Least-Squares Estimation for the Subcritical Heston Model Based on Continuous-Time Observations

Abstract

We prove strong consistency and asymptotic normality of least-squares estimators for the subcritical Heston model based on continuous-time observations. We also present some numerical illustrations of our results.

1 Introduction

Stochastic processes given by solutions to stochastic differential equations (SDEs) have been frequently applied in financial mathematics. So the theory and practice of stochastic analysis and statistical inference for such processes are important topics. In this note, we consider such a model, namely the Heston model

$$\begin{aligned} {\left\{ \begin{array}{l} {\mathrm {d}}Y_t = (a - b Y_t) \, {\mathrm {d}}t + \sigma _1 \sqrt{Y_t} \, {\mathrm {d}}W_t , \\ {\mathrm {d}}X_t = (\alpha - \beta Y_t) \, {\mathrm {d}}t + \sigma _2 \sqrt{Y_t} \bigl (\varrho \, {\mathrm {d}}W_t + \sqrt{1 - \varrho ^2} \, {\mathrm {d}}B_t\bigr ) , \end{array}\right. } \qquad t \geqslant 0 , \end{aligned}$$

(1.1)

where a > 0, $b, \alpha , \beta \in {\mathbb {R}}$, $\sigma _1 > 0$, $\sigma _2 > 0$, $\varrho \in (-1, 1)$, and $(W_t, B_t)_{t\geqslant 0}$ is a 2-dimensional standard Wiener process, see Heston [14]. For interpretation of Y and X in financial mathematics, see, e.g., Hurn et al. [20, Section 4], here we only note that $X_t$ is the logarithm of the asset price at time t and $Y_t$ its volatility for each $t\geqslant 0$. The first coordinate process Y is called a Cox–Ingersoll–Ross (CIR) process (see Cox et al. [9]), square-root process or Feller process.

Parameter estimation for the Heston model (1.1) has a long history, for a short survey of the most recent results, see, e.g., the introduction of Barczy and Pap [5]. The importance of the joint estimation of $(a, b, \alpha , \beta )$ and not only of (a, b) stems from the fact that $X_t$ is the logarithm of the asset price at time t having high importance in finance. In fact, in Barczy and Pap [5], we investigated asymptotic properties of maximum likelihood estimator of $(a, b, \alpha , \beta )$ based on continuous-time observations $(X_t)_{t\in [0,T]}$, $T>0$. In Barczy et al. [6], we studied asymptotic behavior of conditional least-squares estimator of $(a, b, \alpha , \beta )$ based on discrete-time observations $(Y_i,X_i)$, $i=1,\ldots ,n$, starting the process from some known non-random initial value $(y_0,x_0)\in (0,\infty )\times {\mathbb {R}}$. In this note, we study least-squares estimator (LSE) of $(a, b, \alpha , \beta )$ based on continuous-time observations $(X_t)_{t\in [0,T]}$, $T > 0$, starting the process (Y, X) from some known initial value $(Y_0,X_0)$ satisfying ${{\mathbb {P}}}(Y_0\in (0,\infty ))=1$. The investigation of the LSE of $(a,b,\alpha ,\beta )$ based on continuous-time observations $(X_t)_{t\in [0,T]}$, $T>0$, is motivated by the fact that the LSEs of $(a,b,\alpha ,\beta )$ based on appropriate discrete-time observations converge in probability to the LSE of $(a,b,\alpha ,\beta )$ based on continuous-time observations $(X_t)_{t\in [0,T]}$, $T>0$, see Proposition 3.1. We do not suppose that the process $(Y_t)_{t\in [0,T]}$ is observed, since it can be determined using the observations $(X_t)_{t\in [0,T]}$ and the initial value $Y_0$, which follows by a slight modification of Remark 2.5 in Barczy and Pap [5] (replacing $y_0$ by $Y_0$). We do not estimate the parameters $\sigma _1$, $\sigma _2$ and $\varrho$, since these parameters could—in principle, at least—be determined (rather than estimated) using the observations $(X_t)_{t\in [0,T]}$ and the initial value $Y_0$, see Barczy and Pap [5, Remark 2.6]. We investigate only the so-called subcritical case, i.e., when $b>0$, see Definition 2.3.

In Sect. 2, we recall some properties of the Heston model (1.1) such as the existence and uniqueness of a strong solution of the SDE (1.1), the form of conditional expectation of $(Y_t,X_t)$, $t \geqslant 0$, given the past of the process up to time s with $s \in [0, t]$, a classification of the Heston model and the existence of a unique stationary distribution and ergodicity for the first coordinate process of the SDE (1.1). Section 3 is devoted to derive a LSE of $(a, b, \alpha , \beta )$ based on continuous-time observations $(X_t)_{t\in [0,T]}$, $T > 0$, see Proposition 3.1. We note that Overbeck and Rydén [27, Theorems 3.5 and 3.6] have already proved the strong consistency and asymptotic normality of the LSE of (a, b) based on continuous-time observations $(Y_t)_{t\in [0,T]}$, $T>0$, in case of a subcritical CIR process Y with an initial value having distribution as the unique stationary distribution of the model. Overbeck and Rydén [27, page 433] also noted that (without providing a proof) their results are valid for an arbitrary initial distribution using some coupling argument. In Sect. 4, we prove strong consistency and asymptotic normality of the LSE of $(a, b, \alpha , \beta )$ introduced in Sect. 3, so our results for the Heston model (1.1) in Sect. 3 can be considered as generalizations of the corresponding ones in Overbeck and Rydén [27, Theorems 3.5 and 3.6] with the advantage that our proof is presented for an arbitrary initial value $(Y_0,X_0)$ satisfying ${{\mathbb {P}}}(Y_0\in (0,\infty ))=1$, without using any coupling argument. The covariance matrix of the limit normal distribution in question depends on the unknown parameters a and b as well, but somewhat surprisingly not on $\alpha$ and $\beta$. We point out that our proof of technique for deriving the asymptotic normality of the LSE in question is completely different from that of Overbeck and Rydén [27]. We use a limit theorem for continuous martingales (see, Theorem 2.6), while Overbeck and Rydén [27] use a limit theorem for ergodic processes due to Jacod and Shiryaev [21, Theorem VIII.3.79] and the so-called Delta method (see, e.g., Theorem 11.2.14 in Lehmann and Romano [24]). We also remark that the approximation in probability of the LSE of $(a,b,\alpha ,\beta )$ based on continuous-time observations $(X_t)_{t\in [0,T]}$, $T>0$, given in Proposition 3.1 is not at all used for proving the asymptotic behavior of the LSE in question as $T\rightarrow \infty$ in Theorems 4.1 and 4.2. Further, we mention that the covariance matrix of the limit normal distribution in Theorem 3.6 in Overbeck and Rydén [27] is somewhat complicated, while, as a special case of our Theorem 4.2, it turns out that it can be written in a much simpler form by making a simple reparametrization of the SDE (1) in Overbeck and Rydén [27], estimating $-b$ instead of b (with the notations of Overbeck and Rydén [27]), i.e., considering the SDE (1.1) and estimating b (with our notations), see Corollary 4.3. Section 5 is devoted to present some numerical illustrations of our results in Sect. 4.

2 Preliminaries

Let ${\mathbb {N}}$, ${\mathbb {Z}}_+$, ${\mathbb {R}}$, ${\mathbb {R}}_+$, ${\mathbb {R}}_{++}$, ${\mathbb {R}}_-$ and ${\mathbb {R}}_{--}$ denote the sets of positive integers, non-negative integers, real numbers, non-negative real numbers, positive real numbers, non-positive real numbers and negative real numbers, respectively. For $x , y \in {\mathbb {R}}$, we will use the notation $x \wedge y := \min (x, y)$. By $\Vert x\Vert$ and $\Vert A\Vert$, we denote the Euclidean norm of a vector $x \in {\mathbb {R}}^d$ and the induced matrix norm of a matrix $A \in {\mathbb {R}}^{d \times d}$, respectively. By ${\varvec{I}}_d \in {\mathbb {R}}^{d \times d}$, we denote the d-dimensional unit matrix.

Let $\bigl (\Omega , {{\mathcal F}}, {{\mathbb {P}}}\bigr )$ be a probability space equipped with the augmented filtration $({{\mathcal F}}_t)_{t\in {\mathbb {R}}_+}$ corresponding to $(W_t,B_t)_{t\in {\mathbb {R}}_+}$ and a given initial value $(\eta _0,\zeta _0)$ being independent of $(W_t,B_t)_{t\in {\mathbb {R}}_+}$ such that ${{\mathbb {P}}}(\eta _0\in {\mathbb {R}}_+)=1$, constructed as in Karatzas and Shreve [22, Section 5.2]. Note that $({{\mathcal F}}_t)_{t\in {\mathbb {R}}_+}$ satisfies the usual conditions, i.e., the filtration $({{\mathcal F}}_t)_{t\in {\mathbb {R}}_+}$ is right-continuous and ${{\mathcal F}}_0$ contains all the ${\mathbb {P}}$-null sets in ${\mathcal F}$.

By $C^2_c({\mathbb {R}}_+\times {\mathbb {R}}, {\mathbb {R}})$ and $C^{\infty }_c({\mathbb {R}}_+\times {\mathbb {R}}, {\mathbb {R}})$, we denote the set of twice continuously differentiable real-valued functions on ${\mathbb {R}}_+\times {\mathbb {R}}$ with compact support, and the set of infinitely differentiable real-valued functions on ${\mathbb {R}}_+\times {\mathbb {R}}$ with compact support, respectively.

The next proposition is about the existence and uniqueness of a strong solution of the SDE (1.1), see, e.g., Barczy and Pap [5, Proposition 2.1].

Proposition 2.1

Let $(\eta _0, \zeta _0)$ be a random vector independent of $(W_t, B_t)_{t\in {\mathbb {R}}_+}$ satisfying ${{\mathbb {P}}}(\eta _0 \in {\mathbb {R}}_+) = 1$. Then for all $a \in {\mathbb {R}}_{++}$, $b, \alpha , \beta \in {\mathbb {R}}$, $\sigma _1, \sigma _2 \in {\mathbb {R}}_{++}$, and $\varrho \in (-1, 1)$, there is a pathwise unique strong solution $(Y_t, X_t)_{t\in {\mathbb {R}}_+}$ of the SDE (1.1) such that ${{\mathbb {P}}}((Y_0, X_0) = (\eta _0, \zeta _0)) = 1$ and ${\mathbb {P}}(Y_t \in {\mathbb {R}}_+{\text { for all }}t \in {\mathbb {R}}_+) = 1$. Further, for all $s, t \in {\mathbb {R}}_+$ with $s \leqslant t$,

$$\begin{aligned} {\left\{ \begin{array}{l} Y_t = \mathrm {e}^{-b(t-s)} Y_s + a \int _s^t \mathrm {e}^{-b(t-u)} \, {\mathrm {d}}u + \sigma _1 \int _s^t \mathrm {e}^{-b(t-u)} \sqrt{Y_u} \, {\mathrm {d}}W_u , \\ X_t = X_s + \int _s^t (\alpha - \beta Y_u) \, {\mathrm {d}}u + \sigma _2 \int _s^t \sqrt{Y_u} \, {\mathrm {d}}(\varrho W_u + \sqrt{1 - \varrho ^2} B_u) . \end{array}\right. } \end{aligned}$$

(2.1)

Next we present a result about the first moment and the conditional moment of $(Y_t, X_t)_{t\in {\mathbb {R}}_+}$, see Barczy et al. [6, Proposition 2.2].

