Abstract
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 this "truncated" process does not hit zero under the same condition considered for the original nontruncated process.
Highlights
The problem of convergence of discrete-time financial models to the models with continuous time is well developed; see, e.g., [6, 7, 9, 11, 14, 17, 19]
The CIR process was originally proposed by Cox, Ingersoll, and Ross [8] as a model for short-term interest rates
The strong global approximation of CIR process is studied in several articles
Summary
The problem of convergence of discrete-time financial models to the models with continuous time is well developed; see, e.g., [6, 7, 9, 11, 14, 17, 19]. In [10], the author extends the CIR model of the short interest rate by assuming a stochastic reversion level, which better reflects the time dependence caused by the cyclical nature of the economy or by expectations concerning the future impact of monetary policies In this framework, the convergence of the long-term return by using the theory of generalized Bessel-square processes is studied. Consider a Cox–Ingersoll– Ross process with constant parameters on this space This process is described as the unique strong solution of the following stochastic differential equation: dXt = (b − Xt )dt + σ Xt dWt , X0 = x0 > 0, t ≥ 0,.
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