, we show that this differential equation has a unique solution which is continuous and positive up to the time of the first visit to zero. In addition, we prove that it is strictly positive everywhere almost surely for H > 1/2. In the case where H < 1/2, we consider a sequence of increasing functions and we prove that the probability of hitting zero …
In this we paper we recast the Cox--Ingersoll--Ross model of interest rates into the chaotic representation recently introduced by Hughston and Rafailidis. Beginning with the ``squared Gaussian representation'' of the CIR model, we find a simple expression for the fundamental random variable X. By use of techniques from the theory of infinite dimensional Gaussian integration, we derive an explicit formula for the n-th term of the Wiener chaos expansion of the CIR model, for n=0,1,2,.... We then derive a new expression for the price of a zero coupon bond which reveals a connection between Gauss…
Existence results for the Cox-Ingersoll-Ross model with variable exponent diffusion
本文提出扩散系数含状态依赖变指数函数的随机模型，推广经典Cox-Ingersoll-Ross模型，分析了解的存在性、唯一性与高阶矩性质，并通过数值实验及Itô与Stratonovich解释对比验证模型有效性。
In this paper we show the convergence of the long-term return $t^{-μ}\int_0^tX(s)\d s$ for some $μ\geq1$, where $X$ is the short-term interest rate which follows an extension of Cox-Ingersoll-Ross type model with jumps and memory, and, as an application, we also investigate the corresponding behavior of two-factor Cox-Ingersoll-Ross model with jumps and memory
We propose a change detection method for the famous Cox--Ingersoll--Ross model. This model is widely used in financial mathematics and therefore detecting a change in its parameters is of crucial importance. We develop one- and two-sided testing procedures for both drift parameters of the process. The test process is based on estimators that are motivated by the discrete time least-squares estimators, and its asymptotic distribution under the no-change hypothesis is that of a Brownian bridge. We prove the asymptotic weak consistence of the test, and derive the asymptotic properties of the chan…
Ergodicity and Law-of-large numbers for the Volterra Cox-Ingersoll-Ross process
本文研究 Volterra CIR 过程，借助 Volterra Riccati 方程渐近分析与仿射变换证明其有限维分布渐近独立、$L^p$ 大数定律及平稳版本遍历性，并应用于参数估计一致性与极大似然估计。
Estimation of a pure-jump stable Cox-Ingersoll-Ross process
本文研究了纯跳跃稳定CIR过程的参数估计，提出基于似然近似的相合估计量，证明其渐近性质，并通过一步法改进初步估计量。
High order approximations of the Cox-Ingersoll-Ross process semigroup using random grids
本文利用随机网格上的离散化格式组合技术，为CIR过程构造了任意偶数阶的弱逼近格式，并在参数限制下证明了收敛阶数，数值实验表明该方法对CIR过程和Heston模型均有效且能显著节省计算时间。
Strong convergence rates and temporal regularity for Cox-Ingersoll-Ross processes and Bessel processes with accessible boundaries
本文研究了CIR过程和Bessel过程在可访问边界下的强收敛性及时间正则性，提出了一种隐式平方根欧拉方法，并证明其在L^p距离下的强收敛性。
In this paper, we consider a one-dimensional Cox-Ingersoll-Ross (CIR) process whose drift coefficient depends on unknown parameters. Considering the process discretely observed at high frequency, we prove the local asymptotic normality property in the subcritical case, the local asymptotic quadraticity in the critical case, and the local asymptotic mixed normality property in the supercritical case. To obtain these results, we use the Malliavin calculus techniques developed recently for CIR process by Alòs et {\it al.} \cite{AE08} and Altmayer et {\it al.} \cite{AN14} together with the $L^p$-n…
Scaling limit of heavy tailed nearly unstable cumulative INAR($\infty$) processes and rough fractional diffusions
本文研究了重尾近不稳定累积INAR(∞)过程的标度极限，揭示其与集成分数Cox-Ingersoll-Ross过程的关系，并提出高效模拟方法。
The well-known theorem of Dybvig, Ingersoll and Ross shows that the long zero-coupon rate can never fall. This result, which, although undoubtedly correct, has been regarded by many as surprising, stems from the implicit assumption that the long-term discount function has an exponential tail. We revisit the problem in the setting of modern interest rate theory, and show that if the long "simple" interest rate (or Libor rate) is finite, then this rate (unlike the zero-coupon rate) acts viably as a state variable, the value of which can fluctuate randomly in line with other economic indicators. …
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