Proposition 2.2

Let $(Y_t, X_t)_{t\in {\mathbb {R}}_+}$ be the unique strong solution of the SDE (1.1) satisfying ${\mathbb {P}}(Y_0 \in {\mathbb {R}}_+) = 1$ and ${\mathbb {E}}(Y_0) < \infty$, ${\mathbb {E}}(|X_0|) < \infty$. Then for all $s,t\in {\mathbb {R}}_+$ with $s\leqslant t$, we have

$$\begin{aligned}&{\mathbb {E}}(Y_t \,|\,{\mathcal F}_s) = \mathrm {e}^{-b(t-s)} Y_s + a \int _s^t \mathrm {e}^{-b(t-u)} \, {\mathrm {d}}u, \end{aligned}$$

(2.2)

$$\begin{aligned}&{\mathbb {E}}(X_t \,|\,{\mathcal F}_s) = X_s + \int _s^t (\alpha - \beta {\mathbb {E}}(Y_u \,|\,{\mathcal F}_s)) \, {\mathrm {d}}u \nonumber \\&\phantom {{\mathbb {E}}(X_t \,|\,{\mathcal F}_s)\,} = X_s + \alpha (t - s) - \beta Y_s \int _s^t \mathrm {e}^{-b(u-s)}\,{\mathrm {d}}u - a \beta \int _s^t \left( \int _s^u \mathrm {e}^{-b(u-v)} \, {\mathrm {d}}v\right) {\mathrm {d}}u , \end{aligned}$$

(2.3)

and hence

$$ \left[ {\begin{array}{l} {{\mathbb{E}}(Y_{t} )} \\ {{\mathbb{E}}(X_{t} )} \\ \end{array} } \right] = \left[ {\begin{array}{ll} {{\text{e}}^{{ - bt}} } \hfill & 0 \hfill \\ { - \beta \int_{0}^{t} {{\text{e}}^{{ - bu}} } {\mkern 1mu} {\text{d}}u} \hfill & 1 \hfill \\ \end{array} } \right]\left[ {\begin{array}{l} {{\mathbb{E}}(Y_{0} )} \\ {{\mathbb{E}}(X_{0} )} \\ \end{array} } \right] + \left[ {\begin{array}{ll} {\int_{0}^{t} {{\text{e}}^{{ - bu}} } {\mkern 1mu} {\text{d}}u} & 0 \\ { - \beta \int_{0}^{t} {\left( {\int_{0}^{u} {{\text{e}}^{{ - bv}} } {\mkern 1mu} {\text{d}}v} \right)} {\text{d}}u} & t \\ \end{array} } \right]\left[ {\begin{array}{l} a \\ \alpha \\ \end{array} } \right]. $$

Consequently, if $b \in {\mathbb {R}}_{++}$, then

$$\begin{aligned} \lim _{t\rightarrow \infty } {\mathbb {E}}(Y_t) = \frac{a}{b} , \qquad \lim _{t\rightarrow \infty } t^{-1} {\mathbb {E}}(X_t) = \alpha - \frac{\beta a}{b} , \end{aligned}$$

if $b = 0$, then

$$\begin{aligned} \lim _{t\rightarrow \infty } t^{-1} {\mathbb {E}}(Y_t) = a , \qquad \lim _{t\rightarrow \infty } t^{-2} {\mathbb {E}}(X_t) = - \frac{1}{2} \beta a , \end{aligned}$$

if $b \in {\mathbb {R}}_{--}$, then

$$\begin{aligned} \lim _{t\rightarrow \infty } \mathrm {e}^{bt} {\mathbb {E}}(Y_t) = {\mathbb {E}}(Y_0) - \frac{a}{b} , \qquad \lim _{t\rightarrow \infty } \mathrm {e}^{bt} {\mathbb {E}}(X_t) = \frac{\beta }{b} {\mathbb {E}}(Y_0) - \frac{\beta a}{b^2} . \end{aligned}$$

Based on the asymptotic behavior of the expectations $({\mathbb {E}}(Y_t), {\mathbb {E}}(X_t))$ as $t \rightarrow \infty$, we recall a classification of the Heston process given by the SDE (1.1), see, Barczy and Pap [5, Definition 2.3].

Definition 2.3

Let $(Y_t, X_t)_{t\in {\mathbb {R}}_+}$ be the unique strong solution of the SDE (1.1) satisfying ${\mathbb {P}}(Y_0 \in {\mathbb {R}}_+) = 1$. We call $(Y_t, X_t)_{t\in {\mathbb {R}}_+}$ subcritical, critical or supercritical if $b \in {\mathbb {R}}_{++}$, $b = 0$ or $b \in {\mathbb {R}}_{--}$, respectively.

In the sequel ${\mathop {\longrightarrow }\limits ^{{\mathbb {P}}}}$, ${\mathop {\longrightarrow }\limits ^{{\mathcal L}}}$ and ${\mathop {\longrightarrow }\limits ^{{\mathrm {a.s.}}}}$ will denote convergence in probability, in distribution and almost surely, respectively.

The following result states the existence of a unique stationary distribution and the ergodicity for the process $(Y_t)_{t\in {\mathbb {R}}_+}$ given by the first equation in (1.1) in the subcritical case, see, e.g., Cox et al. [9, Equation (20)], Li and Ma [25, Theorem 2.6] or Theorem 3.1 with $\alpha = 2$ and Theorem 4.1 in Barczy et al. [4].

Theorem 2.4

Let $a, b, \sigma _1 \in {\mathbb {R}}_{++}$. Let $(Y_t)_{t\in {\mathbb {R}}_+}$ be the unique strong solution of the first equation of the SDE (1.1) satisfying ${\mathbb {P}}(Y_0 \in {\mathbb {R}}_+) = 1$. Then

• (1) $Y_t\, {\mathop {\longrightarrow }\limits ^{{\mathcal L}}}Y_\infty$ as $t \rightarrow \infty$, and the distribution of $Y_\infty$ is given by

  $$\begin{aligned} {\mathbb {E}}(\mathrm {e}^{-\lambda Y_\infty }) = \left( 1 + \frac{\sigma _1^2}{2b} \lambda \right) ^{-2a/\sigma _1^2} , \qquad \lambda \in {\mathbb {R}}_+ , \end{aligned}$$

  (2.4)

  i.e., $Y_\infty$ has Gamma distribution with parameters $2a / \sigma _1^2$ and $2b / \sigma _1^2$, hence

  $$\begin{aligned} {\mathbb {E}}(Y_\infty ) = \frac{a}{b}, \qquad {\mathbb {E}}(Y_\infty ^2) = \frac{(2a+\sigma _1^2)a}{2b^2} , \qquad {\mathbb {E}}(Y_\infty ^3) = \frac{(2a+\sigma _1^2)(a+\sigma _1^2)a}{2b^3} . \end{aligned}$$

• (2) supposing that the random initial value $Y_0$ has the same distribution as $Y_\infty$, the process $(Y_t)_{t \in {\mathbb {R}}_+}$ is strictly stationary.

• (3) for all Borel measurable functions $f : {\mathbb {R}}\rightarrow {\mathbb {R}}$ such that ${\mathbb {E}}(|f(Y_\infty )|) < \infty$, we have

  $$\begin{aligned} \frac{1}{T} \int _0^T f(Y_s) \, {\mathrm {d}}s\, \mathop {\longrightarrow }\limits ^{{\mathrm {a.s.}}} {\mathbb {E}}(f(Y_\infty )) \qquad {\text {as }}T \rightarrow \infty . \end{aligned}$$

  (2.5)

In what follows we recall some limit theorems for continuous (local) martingales. We will use these limit theorems later on for studying the asymptotic behaviour of least-squares estimators of $(a, b, \alpha , \beta )$. First we recall a strong law of large numbers for continuous local martingales.

Theorem 2.5

(Liptser and Shiryaev [26, Lemma 17.4]) Let $\bigl ( \Omega , {\mathcal F}, ({\mathcal F}_t)_{t\in {\mathbb {R}}_+}, {\mathbb {P}}\bigr )$ be a filtered probability space satisfying the usual conditions. Let $(M_t)_{t\in {\mathbb {R}}_+}$ be a square-integrable continuous local martingale with respect to the filtration $({\mathcal F}_t)_{t\in {\mathbb {R}}_+}$ such that ${\mathbb {P}}(M_0 = 0) = 1$. Let $(\xi _t)_{t\in {\mathbb {R}}_+}$ be a progressively measurable process such that ${\mathbb {P}}\big ( \int _0^t \xi _u^2 \, {\mathrm {d}}\langle M \rangle _u < \infty \big ) = 1$, $t \in {\mathbb {R}}_+$, and

$$\begin{aligned} \int _0^t \xi _u^2 \, {\mathrm {d}} \langle M \rangle _u\, {\mathop {\longrightarrow }\limits ^{{\mathrm {a.s.}}}} \infty \qquad {\text {as} } \,t \rightarrow \infty , \end{aligned}$$

(2.6)

where $(\langle M \rangle _t)_{t\in {\mathbb {R}}_+}$ denotes the quadratic variation process of M. Then

$$\begin{aligned} \frac{\int _0^t \xi _u \, {\mathrm {d}}M_u}{\int _0^t \xi _u^2 \, {\mathrm {d}}\langle M \rangle _u}\, {\mathop {\longrightarrow }\limits ^{{\mathrm {a.s.}}}} 0 \qquad {\text {as} }\, t \rightarrow \infty . \end{aligned}$$

(2.7)

If $(M_t)_{t\in {\mathbb {R}}_+}$ is a standard Wiener process, the progressive measurability of $(\xi _t)_{t\in {\mathbb {R}}_+}$ can be relaxed to measurability and adaptedness to the filtration $({\mathcal F}_t)_{t\in {\mathbb {R}}_+}$.

The next theorem is about the asymptotic behaviour of continuous multivariate local martingales, see van Zanten [28, Theorem 4.1].

Theorem 2.6

(van Zanten [28, Theorem 4.1]) Let $\bigl ( \Omega , {\mathcal F}, ({\mathcal F}_t)_{t\in {\mathbb {R}}_+}, {\mathbb {P}}\bigr )$ be a filtered probability space satisfying the usual conditions. Let $({\varvec{M}}_t)_{t\in {\mathbb {R}}_+}$ be a d-dimensional square-integrable continuous local martingale with respect to the filtration $({\mathcal F}_t)_{t\in {\mathbb {R}}_+}$ such that ${\mathbb {P}}({\varvec{M}}_0 = {\varvec{0}}) = 1$. Suppose that there exists a function ${\varvec{Q}}: {\mathbb {R}}_+ \rightarrow {\mathbb {R}}^{d \times d}$ such that ${\varvec{Q}}(t)$ is an invertible (non-random) matrix for all $t \in {\mathbb {R}}_+$, $\lim _{t\rightarrow \infty } \Vert {\varvec{Q}}(t)\Vert = 0$ and

$$ \varvec{Q}(t)\langle \varvec{M}\rangle _{t} {\text{ }}\varvec{Q}(t)^{{ \top }} \xrightarrow{{\mathbb{P}}}\varvec{\eta \eta }^{{ \top }} \quad {\text{as}}\;t \to \infty , $$

where ${\varvec{\eta }}$ is a $d \times d$ random matrix. Then, for each ${\mathbb {R}}^k$-valued random vector ${\varvec{v}}$ defined on $(\Omega , {\mathcal F}, {\mathbb {P}})$, we have

$$\begin{aligned} ({\varvec{Q}}(t) {\varvec{M}}_t, {\varvec{v}})\, {\mathop {\longrightarrow } \limits^{{\mathcal L}}} ({\varvec{\eta }}{\varvec{Z}}, {\varvec{v}}) \qquad {\text {as} }\, t \rightarrow \infty , \end{aligned}$$

where ${\varvec{Z}}$ is a d-dimensional standard normally distributed random vector independent of $({\varvec{\eta }}, {\varvec{v}})$.

We note that Theorem 2.6 remains true if the function ${\varvec{Q}}$ is defined only on an interval $[t_0, \infty )$ with some $t_0 \in {\mathbb {R}}_{++}$.

