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 reason for such an interest can be explained as follows: from the analytical point of view, it is much simpler to deal with continuous-time models although all real-world models operate in the discrete time. In what concerns the rate of convergence, there can be different approaches to its estimation. Some of this approaches are established in [6, 7, 23–27]. In this paper, we consider the Cox–Ingersoll–Ross process and its approximation on a finite time interval. The CIR process was originally proposed by Cox, Ingersoll, and Ross [8] as a model for short-term interest rates. Nowadays, this model is widely used in financial modeling, for example, as the volatility process in the Heston model [16]. The strong global approximation of CIR process is studied in several articles. Strong convergence (without a rate or with a logarithmic rate) of several discretization schemes is shown by [1, 4, 12, 15, 18]. In [1], a general framework for the analysis of strong approximation of the CIR process is presented along with extensive simulation studies. Nonlogarithmic convergence rates are obtained in [2]. 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. In [28], the authors propose an empirical method that utilizes the conditional density of the state variables to estimate and test a term structure model with known price formula using data on both discount and coupon bonds. The method is applied to an extension of a two-factor model due to Cox, Ingersoll, and Ross. Their results show that estimates based solely on bills imply unreasonably large price errors for longer maturities. The process is also discussed in [5].

In this article, we focus on the regime where the CIR process does not hit zero and study weak approximation of this process. In the first case, the sequence of prelimit markets is modeled as the sequence of the discrete-time additive stochastic processes, whereas in the second case, the sequence of multiplicative stochastic processes is modeled. The additive scheme is widely used, for example, in the papers [1, 4, 13]. The papers [10, 28] are recent examples of modeling a stochastic interest rate by the multiplicative model of CIR process. In [10], the authors say that the model has the “strong convergence property,” whereas they refer to models as having the “weak convergence property” when the returns converge to a constant, which generally depends upon the current economic environment and that may change in a stochastic fashion over time. We construct a discrete approximation scheme for the price of asset that is modeled by the Cox–Ingersoll–Ross process. In order to construct these additive and multiplicative processes, 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 use it to prove the weak convergence of asset prices.

The paper is organized as follows. In Section 2, we present a complete and “truncated” CIR process and establish that the “truncated” CIR process can be described as the unique strong solution to the corresponding stochastic differential equation. We establish that this “truncated” process does not hit zero under the same condition as for the original nontruncated process. In Section 3, we present discrete approximation schemes for both these processes and prove the weak convergence of asset prices for the additive model. In the next section, we prove the weak convergence of asset prices for the multiplicative model. Appendix contains additional and technical results.

Let ΩF=(Ω,F,(Ft,t≥0),P) be a complete filtered probability space, and W={Wt,Ft,t≥0} be an adapted Wiener process. 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: (1) dXt=(b−Xt)dt+σXtdWt,X0=x0>0,t≥0, 0,\hspace{0.2778em}t\ge 0,\]]]> where b>0 0$]]>, σ>0 0$]]>. The integral form of the process X has the following form: Xt=x0+ ∫0t(b−Xs)ds+σ ∫0tXsdWs. According to the paper [8], the condition σ2≤2b is necessary and sufficient for the process X to get positive values and not to hit zero. Further, we will assume that this condition is satisfied.

For the proof of functional limit theorems, we will need a modification of the Cox–Ingersoll–Ross process with bounded coefficients. This process is called a truncated Cox–Ingerssol–Ross process. Let C>0 0$]]>. Consider the following stochastic differential equation with the same coefficients b and σ as in (1): (2) dXtC=(b−XtC∧C)dt+σ(XtC∨0)∧CdWt,X0=x0>0,t≥0. 0,\hspace{0.2778em}t\ge 0.\]]]>

The integral form of the process XC is as follows: XtC=x0+ ∫0t(b−XsC∧C)ds+σ ∫0tXsC∧CdWs.

Now, let T>0 0$]]> be fixed. Lemma 2.3.