3 Existence of LSE Based on Continuous-Time Observations

First, we define the LSE of $(a,b,\alpha ,\beta )$ based on discrete-time observations $(Y_{\frac{i}{n}},X_{\frac{i}{n}})_{i\in \{0,1,\ldots ,{\lfloor nT\rfloor }\}}$, $n\in \mathbb {N}$, $T\in {\mathbb {R}}_{++}$ [see (3.1)] by pointing out that the sum appearing in this definition of LSE can be considered as an approximation of the corresponding sum of the conditional LSE of $(a,b,\alpha ,\beta )$ based on discrete-time observations $(Y_{\frac{i}{n}},X_{\frac{i}{n}})_{i\in \{0,1,\ldots ,{\lfloor nT\rfloor }\}}$, $n\in \mathbb {N}$, $T\in {\mathbb {R}}_{++}$ (which was investigated in Barczy et al. [6]). Then, we introduce the LSE of $(a,b,\alpha ,\beta )$ based on continuous-time observations $(X_t)_{t\in [0,T]}$, $T\in {\mathbb {R}}_{++}$ [see (3.4) and (3.5)] as the limit in probability of the LSE of $(a,b,\alpha ,\beta )$ based on discrete-time observations $(Y_{\frac{i}{n}},X_{\frac{i}{n}})_{i\in \{0,1,\ldots ,{\lfloor nT\rfloor }\}}$, $n\in \mathbb {N}$, $T\in {\mathbb {R}}_{++}$ (see Proposition 3.1).

A LSE of $(a, b, \alpha , \beta )$ based on discrete-time observations $(Y_{\frac{i}{n}},X_{\frac{i}{n}})_{i\in \{0,1,\ldots ,{\lfloor nT\rfloor }\}}$, $n\in \mathbb {N}$, $T\in {\mathbb {R}}_{++}$, can be obtained by solving the extremum problem

$$\begin{aligned} \begin{aligned}&\left( \widehat{a}_{T,n}^{\mathrm {LSE,D}}, \widehat{b}_{T,n}^{\mathrm {LSE,D}}, \widehat{\alpha }_{T,n}^{\mathrm {LSE,D}}, \widehat{\beta }_{T,n}^{\mathrm {LSE,D}}\right) \\&:= \begin{array}{c} {{\mathrm{arg\,min}}}\\ {(a,b,\alpha ,\beta )\in {\mathbb {R}}^4} \end{array} \sum _{i=1}^{{\lfloor nT\rfloor }} \left[ \left( Y_{\frac{i}{n}} - Y_{\frac{i-1}{n}} - \frac{1}{n} \left( a - b Y_{\frac{i-1}{n}}\right) \right) ^2 \right. \\&\qquad \quad \left. +\, \left( X_{\frac{i}{n}} - X_{\frac{i-1}{n}} - \frac{1}{n} \left( \alpha - \beta Y_{\frac{i-1}{n}}\right) \right) ^2 \right] . \end{aligned} \end{aligned}$$

(3.1)

Here in the notations the letter $\mathrm {D}$ refers to discrete-time observations. This definition of LSE can be considered as the corresponding one given in Hu and Long [17, formula (1.2)] for generalized Ornstein-Uhlenbeck processes driven by $\alpha$-stable motions, see also Hu and Long [18, formula (3.1)]. For a heuristic motivation of the LSE (3.1) based on the discrete observations, see, e.g., Hu and Long [16, page 178] (formulated for Langevin equations), and for a mathematical one, see as follows. By (2.2), for all $i\in \mathbb {N}$,

$$\begin{aligned} Y_{\frac{i}{n}} - {\mathbb {E}}\left( Y_{\frac{i}{n}} \,|\,{\mathcal F}_{\frac{i-1}{n}}\right)&= Y_{\frac{i}{n}} - \mathrm {e}^{-{\frac{b}{n}}} Y_{\frac{i-1}{n}} - a\int _{\frac{i-1}{n}}^{\frac{i}{n}} \mathrm {e}^{-b({\frac{i}{n}}-u)}\,{\mathrm {d}}u\\&= Y_{\frac{i}{n}} - \mathrm {e}^{-{\frac{b}{n}}} Y_{\frac{i-1}{n}} - a\int _0^{\frac{1}{n}} \mathrm {e}^{-bv}\,{\mathrm {d}}v\\&= {\left\{ \begin{array}{ll} Y_{\frac{i}{n}} - Y_{\frac{i-1}{n}} - \frac{a}{n} &{}\quad {\text {if} }\, b=0,\\ Y_{\frac{i}{n}} - \mathrm {e}^{-{\frac{b}{n}}} Y_{\frac{i-1}{n}} + \frac{a}{b}(\mathrm {e}^{-{\frac{b}{n}}} - 1) &{}\quad {\text {if} }\, b\ne 0. \end{array}\right. } \end{aligned}$$

Using first-order Taylor approximation of $\mathrm {e}^{-{\frac{b}{n}}}$ at $b = 0$ by $1-{\frac{b}{n}}$, and that of $\frac{a}{b}(\mathrm {e}^{-{\frac{b}{n}}} - 1)$ at $(a, b) = (0, 0)$ by $-{\frac{a}{n}}$, the random variable $Y_{\frac{i}{n}} - Y_{\frac{i-1}{n}} - \frac{1}{n} (a - b Y_{\frac{i-1}{n}})$ in the definition (3.1) of the LSE of $(a,b,\alpha ,\beta )$ can be considered as a first-order Taylor approximation of

$$\begin{aligned} Y_{\frac{i}{n}} - {\mathbb {E}}\left( Y_{\frac{i}{n}} \,|\,Y_0,X_0,Y_{\frac{1}{n}},X_{\frac{1}{n}},\ldots ,Y_{\frac{i-1}{n}},X_{\frac{i-1}{n}}\right) = Y_{\frac{i}{n}} - {\mathbb {E}}\left( Y_{\frac{i}{n}}\,|\,{\mathcal F}_{\frac{i-1}{n}}\right) , \end{aligned}$$

which appears in the definition of the conditional LSE of $(a,b,\alpha ,\beta )$ based on discrete-time observations $(Y_{\frac{i}{n}},X_{\frac{i}{n}})_{i\in \{0,1,\ldots ,{\lfloor nT\rfloor }\}}$, $n\in \mathbb {N}$, $T\in {\mathbb {R}}_{++}$. Similarly, by (2.3), for all $i\in \mathbb {N}$,

$$\begin{aligned} X_{\frac{i}{n}} - {\mathbb {E}}(X_{\frac{i}{n}} \,|\,{\mathcal F}_{\frac{i-1}{n}})&= X_{\frac{i}{n}} - X_{\frac{i-1}{n}} - {\frac{\alpha }{n}}\\ {}&\quad + \beta Y_{\frac{i-1}{n}} \int _{\frac{i-1}{n}}^{\frac{i}{n}} \mathrm {e}^{-b(u - \frac{i-1}{n})}\,{\mathrm {d}}u + a\beta \int _{\frac{i-1}{n}}^{\frac{i}{n}}\left( \int _{\frac{i-1}{n}}^u \mathrm {e}^{-b(u-v)}\,{\mathrm {d}}v\right) {\mathrm {d}}u\\&= X_{\frac{i}{n}} - X_{\frac{i-1}{n}} - \frac{\alpha }{n} + \beta Y_{\frac{i-1}{n}} \int _0^{\frac{1}{n}} \mathrm {e}^{-bu}\,{\mathrm {d}}u + a\beta \int _0^{\frac{1}{n}}\left( \int _0^u \mathrm {e}^{-bv}\,{\mathrm {d}}v\right) {\mathrm {d}}u\\&= {\left\{ \begin{array}{ll} X_{\frac{i}{n}} - X_{\frac{i-1}{n}} - \frac{\alpha }{n} + \frac{\beta }{n} Y_{\frac{i-1}{n}} + \frac{a\beta }{2n^2} &{}\quad {\text {if} } \, b=0,\\ X_{\frac{i}{n}} - X_{\frac{i-1}{n}} - \frac{\alpha }{n} + \frac{\beta }{b}(1-\mathrm {e}^{-\frac{b}{n}}) Y_{\frac{i-1}{n}} + \frac{a\beta }{b}\bigl (\frac{1}{n} - \frac{1-\mathrm {e}^{-\frac{b}{n}}}{b}\bigr ) &{}\quad {\text {if }} b\ne 0. \end{array}\right. } \end{aligned}$$

Using first-order Taylor approximation of $\frac{a\beta }{2n^2}$ at $(a, \beta ) = (0, 0)$ by 0, that of $\frac{\beta }{b}(1-\mathrm {e}^{-\frac{b}{n}})$ at $(b, \beta ) = (0, 0)$ by $\frac{\beta }{n}$, and that of $\frac{a\beta }{b}\bigl (\frac{1}{n} - \frac{1-\mathrm {e}^{-\frac{b}{n}}}{b}\bigr ) = \frac{a\beta }{n^2}\sum _{k=0}^\infty (-1)^k \frac{(b/n)^k}{(k+2)!}$ at $(a, b, \beta ) = (0, 0, 0)$ by 0, the random variable $X_{\frac{i}{n}} - X_{\frac{i-1}{n}} - \frac{1}{n} (\alpha - \beta Y_{\frac{i-1}{n}})$ in the definition (3.1) of the LSE of $(a,b,\alpha ,\beta )$ can be considered as a first-order Taylor approximation of

$$\begin{aligned} X_{\frac{i}{n}} - {\mathbb {E}}(X_{\frac{i}{n}} \,|\,Y_0,X_0,Y_{\frac{1}{n}},X_{\frac{1}{n}},\ldots ,Y_{\frac{i-1}{n}},X_{\frac{i-1}{n}} ) = X_{\frac{i}{n}} - {\mathbb {E}}(X_{\frac{i}{n}} \,|\,{\mathcal F}_{\frac{i-1}{n}}) , \end{aligned}$$

which appears in the definition of the conditional LSE of $(a,b,\alpha ,\beta )$ based on discrete-time observations $(Y_{\frac{i}{n}},X_{\frac{i}{n}})_{i\in \{0,1,\ldots ,{\lfloor nT\rfloor }\}}$, $n\in \mathbb {N}$, $T\in {\mathbb {R}}_{++}$.

We note that in Barczy et al. [6] we proved strong consistency and asymptotic normality of conditional LSE of $(a,b,\alpha ,\beta )$ based on discrete-time observations $(Y_i,X_i)_{i\in \{1,\ldots ,n\}}$, $n\in \mathbb {N}$, starting the process from some known non-random initial value $(y_0,x_0)\in {\mathbb {R}}_{++}\times {\mathbb {R}}$, as the sample size n tends to infinity in the subcritical case.

Solving the extremum problem (3.1), we have

$$\begin{aligned} \bigl (\widehat{a}_{T,n}^{\mathrm {LSE,D}}, \widehat{b}_{T,n}^{\mathrm {LSE,D}}\bigr )&= \begin{array}{c} {{\mathrm{arg\,min}}}\\ {(a,b)\in {\mathbb {R}}^2} \end{array} \sum _{i=1}^{{\lfloor nT\rfloor }} \left( Y_{\frac{i}{n}} - Y_{\frac{i-1}{n}} - \frac{1}{n}\left( a - b Y_{\frac{i-1}{n}}\right) \right) ^2 , \\ \bigl (\widehat{\alpha }_{T,n}^{\mathrm {LSE,D}}, \widehat{\beta }_{T,n}^{\mathrm {LSE,D}}\bigr )&=\begin{array}{c} {{\mathrm{arg\,min}}}\\ {(\alpha ,\beta )\in {\mathbb {R}}^2} \end{array} \sum _{i=1}^{{\lfloor nT\rfloor }} \left( X_{\frac{i}{n}} - X_{\frac{i-1}{n}} - \frac{1}{n}\left( \alpha - \beta Y_{\frac{i-1}{n}}\right) \right) ^2 , \end{aligned}$$

hence, similarly as on page 675 in Barczy et al. [3], we get

$$ \left[ {\begin{array}{l} {\hat{a}_{{T,n}}^{{LSE,D}} } \hfill \\ {\hat{b}_{{T,n}}^{{LSE,D}} } \hfill \\ \end{array} } \right] = n\,\left[ {\begin{array}{ll} {{ \lfloor }nT{ \rfloor }} \hfill & { - \sum\nolimits_{{i = 1}}^{{{ \lfloor }nT{ \rfloor }}} {Y_{{\frac{{i - 1}}{n}}} } } \hfill \\ { - \sum\nolimits_{{i = 1}}^{{{ \lfloor }nT{ \rfloor }}} {Y_{{\frac{{i - 1}}{n}}} } } \hfill & {\sum\nolimits_{{i = 1}}^{{{ \lfloor }nT{ \rfloor }}} {Y_{{\frac{{i - 1}}{n}}}^{2} } } \hfill \\ \end{array} } \right]^{{ - 1}} \left[ {\begin{array}{l} {Y_{\frac{\lfloor nT\rfloor}{n}} - Y_{0} } \hfill \\ { - \sum\nolimits_{{i = 1}}^{{{ \lfloor }nT{ \rfloor }}} {(Y_{{\frac{i}{n}}} - Y_{{\frac{{i - 1}}{n}}} )} Y_{{\frac{{i - 1}}{n}}} } \hfill \\ \end{array} } \right], $$

(3.2)

and

$$ \left[ {\begin{array}{l} {\hat{\alpha }_{{T,n}}^{{{\text{LSE}},{\text{D}}}} } \\ {\hat{\beta }_{{T,n}}^{{{\text{LSE}},{\text{D}}}} } \\ \end{array} } \right] = n\,\left[ {\begin{array}{ll} {{ \lfloor }nT{ \rfloor }} \hfill & { - \sum\nolimits_{{i = 1}}^{{{ \lfloor }nT{ \rfloor }}} {Y_{{\frac{{i - 1}}{n}}} } } \hfill \\ { - \sum\nolimits_{{i = 1}}^{{{ \lfloor }nT{ \rfloor }}} {Y_{{\frac{{i - 1}}{n}}} } } \hfill & {\sum\nolimits_{{i = 1}}^{{{ \lfloor }nT{ \rfloor }}} {Y_{{\frac{{i - 1}}{n}}}^{2} } } \hfill \\ \end{array} } \right]^{{ - 1}} \left[ {\begin{array}{ll} {X_{{\frac{{{ \lfloor }nT{ \rfloor }}}{n}}} - X_{0} } \hfill \\ { - \sum\nolimits_{{i = 1}}^{{{ \lfloor }nT{ \rfloor }}} {(X_{{\frac{i}{n}}} - X_{{\frac{{i - 1}}{n}}} )} Y_{{\frac{{i - 1}}{n}}} } \hfill \\ \end{array} } \right], $$

(3.3)

provided that the inverse exists, i.e., ${\lfloor nT\rfloor }\sum _{i=1}^{{\lfloor nT\rfloor }} Y_{\frac{i-1}{n}}^2 > \left( \sum _{i=1}^{{\lfloor nT\rfloor }} Y_{\frac{i-1}{n}}\right) ^2$. By Lemma 3.1 in Barczy et al. [6], for all $n\in \mathbb {N}$ and $T\in {\mathbb {R}}_{++}$ with ${\lfloor nT\rfloor }\geqslant 2$, we have ${\mathbb {P}}\left( {\lfloor nT\rfloor }\sum _{i=1}^{{\lfloor nT\rfloor }} Y_{\frac{i-1}{n}}^2 > \left( \sum _{i=1}^{{\lfloor nT\rfloor }} Y_{\frac{i-1}{n}}\right) ^2\right) =1$.

Proposition 3.1

If $a \in {\mathbb {R}}_{++}$, $b \in {\mathbb {R}}$, $\alpha ,\beta \in {\mathbb {R}}$, $\sigma _1,\sigma _2\in {\mathbb {R}}_{++}$, $\varrho \in (-1,1)$, and ${\mathbb {P}}(Y_0 \in {\mathbb {R}}_{++})=1$, then for any $T\in {\mathbb {R}}_{++}$, we have

$$ \left[ {\begin{array}{l} {\hat{a}_{{T,n}}^{{{\text{LSE,D}}}} } \hfill \\ {\hat{b}_{{T,n}}^{{{\text{LSE,D}}}} } \hfill \\ {\hat{\alpha }_{{T,n}}^{{{\text{LSE}},{\text{D}}}} } \hfill \\ {\hat{\beta }_{{T,n}}^{{{\text{LSE}},{\text{D}}}} } \hfill \\ \end{array} } \right]\mathop \to \limits^{{\mathbb{P}}} \left[ {\begin{array}{l} {\hat{a}_{T}^{{{\text{LSE}}}} } \hfill \\ {\hat{b}_{T}^{{{\text{LSE}}}} } \hfill \\ {\hat{\alpha }_{T}^{{{\text{LSE}}}} } \hfill \\ {\hat{\beta }_{T}^{{{\text{LSE}}}} } \hfill \\ \end{array} } \right]\qquad {\text{as }}n \to \infty , $$

where

$$ \begin{aligned} \left[ {\begin{array}{l} {\hat{a}_{T}^{{{\text{LSE}}}} } \\ {\hat{b}_{T}^{{{\text{LSE}}}} } \\ \end{array} } \right] & : = \left[ {\begin{array}{ll} T \hfill & { - \int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s} \hfill \\ { - \int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s} \hfill & {\int_{0}^{T} {Y_{s}^{2} } {\mkern 1mu} {\text{d}}s} \hfill \\ \end{array} } \right]^{{ - 1}} \left[ {\begin{array}{l} {Y_{T} - Y_{0} } \\ { - \int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}Y_{s} } \\ \end{array} } \right] \\ & = \frac{1}{{T\int_{0}^{T} {Y_{s}^{2} } {\mkern 1mu} {\text{d}}s - \left( {\int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s} \right)^{2} }}\left[ {\begin{array}{l} {(Y_{T} - Y_{0} )\int_{0}^{T} {Y_{s}^{2} } {\mkern 1mu} {\text{d}}s - \int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s\int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}Y_{s} } \\ {(Y_{T} - Y_{0} )\int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s - T\int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}Y_{s} } \\ \end{array} } \right], \\ \end{aligned} $$

(3.4)

and

$$ \begin{aligned} \left[ {\begin{array}{l} {\hat{\alpha }_{T}^{{{\text{LSE}}}} } \\ {\hat{\beta }_{T}^{{{\text{LSE}}}} } \\ \end{array} } \right] & : = \left[ {\begin{array}{ll} T & { - \int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s} \\ { - \int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s} & {\int_{0}^{T} {Y_{s}^{2} } {\mkern 1mu} {\text{d}}s} \\ \end{array} } \right]^{{ - 1}} \left[ {\begin{array}{l} {X_{T} - X_{0} } \\ { - \int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}X_{s} } \\ \end{array} } \right] \\ & = \frac{1}{{T\int_{0}^{T} {Y_{s}^{2} } {\mkern 1mu} {\text{d}}s - \left( {\int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s} \right)^{2} }}\left[ {\begin{array}{l} {(X_{T} - X_{0} )\int_{0}^{T} {Y_{s}^{2} } {\mkern 1mu} {\text{d}}s - \int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s\int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}X_{s} } \\ {(X_{T} - X_{0} )\int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s - T\int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}X_{s} } \\ \end{array} } \right], \\ \end{aligned} $$

(3.5)

which exist almost surely, since

$$\begin{aligned} {\mathbb {P}}\left( T \int _0^T Y_s^2 \, {\mathrm {d}}s > \left( \int _0^T Y_s \, {\mathrm {d}}s\right) ^2 \right) = 1 \qquad \text {for} \,\, \text{all }T \in {\mathbb {R}}_{++}. \end{aligned}$$

(3.6)

By definition, we call $\bigl (\widehat{a}_T^{\mathrm {LSE}}, \widehat{b}_T^{\mathrm {LSE}}, \widehat{\alpha }_T^{\mathrm {LSE}}, \widehat{\beta }_T^{\mathrm {LSE}}\bigr )$ the LSE of $(a, b, \alpha , \beta )$ based on continuous-time observations $(X_t)_{t\in [0,T]}$, $T\in {\mathbb {R}}_{++}$.

Proof

First, we check (3.6). Note that ${\mathbb {P}}(\int _0^T Y_s \, {\mathrm {d}}s < \infty ) = 1$ and ${\mathbb {P}}(\int _0^T Y_s^2 \, {\mathrm {d}}s < \infty ) = 1$ for all $T \in {\mathbb {R}}_+$, since Y has continuous trajectories almost surely. For each $T \in {\mathbb {R}}_{++}$, put

$$\begin{aligned} A_T := \{ \omega \in \Omega : t \mapsto Y_t(\omega ) \ \text {is continuous and non-negative on }[0,T] \} . \end{aligned}$$

Then $A_T \in {\mathcal F}$, ${\mathbb {P}}(A_T) = 1$, and for all $\omega \in A_T$, by the Cauchy–Schwarz’s inequality, we have

$$\begin{aligned} T \int _0^T Y_s(\omega )^2 \, {\mathrm {d}}s \geqslant \left( \int _0^T Y_s(\omega ) \, {\mathrm {d}}s\right) ^2 , \end{aligned}$$

and $T \int _0^T Y_s(\omega )^2 \, {\mathrm {d}}s - \left( \int _0^T Y_s(\omega ) \, {\mathrm {d}}s\right) ^2 = 0$ if and only if $Y_s(\omega ) = K_T(\omega )$ for almost every $s \in [0, T]$ with some $K_T(\omega ) \in {\mathbb {R}}_+$. Hence $Y_s(\omega ) = Y_0(\omega )$ for all $s \in [0, T]$ if $\omega \in A_T$ and $T \int _0^T Y_s^2(\omega ) \, {\mathrm {d}}s - \left( \int _0^T Y_s(\omega ) \, {\mathrm {d}}s\right) ^2 = 0$. Consequently, using that ${\mathbb {P}}(A_T)=1$, we have

$$\begin{aligned} {\mathbb {P}}\left( T \int _0^T Y_s^2 \, {\mathrm {d}}s - \left( \int _0^T Y_s \, {\mathrm {d}}s\right) ^2 = 0 \right)&= {\mathbb {P}}\left( \left\{ T \int _0^T Y_s^2 \, {\mathrm {d}}s - \left( \int _0^T Y_s \, {\mathrm {d}}s\right) ^2 = 0 \right\} \cap A_T\right) \\&\leqslant {\mathbb {P}}(Y_s = Y_0,\;\forall \,s\in [0,T]) \leqslant {\mathbb {P}}(Y_T = Y_0) =0, \end{aligned}$$

where the last equality follows by the fact that $Y_T$ is absolutely continuous (see, e.g., Alfonsi [2, Proposition 1.2.11]) together with the law of total probability. Hence ${\mathbb {P}}\Big (T \int _0^T Y_s^2 \, {\mathrm {d}}s - \Big (\int _0^T Y_s \, {\mathrm {d}}s\Big )^2 = 0 \Big ) = 0$, yielding (3.6).