P{∃t∈[0,T]:Xt≠XtC}→0

as C→∞ .

Consider the following discrete approximation scheme for the process X. Assume that we have a sequence of the probability spaces (Ω(n),F(n),P(n)) , n≥1 . Let {qk(n),n≥1 , 0≤k≤n} be the sequence of symmetric iid random variables defined on the corresponding probability space and taking values  ±Tn , that is, Pn(qk(n)=±Tn)=12 . Let further n>T T$]]>. We construct discrete approximation schemes for the stochastic processes X and XC as follows. Consider the following approximation for the complete process: (6) X0(n)=x0>0,Xk(n)=Xk−1(n)+(b−Xk−1(n))Tn+σqk(n)Xk−1(n),Qk(n):=Xk(n)−Xk−1(n)=(b−Xk−1(n))Tn+σqk(n)Xk−1(n),1≤k≤n, 0,\hspace{2em}{X_{k}^{(n)}}={X_{k-1}^{(n)}}+\frac{(b-{X_{k-1}^{(n)}})T}{n}+\sigma {q_{k}^{(n)}}\sqrt{{X_{k-1}^{(n)}}},\\{} & \displaystyle {Q_{k}^{(n)}}:={X_{k}^{(n)}}-{X_{k-1}^{(n)}}=\frac{(b-{X_{k-1}^{(n)}})T}{n}+\sigma {q_{k}^{(n)}}\sqrt{{X_{k-1}^{(n)}}},\hspace{1em}1\le k\le n,\end{array}\]]]> and the corresponding approximations for XC given by (7) X0(n,C)=x0>0,Xk(n,C)=Xk−1(n,C)+(b−(Xk−1(n,C)∧C))Tn+σqk(n)Xk−1(n,C)∧C,Qk(n,C):=Xk(n,C)−Xk−1(n,C)=(b−(Xk−1(n,C)∧C))Tn+σqk(n)Xk−1(n,C)∧C,1≤k≤n. 0,\\{} \displaystyle {X_{k}^{(n,C)}}& \displaystyle ={X_{k-1}^{(n,C)}}+\frac{(b-({X_{k-1}^{(n,C)}}\wedge C))T}{n}+\sigma {q_{k}^{(n)}}\sqrt{{X_{k-1}^{(n,C)}}\wedge C},\\{} \displaystyle {Q_{k}^{(n,C)}}:& \displaystyle ={X_{k}^{(n,C)}}-{X_{k-1}^{(n,C)}}\\{} & \displaystyle =\frac{(b-({X_{k-1}^{(n,C)}}\wedge C))T}{n}+\sigma {q_{k}^{(n)}}\sqrt{{X_{k-1}^{(n,C)}}\wedge C},1\le k\le n.\end{array}\]]]> The following lemma confirms the correctness of the construction of these approximations.

Consider the sequences of step processes corresponding to these schemes: Xt(n)=Xk(n)for kTn≤t<(k+1)Tn and Xt(n,C)=Xk(n,C)for kTn≤t<(k+1)Tn. Thus, the trajectories of the processes X(n) and X(n,C) have jumps at the points kT/n,k=0,…,n , and are constant on the interior intervals. Consider the filtrations Fkn=σ(Xt(n),t≤kTn) . The processes X(n,C) are adapted with respect to them. Therefore, we can consider the same filtrations for all discrete approximation schemes. So, we can identify Ftn with Fkn for kTn≤t<(k+1)Tn .

Denote by Q and Qn,n≥1 , the measures corresponding to the processes X and X(n),n≥1 , respectively, and by QC and Qn,C,n≥1 , the measures corresponding to the processes XC and X(n,C),n≥1 , respectively. Denote by ⟶W the weak convergence of measures corresponding to stochastic processes. We apply Theorem 3.2 from [23] to prove the weak convergence of measures Qn,C to the measure QC . This theorem can be formulated as follows.

Using Theorem 3.1, we prove the following result. Theorem 3.2.

Qn,C⟶WQC .