Further, we have

$$\frac{1}{n}\left[ {\begin{array}{ll} {\left\lfloor {nT} \right\rfloor } \hfill & { - \sum\nolimits_{{i = 1}}^{{\left\lfloor {nT} \right\rfloor }} {Y_{{\frac{{i - 1}}{n}}} } } \hfill \\ { - \sum\nolimits_{{i = 1}}^{{\left\lfloor {nT} \right\rfloor }} {Y_{{\frac{{i - 1}}{n}}} } } \hfill & {\sum\nolimits_{{i = 1}}^{{\left\lfloor {nT} \right\rfloor }} {Y_{{\frac{{i - 1}}{n}}}^{2} } } \hfill \\ \end{array} } \right]\;\underrightarrow {a.s.}\;\left[ {\begin{array}{ll} T \hfill & { - \int_{0}^{T} {Y_{s} } {\mkern 1mu} ds} \hfill \\ { - \int_{0}^{T} {Y_{s} } {\mkern 1mu} ds} \hfill & {\int_{0}^{T} {Y_{s}^{2} } {\mkern 1mu} ds} \hfill \\ \end{array} } \right]\quad as{\text{ }}n \to \infty , $$

since $(Y_t)_{t\in {\mathbb {R}}_+}$ is almost surely continuous. By Proposition I.4.44 in Jacod and Shiryaev [21] with the Riemann sequence of deterministic subdivisions $\bigl (\frac{i}{n} \wedge T\bigr )_{i\in \mathbb {N}}$, $n \in \mathbb {N}$, and using the almost sure continuity of $(Y_t, X_t)_{t\in {\mathbb {R}}_+}$, we obtain

$$\begin{aligned} & \left[ {\begin{array}{l} {Y_{{\frac{{\left\lfloor {nT} \right\rfloor }}{n}}} - Y_{0} } \hfill \\ { - \sum\nolimits_{{i = 1}}^{{\left\lfloor {nT} \right\rfloor }} {(Y_{{\frac{i}{n}}} - Y_{{\frac{{i - 1}}{n}}} )} Y_{{\frac{{i - 1}}{n}}} } \hfill \\ \end{array} } \right] \underrightarrow {\mathbb{P}}\left[ {\begin{array}{l} {Y_{T} - Y_{0} } \hfill \\ { - \int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}Y_{s} } \hfill \\ \end{array} } \right]\quad {\text{as }}n \to \infty , \\ & \left[ {\begin{array}{l} {X_{{\frac{{\left\lfloor {nT} \right\rfloor }}{n}}} - X_{0} } \hfill \\ { - \sum\nolimits_{{i = 1}}^{{\left\lfloor {nT} \right\rfloor }} {(X_{{\frac{i}{n}}} - X_{{\frac{{i - 1}}{n}}} )} Y_{{\frac{{i - 1}}{n}}} } \hfill \\ \end{array} } \right] \underrightarrow {\mathbb{P}}\left[ {\begin{array}{l} {X_{T} - X_{0} } \hfill \\ { - \int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}X_{s} } \hfill \\ \end{array} } \right]\quad {\text{as }}n \to \infty . \\ \end{aligned}$$

By Slutsky’s lemma, using also (3.2), (3.3) and (3.6), we obtain the assertion.□

Note that Proposition 3.1 is valid for all $b\in {\mathbb {R}}$, i.e., not only for subcritical Heston models.

We call the attention that $(\widehat{a}_T^{\mathrm {LSE}} , \widehat{b}_T^{\mathrm {LSE}} , \widehat{\alpha }_T^{\mathrm {LSE}} , \widehat{\beta }_T^{\mathrm {LSE}})$ can be considered to be based only on $(X_t)_{t\in [0,T]}$, since the process $(Y_t)_{t\in [0,T]}$ can be determined using the observations $(X_t)_{t\in [0,T]}$ and the initial value $Y_0$, see Barczy and Pap [5, Remark 2.5]. We also point out that Overbeck and Rydén [27, formulae (22) and (23)] have already come up with the definition of LSE $(\widehat{a}_T^{\mathrm {LSE}} , \widehat{b}_T^{\mathrm {LSE}} )$ of (a, b) based on continuous-time observations $(Y_t)_{t\in [0,T]}$, $T \in {\mathbb {R}}_{++}$, for the CIR process Y. They investigated only the CIR process Y, so our definitions (3.4) and (3.5) can be considered as generalizations of formulae (22) and (23) in Overbeck and Rydén [27] for the Heston model (1.1). Overbeck and Rydén [27, Theorem 3.4] also proved that the LSE of (a, b) based on continuous-time observations can be approximated in probability by conditional LSEs of (a, b) based on appropriate discrete-time observations.

In the next remark, we point out that the LSE of $(a,b,\alpha ,\beta )$ given in (3.4) and (3.5) can be approximated using discrete-time observations for X, which can be reassuring for practical applications, where data in continuous record is not available.

Remark 3.2

The stochastic integral $\int _0^T Y_s \, {\mathrm {d}}Y_s$ in (3.4) is a measurable function of $(X_s)_{s\in [0,T]}$ and $Y_0$. Indeed, for all $t\in [0,T]$, $Y_t$ and $\int _0^t Y_s\,{\mathrm {d}}s$ are measurable functions of $(X_s)_{s\in [0,T]}$ and $Y_0$, i.e., they can be determined from a sample $(X_s)_{s\in [0,T]}$ and $Y_0$ following from a slight modification of Remark 2.5 in Barczy and Pap [5] (replacing $y_0$ by $Y_0$), and, by Itô’s formula, we have ${\mathrm {d}}(Y_t^2) = 2 Y_t \,{\mathrm {d}}Y_t + \sigma _1^2 Y_t \, {\mathrm {d}}t$, $t \in {\mathbb {R}}_+$, implying that $\int _0^T Y_s \, {\mathrm {d}}Y_s = \frac{1}{2} \big ( Y_T^2 - Y_0^2 - \sigma _1^2 \int _0^T Y_s \, {\mathrm {d}}s \big )$, $T\in {\mathbb {R}}_+$. For the stochastic integral $\int _0^T Y_s \, {\mathrm {d}}X_s$ in (3.5), we have

$$\begin{aligned} \sum _{i=1}^{\lfloor nT\rfloor }Y_{\frac{i-1}{n}} (X_{\frac{i}{n}} - X_{\frac{i-1}{n}})\, {\mathop {\longrightarrow }\limits ^{{\mathbb {P}}}}\int _0^T Y_s \, {\mathrm {d}}X_s \qquad \text {as } n \rightarrow \infty , \end{aligned}$$

(3.7)

following from Proposition I.4.44 in Jacod and Shiryaev [21] with the Riemann sequence of deterministic subdivisions $\left( \frac{i}{n} \wedge T\right) _{i\in \mathbb {N}}$, $n \in \mathbb {N}$. Thus, there exists a measurable function $\Phi : C([0,T],{\mathbb {R}})\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ such that $\int _0^T Y_s \, {\mathrm {d}}X_s = \Phi ((X_s)_{s\in [0,T]},Y_0)$, since the convergence in (3.7) holds almost surely along a suitable subsequence, for each $n \in \mathbb {N}$, the members of the sequence in (3.7) are measurable functions of $(X_s)_{s\in [0,T]}$ and $Y_0$, and one can use Theorems 4.2.2 and 4.2.8 in Dudley [13]. Hence, the right-hand sides of (3.4) and (3.5) are measurable functions of $(X_s)_{s\in [0,T]}$ and $Y_0$, i.e., they are statistics.□

Using the SDE (1.1) and Corollary 3.2.20 in Karatzas and Shreve [22], one can check that

$$ \begin{aligned} \left[ {\begin{array}{c} {\hat{a}_{T}^{{{\text{LSE}}}} - a} \\ {\hat{b}_{T}^{{{\text{LSE}}}} - b} \\ \end{array} } \right] & = \left[ {\begin{array}{ll} T \hfill & { - \int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s} \hfill \\ { - \int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s} \hfill & {\int_{0}^{T} {Y_{s}^{2} } {\mkern 1mu} {\text{d}}s} \hfill \\ \end{array} } \right]^{{ - 1}} \left[ {\begin{array}{l} {\sigma _{1} \int_{0}^{T} {Y_{s}^{{1/2}} } {\mkern 1mu} {\text{d}}W_{s} } \hfill \\ { -\, \sigma _{1} \int_{0}^{T} {Y_{s}^{{3/2}} } {\mkern 1mu} {\text{d}}W_{s} } \hfill \\ \end{array} } \right], \\ \left[ {\begin{array}{c} {\hat{\alpha }_{T}^{{{\text{LSE}}}} - \alpha } \\ {\hat{\beta }_{T}^{{{\text{LSE}}}} - \beta } \\ \end{array} } \right] & = \left[ {\begin{array}{ll} T \hfill & { - \int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s} \hfill \\ { - \int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s} \hfill & {\int_{0}^{T} {Y_{s}^{2} } {\mkern 1mu} {\text{d}}s} \hfill \\ \end{array} } \right]^{{ - 1}} \left[ {\begin{array}{l} {\sigma _{2} \int_{0}^{T} {Y_{s}^{{1/2}} } {\mkern 1mu} {\text{d}}\widetilde{W}_s} \hfill \\ { -\,\sigma _{2} \int_{0}^{T} {Y_{s}^{{3/2}} } {\mkern 1mu} {\text{d}}\widetilde{W}_s} \hfill \\ \end{array} } \right], \\ \end{aligned} $$

provided that $T \int _0^T Y_s^2 \, {\mathrm {d}}s > \left( \int _0^T Y_s \, {\mathrm {d}}s\right) ^2$, where $\widetilde{W}_t:=\varrho W_t + \sqrt{1-\varrho ^2} B_t$, $t\in {\mathbb {R}}_+$, and hence

$$\begin{aligned} \begin{aligned} \widehat{a}_T^{\mathrm {LSE}} - a&= \frac{\sigma _1 \left( \int _0^T Y_s^{1/2} \, {\mathrm {d}}W_s\right) \left( \int _0^T Y_s^2 \, {\mathrm {d}}s\right) - \sigma _1 \left( \int _0^T Y_s \, {\mathrm {d}}s\right) \left( \int _0^T Y_s^{3/2} \, {\mathrm {d}}W_s\right) }{T \int _0^T Y_s^2 \, {\mathrm {d}}s - \left( \int _0^T Y_s \, {\mathrm {d}}s\right) ^2} , \\ \widehat{b}_T^{\mathrm {LSE}} - b&= \frac{\sigma _1 \left( \int _0^T Y_s^{1/2} \, {\mathrm {d}}W_s\right) \left( \int _0^T Y_s \, {\mathrm {d}}s\right) - \sigma _1 T \int _0^T Y_s^{3/2} \, {\mathrm {d}}W_s}{T \int _0^T Y_s^2 \, {\mathrm {d}}s - \left( \int _0^T Y_s \, {\mathrm {d}}s\right) ^2} , \\ \widehat{\alpha }_T^{\mathrm {LSE}} - \alpha&= \frac{\sigma _2 \left( \int _0^T Y_s^{1/2} \, {\mathrm {d}}\widetilde{W}_s\right) \left( \int _0^T Y_s^2 \, {\mathrm {d}}s\right) - \sigma _2 \left( \int _0^T Y_s \, {\mathrm {d}}s\right) \left( \int _0^T Y_s^{3/2} \, {\mathrm {d}}\widetilde{W}_s\right) }{T \int _0^T Y_s^2 \, {\mathrm {d}}s - \left( \int _0^T Y_s \, {\mathrm {d}}s\right) ^2} , \\ \widehat{\beta }_T^{\mathrm {LSE}} - \beta&= \frac{\sigma _2 \left( \int _0^T Y_s^{1/2} \, {\mathrm {d}}\widetilde{W}_s\right) \left( \int _0^T Y_s \, {\mathrm {d}}s\right) - \sigma _2 T \int _0^T Y_s^{3/2} \, {\mathrm {d}}\widetilde{W}_s}{T \int _0^T Y_s^2 \, {\mathrm {d}}s - \left( \int _0^T Y_s \, {\mathrm {d}}s\right) ^2} , \end{aligned} \end{aligned}$$

(3.8)

provided that $T \int _0^T Y_s^2 \, {\mathrm {d}}s > \left( \int _0^T Y_s \, {\mathrm {d}}s\right) ^2$.

4 Consistency and Asymptotic Normality of LSE

Our first result is about the consistency of LSE in case of subcritical Heston models.

Theorem 4.1

If $a, b, \sigma _1, \sigma _2 \in {\mathbb {R}}_{++}$, $\alpha , \beta \in {\mathbb {R}}$, $\varrho \in (-1, 1)$, and ${\mathbb {P}}((Y_0,X_0) \in {\mathbb {R}}_{++}\times {\mathbb {R}})=1$, then the LSE of $(a, b, \alpha , \beta )$ is strongly consistent, i.e., $\bigl (\widehat{a}_T^{\mathrm {LSE}}, \widehat{b}_T^{\mathrm {LSE}}, \widehat{\alpha }_T^{\mathrm {LSE}}, \widehat{\beta }_T^{\mathrm {LSE}}\bigr )\, {\mathop {\longrightarrow }\limits ^{{\mathrm {a.s.}}}}(a, b, \alpha , \beta )$ as $T \rightarrow \infty$.

Proof

By Proposition 3.1, there exists a unique LSE $\bigl (\widehat{a}^{\mathrm {LSE}}_T, \widehat{b}^{\mathrm {LSE}}_T, \widehat{\alpha }^{\mathrm {LSE}}_T, \widehat{\beta }^{\mathrm {LSE}}_T\bigr )$ of $(a, b, \alpha , \beta )$ for all $T\in {\mathbb {R}}_{++}$. By (3.8), we have

$$\begin{aligned} \widehat{a}^{\mathrm {LSE}}_T - a = \frac{\sigma _1 \cdot \frac{1}{T} \int _0^T Y_s \, {\mathrm {d}}s \cdot \frac{1}{T} \int _0^T Y_s^2 \, {\mathrm {d}}s \cdot \frac{\int _0^T Y_s^{1/2} \, {\mathrm {d}}W_s}{\int _0^T Y_s \, {\mathrm {d}}s} - \sigma _1\cdot \frac{1}{T} \int _0^T Y_s \, {\mathrm {d}}s \cdot \frac{1}{T} \int _0^T Y_s^3 \, {\mathrm {d}}s \cdot \frac{\int _0^T Y_s^{3/2} \, {\mathrm {d}}W_s}{\int _0^T Y_s^3 \, {\mathrm {d}}s}}{ \frac{1}{T} \int _0^T Y_s^2 \, {\mathrm {d}}s - \left( \frac{1}{T} \int _0^T Y_s \, {\mathrm {d}}s\right) ^2} \end{aligned}$$

provided that $\int _0^T Y_s \, {\mathrm {d}}s \in {\mathbb {R}}_{++}$, which holds almost surely, see the proof of Proposition 3.1. Since, by part (1) of Theorem 2.4, ${\mathbb {E}}(Y_\infty )$, ${\mathbb {E}}(Y_\infty ^2)$, ${\mathbb {E}}(Y_\infty ^3) \in {\mathbb {R}}_{++}$, part (3) of Theorem 2.4 yields

$$\begin{aligned} \frac{1}{T} \int _0^T Y_s \, {\mathrm {d}}s\, {\mathop {\longrightarrow }\limits ^{{\mathrm {a.s.}}}}{\mathbb {E}}(Y_\infty ) , \qquad \frac{1}{T} \int _0^T Y_s^2 \, {\mathrm {d}}s\, {\mathop {\longrightarrow }\limits ^{{\mathrm {a.s.}}}}{\mathbb {E}}(Y_\infty ^2) , \qquad \frac{1}{T} \int _0^T Y_s^3 \, {\mathrm {d}}s\, {\mathop {\longrightarrow }\limits ^{{\mathrm {a.s.}}}}{\mathbb {E}}(Y_\infty ^3) \end{aligned}$$

as $T \rightarrow \infty$, and then

$$\begin{aligned} \int _0^T Y_s \, {\mathrm {d}}s\, {\mathop {\longrightarrow }\limits ^{{\mathrm {a.s.}}}}\infty , \qquad \int _0^T Y_s^2 \, {\mathrm {d}}s\, {\mathop {\longrightarrow }\limits ^{{\mathrm {a.s.}}}}\infty , \qquad \int _0^T Y_s^3 \, {\mathrm {d}}s\, {\mathop {\longrightarrow }\limits ^{{\mathrm {a.s.}}}}\infty \end{aligned}$$

as $T \rightarrow \infty$. Hence, by a strong law of large numbers for continuous local martingales (see, e.g., Theorem 2.5), we obtain

$$\begin{aligned} \widehat{a}^{\mathrm {LSE}}_T - a\, {\mathop {\longrightarrow }\limits ^{{\mathrm {a.s.}}}}\frac{\sigma _1 \cdot {\mathbb {E}}(Y_\infty ) \cdot {\mathbb {E}}(Y_\infty ^2) \cdot 0 - \sigma _1 \cdot {\mathbb {E}}(Y_\infty ) \cdot {\mathbb {E}}(Y_\infty ^3) \cdot 0}{{\mathbb {E}}(Y_\infty ^2) - ({\mathbb {E}}(Y_\infty ))^2} = 0 \qquad \text {as }T \rightarrow \infty , \end{aligned}$$

where for the last step we also used that ${\mathbb {E}}(Y_\infty ^2) - ({\mathbb {E}}(Y_\infty ))^2 = \frac{a\sigma _1^2}{2b^2} \in {\mathbb {R}}_{++}$.

Similarly, by (3.8),

$$\begin{aligned} \widehat{b}^{\mathrm {LSE}}_T - b&= \frac{\sigma _1 \cdot \left( \frac{1}{T} \int _0^T Y_s \, {\mathrm {d}}s\right) ^2 \cdot \frac{\int _0^T Y_s^{1/2} \, {\mathrm {d}}W_s}{\int _0^T Y_s \, {\mathrm {d}}s} - \sigma _1 \cdot \frac{1}{T} \int _0^T Y_s^3 \, {\mathrm {d}}s \cdot \frac{\int _0^T Y_s^{3/2} \, {\mathrm {d}}W_s}{\int _0^T Y_s^3 \, {\mathrm {d}}s}}{\frac{1}{T} \int _0^T Y_s^2 \, {\mathrm {d}}s - \left( \frac{1}{T} \int _0^T Y_s \, {\mathrm {d}}s\right) ^2} \\&{\mathop {\longrightarrow }\limits ^{{\mathrm {a.s.}}}}\frac{\sigma _1 \cdot ({\mathbb {E}}(Y_\infty ))^2 \cdot 0 - \sigma _1 \cdot {\mathbb {E}}(Y_\infty ^3) \cdot 0}{{\mathbb {E}}(Y_\infty ^2) - ({\mathbb {E}}(Y_\infty ))^2} = 0 \qquad \text {as }T \rightarrow \infty . \end{aligned}$$

One can prove

$$\begin{aligned} \widehat{\alpha }^{\mathrm {LSE}}_T - \alpha\, {\mathop {\longrightarrow }\limits ^{{\mathrm {a.s.}}}}0 \qquad \text {and} \qquad \widehat{\beta }^{\mathrm {LSE}}_T - \beta\, {\mathop {\longrightarrow }\limits ^{{\mathrm {a.s.}}}}0 \qquad \text {as }T \rightarrow \infty \end{aligned}$$

in a similar way.□

Our next result is about the asymptotic normality of LSE in case of subcritical Heston models.

Theorem 4.2

If $a, b, \sigma _1, \sigma _2 \in {\mathbb {R}}_{++}$, $\alpha , \beta \in {\mathbb {R}}$, $\varrho \in (-1, 1)$ and ${\mathbb {P}}((Y_0,X_0) \in {\mathbb {R}}_{++}\times {\mathbb {R}})=1$, then the LSE of $(a, b, \alpha , \beta )$ is asymptotically normal, i.e.,

$$ T^{{\frac{1}{2}}} \left[ {\begin{array}{l} {\hat{a}_{T}^{{{\text{LSE}}}} - a} \hfill \\ {\hat{b}_{T}^{{{\text{LSE}}}} - b} \hfill \\ {\hat{\alpha }_{T}^{{{\text{LSE}}}} - \alpha } \hfill \\ {\hat{\beta }_{T}^{{{\text{LSE}}}} - \beta } \hfill \\ \end{array} } \right]\mathop \to \limits^{{\mathcal{L}}} {\mathcal{N}}_{4} \left( {{\mathbf{0}},\varvec{S} \otimes \left[ \begin{array}{ll} {\frac{{(2a + \sigma _{1}^{2} )a}}{{\sigma _{1}^{2} b}}} \hfill & {\frac{{2a + \sigma _{1}^{2} }}{{\sigma _{1}^{2} }}} \hfill \\ {\frac{{2a + \sigma _{1}^{2} }}{{\sigma _{1}^{2} }}} \hfill & {\frac{{2b(a + \sigma _{1}^{2} )}}{{\sigma _{1}^{2} a}}} \hfill \\ \end{array} \right] } \right)\qquad {\text{as }}T \to \infty , $$

(4.1)

where $\otimes$ denotes the tensor product of matrices, and

$$\varvec{S}: = \left[ {\begin{array}{ll} {\sigma _{1}^{2} } & {{\varrho }\sigma _{1} \sigma _{2} } \\ {{\varrho }\sigma _{1} \sigma _{2} } & {\sigma _{2}^{2} } \\ \end{array} } \right].$$

With a random scaling, we have

$$ E_{{1,T}}^{{ - \frac{1}{2}}} I_{2} \otimes \left[ {\begin{array}{ll} {(TE_{{2,T}} - E_{{1,T}}^{2} )(E_{{1,T}} E_{{3,T}} - E_{{2,T}}^{2} )^{{ - \frac{1}{2}}} } \hfill & 0 \hfill \\ { - T} \hfill & {E_{{1,T}} } \hfill \\ \end{array} } \right]{\text{ }}\left[ {\begin{array}{c} {\hat{a}_{T}^{{{\text{LSE}}}} - a} \\ {\hat{b}_{T}^{{{\text{LSE}}}} - b} \\ {\hat{\alpha }_{T}^{{{\text{LSE}}}} - \alpha } \\ {\hat{\beta }_{T}^{{{\text{LSE}}}} - \beta } \\ \end{array} } \right]\mathop \to \limits^{{\mathcal{L}}} {\mathcal{N}}_{4} \left( {{\mathbf{0}},\varvec{S} \otimes \varvec{I}_{2} } \right) $$

(4.2)

as $T \rightarrow \infty$, where $E_{i,T}:=\int _0^T Y^i_s\,{\mathrm {d}}s$, $T\in {\mathbb {R}}_{++}$, $i=1,2,3$.

Proof

By Proposition 3.1, there exists a unique LSE $\bigl (\widehat{a}^{\mathrm {LSE}}_T, \widehat{b}^{\mathrm {LSE}}_T, \widehat{\alpha }^{\mathrm {LSE}}_T, \widehat{\beta }^{\mathrm {LSE}}_T\bigr )$ of $(a, b, \alpha , \beta )$. By (3.8), we have

$$\begin{aligned} \sqrt{T} (\widehat{a}^{\mathrm {LSE}}_T - a)&= \frac{\frac{1}{T} \int _0^T Y_s^2 \, {\mathrm {d}}s \, \cdot \frac{\sigma _1}{\sqrt{T}} \int _0^T Y_s^{1/2} \, {\mathrm {d}}W_s - \frac{1}{T} \int _0^T Y_s \, {\mathrm {d}}s \, \cdot \frac{\sigma _1}{\sqrt{T}} \int _0^T Y_s^{3/2} \, {\mathrm {d}}W_s}{\frac{1}{T} \int _0^T Y_s^2 \, {\mathrm {d}}s - \left( \frac{1}{T} \int _0^T Y_s \, {\mathrm {d}}s\right) ^2} , \\ \sqrt{T} (\widehat{b}^{\mathrm {LSE}}_T - b)&= \frac{\frac{1}{T} \int _0^T Y_s \, {\mathrm {d}}s \, \cdot \frac{\sigma _1}{\sqrt{T}} \int _0^T Y_s^{1/2} \, {\mathrm {d}}W_s - \frac{\sigma _1}{\sqrt{T}} \int _0^T Y_s^{3/2} \, {\mathrm {d}}W_s}{\frac{1}{T} \int _0^T Y_s^2 \, {\mathrm {d}}s - \left( \frac{1}{T} \int _0^T Y_s \, {\mathrm {d}}s\right) ^2} , \\ \sqrt{T} (\widehat{\alpha }^{\mathrm {LSE}}_T - \alpha )&= \frac{ \frac{1}{T} \int _0^T Y_s^2 \, {\mathrm {d}}s \, \cdot \frac{\sigma _2}{\sqrt{T}} \int _0^T Y_s^{1/2} \, {\mathrm {d}}\widetilde{W}_s - \frac{1}{T} \int _0^T Y_s \, {\mathrm {d}}s \, \cdot \frac{\sigma _2}{\sqrt{T}} \int _0^T Y_s^{3/2} \, {\mathrm {d}}\widetilde{W}_s}{\frac{1}{T} \int _0^T Y_s^2 \, {\mathrm {d}}s - \left( \frac{1}{T} \int _0^T Y_s \, {\mathrm {d}}s\right) ^2} , \\ \sqrt{T} (\widehat{\beta }^{\mathrm {LSE}}_T - \beta )&= \frac{ \frac{1}{T} \int _0^T Y_s \, {\mathrm {d}}s \, \cdot \frac{\sigma _2}{\sqrt{T}} \int _0^T Y_s^{1/2} \, {\mathrm {d}}\widetilde{W}_s - \frac{\sigma _2}{\sqrt{T}} \int _0^T Y_s^{3/2} \, {\mathrm {d}}\widetilde{W}_s}{\frac{1}{T} \int _0^T Y_s^2 \, {\mathrm {d}}s - \left( \frac{1}{T} \int _0^T Y_s \, {\mathrm {d}}s\right) ^2}, \end{aligned}$$

provided that $T \int _0^T Y_s^2 \, {\mathrm {d}}s > \left( \int _0^T Y_s \, {\mathrm {d}}s\right) ^2$, which holds almost surely. Consequently,

$$ \begin{aligned} \sqrt T \left[ {\begin{array}{l} {\hat{a}_{T}^{{{\text{LSE}}}} - a} \hfill \\ {\hat{b}_{T}^{{{\text{LSE}}}} - b} \hfill \\ {\hat{\alpha }_{T}^{{{\text{LSE}}}} - \alpha } \hfill \\ {\hat{\beta }_{T}^{{{\text{LSE}}}} - \beta } \hfill \\ \end{array} } \right] = & \frac{1}{{\frac{1}{T}\int_{0}^{T} {Y_{s}^{2} } {\mkern 1mu} {\text{d}}s - \left( {\frac{1}{T}\int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s} \right)^{2} }}\left( {\varvec{I}_{2} \otimes \left[\begin{array}{ll} \frac{1}{T}{\int_{0}^{T} {Y_{s}^{2} } {\mkern 1mu} {\text{d}}s} \hfill & {\frac{1}{T}\int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s} \hfill \\ {\frac{1}{T}\int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s} \hfill & 1 \hfill \\ \end{array} \right]} \right)\frac{1}{{\sqrt T }}\varvec{M}_{T} \\ & = \left( {\varvec{I}_{2} \otimes \left[ \begin{array}{ll} 1 \hfill & { - \frac{1}{T}\int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s} \hfill \\ { - \frac{1}{T}\int_{0}^{T} {Y_{s} } {\mkern 1mu} {\text{d}}s} \hfill & {\frac{1}{T}\int_{0}^{T} {Y_{s}^{2} } {\mkern 1mu} {\text{d}}s} \hfill \\ \end{array}\right] ^{{ - 1}} } \right)\frac{1}{{\sqrt T }}\varvec{M}_{T} , \\ \end{aligned} $$

(4.3)

provided that $T \int _0^T Y_s^2 \, {\mathrm {d}}s > \left( \int _0^T Y_s \, {\mathrm {d}}s\right) ^2$, which holds almost surely, where

$$ \varvec{M}_{t} : = \left[ {\begin{array}{l} {\sigma _{1}}{\int_{0}^{t} {Y_{s}^{{1/2}} } {\mkern 1mu} {\text{d}}W_{s} } \hfill \\ { - \sigma _{1} \int_{0}^{t} {Y_{s}^{{3/2}} } {\mkern 1mu} {\text{d}}W_{s} } \hfill \\ {\sigma _{2} \int_{0}^{t} {Y_{s}^{{1/2}} } {\mkern 1mu} {\text{d}}\tilde{W}_{s} } \hfill \\ { - \sigma _{2} \int_{0}^{t} {Y_{s}^{{3/2}} } {\mkern 1mu} {\text{d}}\tilde{W}_{s} } \hfill \\ \end{array} } \right],\qquad t \in \mathbb{R}_{ + } , $$

is a 4-dimensional square-integrable continuous local martingale due to $\int _0^t{\mathbb {E}}(Y_s)\,{\mathrm {d}}s <\infty$ and $\int _0^t{\mathbb {E}}(Y_s^3)\,{\mathrm {d}}s <\infty$, $t\in {\mathbb {R}}_+$. Next, we show that

$$\begin{aligned} \frac{1}{\sqrt{T}} {\varvec{M}}_T\, {\mathop {\longrightarrow }\limits ^{{\mathcal L}}}{\varvec{\eta }}{\varvec{Z}}\qquad \text {as }T \rightarrow \infty , \end{aligned}$$

(4.4)

where ${\varvec{Z}}$ is a 4-dimensional standard normally distributed random vector and ${\varvec{\eta }}\in {\mathbb {R}}^{4 \times 4}$ such that

$$ \varvec{\eta \eta }^{{ \top }} = \varvec{S} \otimes \left[ {\begin{array}{ll} {{\mathbb{E}}(Y_{\infty } )} \hfill & { - {\mathbb{E}}(Y_{\infty }^{2} )} \hfill \\ { - {\mathbb{E}}(Y_{\infty }^{2} )} \hfill & {{\mathbb{E}}(Y_{\infty }^{3} )} \hfill \\ \end{array} } \right]. $$

Here, the two symmetric matrices on the right-hand side are positive definite, since $\sigma _1,\sigma _2\in {\mathbb {R}}_{++}$, $\varrho \in (-1,1)$, ${\mathbb {E}}(Y_\infty )=\frac{a}{b} \in {\mathbb {R}}_{++}$ and

$$\begin{aligned} {\mathbb {E}}(Y_\infty ) {\mathbb {E}}(Y_\infty ^3) - \left( -{\mathbb {E}}\left( Y_\infty ^2\right) \right) ^2 = \frac{a^2\sigma _1^2}{4b^4}(2a+\sigma _1^2) \in {\mathbb {R}}_{++}, \end{aligned}$$

and, so is their Kronecker product. Hence ${\varvec{\eta }}$ can be chosen, for instance, as the uniquely defined symmetric positive definite square root of the Kronecker product of the two matrices in question. We have

$$ \langle \varvec{M}\rangle _{t} = \varvec{S} \otimes \left[ {\begin{array}{ll} {\int_{0}^{t} {Y_{s} } {\mkern 1mu} {\text{d}}s} \hfill & { - \int_{0}^{t} {Y_{s}^{2} } {\mkern 1mu} {\text{d}}s} \hfill \\ { - \int_{0}^{t} {Y_{s}^{2} } {\mkern 1mu} {\text{d}}s} \hfill & {\int_{0}^{t} {Y_{s}^{3} } {\mkern 1mu} {\text{d}}s} \hfill \\ \end{array} } \right],\qquad t \in \mathbb{R}_{ + } . $$

By Theorem 2.4, we have

$$ \varvec{Q}(t)\langle \varvec{M}\rangle _{t} \varvec{Q}(t)^{{ \top }} \xrightarrow{{{\text{a}}.{\text{s}}.}}\varvec{S} \otimes \left[ {\begin{array}{ll} {{\mathbb{E}}(Y_{\infty } )} \hfill & { - {\mathbb{E}}(Y_{\infty }^{2} )} \hfill \\ { - {\mathbb{E}}(Y_{\infty }^{2} )} \hfill & {{\mathbb{E}}(Y_{\infty }^{3} )} \hfill \\ \end{array} } \right]\qquad {\text{as}}\;t \to \infty $$

with ${\varvec{Q}}(t) := t^{-1/2} {\varvec{I}}_4$, $t\in {\mathbb {R}}_{++}$. Hence, Theorem 2.6 yields (4.4). Then, by (4.3), Slutsky’s lemma yields

$$ \sqrt T {\text{ }}\left[ {\begin{array}{l} {\hat{a}_{T}^{{{\text{LSE}}}} - a} \hfill \\ {\hat{b}_{T}^{{{\text{LSE}}}} - b} \hfill \\ {\hat{\alpha }_{T}^{{{\text{LSE}}}} - \alpha } \hfill \\ {\hat{\beta }_{T}^{{{\text{LSE}}}} - \beta } \hfill \\ \end{array} } \right]\mathop \to \limits^{{\mathcal{L}}} \left( {\varvec{I}_{2} \otimes \left[ {\begin{array}{ll} 1 & { - {\mathbb{E}}(Y_{\infty } )} \\ { - {\mathbb{E}}(Y_{\infty } )} & {{\mathbb{E}}(Y_{\infty }^{2} )} \\ \end{array} } \right]^{{ - 1}} } \right)\varvec{\eta Z}\mathop = \limits^{{\mathcal{L}}} {\mathcal{N}}_{4} ({\mathbf{0}},{\mathbf{\Sigma }})\quad {\text{as }}\,T \to \infty , $$

where (applying the identities $({\varvec{A}}\otimes {\varvec{B}})^\top = {\varvec{A}}^\top \otimes {\varvec{B}}^\top$ and $({\varvec{A}}\otimes {\varvec{B}})({\varvec{C}}\otimes {\varvec{D}}) = ({\varvec{A}}{\varvec{C}})\otimes ({\varvec{B}}{\varvec{D}})$)

$$ \begin{aligned} {\mathbf{\Sigma }}&: = \left( {\varvec{I}_{2} \otimes \left[ {\begin{array}{ll} 1 \hfill & { - {\mathbb{E}}(Y_{\infty } )} \hfill \\ { - {\mathbb{E}}(Y_{\infty } )} \hfill & {{\mathbb{E}}(Y_{\infty }^{2} )} \hfill \\ \end{array} } \right]^{{ - 1}} } \right)\varvec{\eta }{\mathbb{E}}(\varvec{ZZ}^{{ \top }} )\varvec{\eta }^{{ \top }} \left( {\varvec{{\rm I}}_{2} \otimes \left[ {\begin{array}{ll} 1 \hfill & { - {\mathbb{E}}(Y_{\infty } )} \hfill \\ { - {\mathbb{E}}(Y_{\infty } )} \hfill & {{\mathbb{E}}(Y_{\infty }^{2} )} \hfill \\ \end{array} } \right]^{{ - 1}} } \right)^{{ \top }} \\ &= \left( {\varvec{I}_{2} \otimes \left[ {\begin{array}{ll} 1 \hfill & { - {\mathbb{E}}(Y_{\infty } )} \hfill \\ { - {\mathbb{E}}(Y_{\infty } )} \hfill & {{\mathbb{E}}(Y_{\infty }^{2} )} \hfill \\ \end{array} } \right]^{{ - 1}} } \right)\;\left( {\varvec{S} \otimes \left[ {\begin{array}{ll} {{\mathbb{E}}(Y_{\infty } )} & { - {\mathbb{E}}(Y_{\infty }^{2} )} \\ { - {\mathbb{E}}(Y_{\infty }^{2} )} & {{\mathbb{E}}(Y_{\infty }^{3} )} \\ \end{array} } \right]} \right)\;\left( {\varvec{I}_{2} \otimes \left[ {\begin{array}{ll} 1 \hfill & { - {\mathbb{E}}(Y_{\infty } )} \hfill \\ { - {\mathbb{E}}(Y_{\infty } )} \hfill & {{\mathbb{E}}(Y_{\infty }^{2} )} \hfill \\ \end{array} } \right]^{{ - 1}} } \right) \\ &= \left( {\varvec{I}_{2} \varvec{SI}_{2} } \right) \otimes \left( {\left[ {\begin{array}{ll} 1 \hfill & { - {\mathbb{E}}(Y_{\infty } )} \hfill \\ { - {\mathbb{E}}(Y_{\infty } )} \hfill & {{\mathbb{E}}(Y_{\infty }^{2} )} \hfill \\ \end{array} } \right]^{{ - 1}} \left[ {\begin{array}{ll} {{\mathbb{E}}(Y_{\infty } )} \hfill & { - {\mathbb{E}}(Y_{\infty }^{2} )} \hfill \\ { - {\mathbb{E}}(Y_{\infty }^{2} )} \hfill & {{\mathbb{E}}(Y_{\infty }^{3} )} \hfill \\ \end{array} } \right]\left[ {\begin{array}{ll} 1 \hfill & { - {\mathbb{E}}(Y_{\infty } )} \hfill \\ { - {\mathbb{E}}(Y_{\infty } )} \hfill & {{\mathbb{E}}(Y_{\infty }^{2} )} \hfill \\ \end{array} } \right]^{{ - 1}} } \right) \\ &= \frac{1}{{({\mathbb{E}}(Y_{\infty }^{2} ) - ({\mathbb{E}}(Y_{\infty } ))^{2} )^{2} }} \\ &\quad \times \varvec{S} \otimes \left[ {\begin{array}{ll} ({{\mathbb{E}}(Y_{\infty } ){\mathbb{E}}(Y_{\infty }^{3} ) - ({\mathbb{E}}(Y_{\infty }^{2} ))^{2}){{\mathbb{E}}(Y_{\infty } )} } \hfill & {{\mathbb{E}}(Y_{\infty } ){\mathbb{E}}(Y_{\infty }^{3} ) - ({\mathbb{E}}(Y_{\infty }^{2} ))^{2} } \hfill \\ {{\mathbb{E}}(Y_{\infty } ){\mathbb{E}}(Y_{\infty }^{3} ) - ({\mathbb{E}}(Y_{\infty }^{2} ))^{2} } \hfill & {{\mathbb{E}}(Y_{\infty }^{3} ) - 2{\mathbb{E}}(Y_{\infty } ){\mathbb{E}}(Y_{\infty }^{2} ) + ({\mathbb{E}}(Y_{\infty } ))^{3} } \hfill \\ \end{array} } \right], \\ \end{aligned} $$

which yields (4.1). Indeed, by Theorem 2.4, an easy calculation shows that

$$\begin{aligned} \begin{aligned}&\left( {\mathbb {E}}(Y_\infty ) {\mathbb {E}}(Y_\infty ^3) - ({\mathbb {E}}(Y_\infty ^2))^2\right) {\mathbb {E}}(Y_\infty ) = \frac{a^3\sigma _1^2}{4b^5}(2a+\sigma _1^2),\\&{\mathbb {E}}(Y_\infty ) {\mathbb {E}}(Y_\infty ^3) - ({\mathbb {E}}(Y_\infty ^2))^2 = \frac{a^2\sigma _1^2}{4b^4}(2a+\sigma _1^2),\\&{\mathbb {E}}(Y_\infty ^3) - 2 {\mathbb {E}}(Y_\infty ) {\mathbb {E}}(Y_\infty ^2) + ({\mathbb {E}}(Y_\infty ))^3 = \frac{a\sigma _1^2}{2b^3}(a+\sigma _1^2),\\&{\mathbb {E}}(Y_\infty ^2) - \left( {\mathbb {E}}(Y_\infty )\right) ^2 = \frac{a\sigma _1^2}{2b^2}. \end{aligned} \end{aligned}$$

(4.5)

Now we turn to prove (4.2). Slutsky’s lemma, (4.1) and (4.5) yield

$$ \begin{aligned} E_{{1,T}}^{{ - \frac{1}{2}}} {\mkern 1mu} \varvec{I}_{2} & \otimes \left[ {\begin{array}{ll} {(TE_{{2,T}} - E_{{1,T}}^{2} )(E_{{1,T}} E_{{3,T}} - E_{{2,T}}^{2} )^{{ - \frac{1}{2}}} } \hfill & 0 \hfill \\ { - T} \hfill & {E_{{1,T}} } \hfill \\ \end{array} } \right]\left[ {\begin{array}{c} {\hat{a}_{T}^{{{\text{LSE}}}} - a} \\ {\hat{b}_{T}^{{{\text{LSE}}}} - b} \\ {\hat{\alpha }_{T}^{{{\text{LSE}}}} - \alpha } \\ {\hat{\beta }_{T}^{{{\text{LSE}}}} - \beta } \\ \end{array} } \right] \\ & = \bar{E}_{{1,T}}^{{ - \frac{1}{2}}} {\mkern 1mu} \varvec{I}_{2} \otimes \left[ {\begin{array}{ll} {(\bar{E}_{{2,T}} - \bar{E}_{{1,T}}^{2} )(\bar{E}_{{1,T}} \bar{E}_{{3,T}} - \bar{E}_{{2,T}}^{2} )^{{ - \frac{1}{2}}} } \hfill & 0 \hfill \\ { - 1} \hfill & {\bar{E}_{{1,T}} } \hfill \\ \end{array} } \right]\sqrt T \left[ {\begin{array}{c} {\hat{a}_{T}^{{{\text{LSE}}}} - a} \\ {\hat{b}_{T}^{{{\text{LSE}}}} - b} \\ {\hat{\alpha }_{T}^{{{\text{LSE}}}} - \alpha } \\ {\hat{\beta }_{T}^{{{\text{LSE}}}} - \beta } \\ \end{array} } \right] \\ & \mathop \to \limits^{{\mathcal{L}}} ({\mathbb{E}}(Y_{\infty } ))^{{ - \frac{1}{2}}} \varvec{I}_{2} \otimes \left[ {\begin{array}{ll} {({\mathbb{E}}(Y_{\infty }^{2} ) - ({\mathbb{E}}(Y_{\infty } ))^{2} )({\mathbb{E}}(Y_{\infty } ){\mathbb{E}}(Y_{\infty }^{3} ) - ({\mathbb{E}}(Y_{\infty }^{2} ))^{2} )^{{ - \frac{1}{2}}} } \hfill & 0 \hfill \\ { - 1} \hfill & {{\mathbb{E}}(Y_{\infty } )} \hfill \\ \end{array} } \right] \\ & \quad \times {\mathcal{N}}_{4} \left( {{\mathbf{0}},\varvec{S} \otimes \left[ {\begin{array}{ll} {\frac{{(2a + \sigma _{1}^{2} )a}}{{\sigma _{1}^{2} b}}} \hfill & {\frac{{2a + \sigma _{1}^{2} }}{{\sigma _{1}^{2} }}} \hfill \\ {\frac{{2a + \sigma _{1}^{2} }}{{\sigma _{1}^{2} }}} \hfill & {\frac{{2b(a + \sigma _{1}^{2} )}}{{\sigma _{1}^{2} a}}} \hfill \\ \end{array} } \right]} \right) \\ & \mathop = \limits^{{\mathcal{L}}} {\mathcal{N}}_{4} (0,{\mathbf{\Xi }})\qquad {\text{as }}T \to \infty , \\ \end{aligned} $$

where $\overline{E}_{i,T}:=\frac{1}{T}\int _0^T Y_s^i\,{\mathrm {d}}s$, $T\in {\mathbb {R}}_{++}$, $i=1,2,3$, and, applying the identities $({\varvec{A}}\otimes {\varvec{B}})^\top = {\varvec{A}}^\top \otimes {\varvec{B}}^\top$, $({\varvec{A}}\otimes {\varvec{B}})({\varvec{C}}\otimes {\varvec{D}}) = ({\varvec{A}}{\varvec{C}})\otimes ({\varvec{B}}{\varvec{D}})$, and using (4.5),

$$ \begin{aligned} {\mathbf{\Xi }}: = & \frac{1}{{{\mathbb{E}}(Y_{\infty } )}}\left( {{\varvec{I}}_{2} \otimes \left[ {\begin{array}{ll} ({\mathbb{E}}(Y_{\infty }^{2} ) - ({\mathbb{E}}(Y_{\infty } ))^{2} ) {( {{\mathbb{E}}(Y_{\infty } ){\mathbb{E}}(Y_{\infty }^{3} ) - ( {{\mathbb{E}}(Y_{\infty }^{2} )} )^{2} } )^{{ - \frac{1}{2}}} } & 0 \\ { - 1} & {{\mathbb{E}}(Y_{\infty } )} \\ \end{array} } \right]} \right) \\ & \quad \times \left( {\varvec{S} \otimes \left[\begin{array}{ll} {\frac{{(2a + \sigma _{1}^{2} )a}}{{\sigma _{1}^{2} b}}} & {\frac{{2a + \sigma _{1}^{2} }}{{\sigma _{1}^{2} }}} \\ {\frac{{2a + \sigma _{1}^{2} }}{{\sigma _{1}^{2} }}} & {\frac{{2b(a + \sigma _{1}^{2} )}}{{\sigma _{1}^{2} a}}} \\ \end{array}\right] } \right) \\ & \quad \times \left( {\varvec{I}_{2} \otimes \left[ {\begin{array}{ll} {({\mathbb{E}}(Y_{\infty }^{2} ) - ({\mathbb{E}}(Y_{\infty } ))^{2} )({\mathbb{E}}(Y_{\infty } ){\mathbb{E}}(Y_{\infty }^{3} ) - ({\mathbb{E}}(Y_{\infty }^{2} ))^{2} )^{{ - \frac{1}{2}}} } &; 0 \\ { - 1} & {{\mathbb{E}}(Y_{\infty } )} \\ \end{array} } \right]} \right)^{{ \top }} \\ & = \frac{1}{{{\mathbb{E}}(Y_{\infty } )}}(\varvec{I}_{2} \varvec{SI}_{2} ) \otimes \left( {\left[ {\begin{array}{ll} {({\mathbb{E}}(Y_{\infty }^{2} ) - ({\mathbb{E}}(Y_{\infty } ))^{2} )({\mathbb{E}}(Y_{\infty } ){\mathbb{E}}(Y_{\infty }^{3} ) - ({\mathbb{E}}(Y_{\infty }^{2} ))^{2} )^{{ - \frac{1}{2}}} } & 0 \\ { - 1} & {{\mathbb{E}}(Y_{\infty } )} \\ \end{array} } \right]} \right. \\ &\quad \times \left[ {\begin{array}{ll} {\frac{{(2a + \sigma _{1}^{2} )a}}{{\sigma _{1}^{2} b}}} & {\frac{{2a + \sigma _{1}^{2} }}{{\sigma _{1}^{2} }}} \\ {\frac{{2a + \sigma _{1}^{2} }}{{\sigma _{1}^{2} }}} & {\frac{{2b(a + \sigma _{1}^{2} )}}{{\sigma _{1}^{2} a}}} \\ \end{array} } \right]{\mkern 1mu} \\ & \quad \times \left. {\left[ {\begin{array}{ll} {({\mathbb{E}}(Y_{\infty }^{2} ) - ({\mathbb{E}}(Y_{\infty } ))^{2} )({\mathbb{E}}(Y_{\infty } ){\mathbb{E}}(Y_{\infty }^{3} ) - ({\mathbb{E}}(Y_{\infty }^{2} ))^{2} )^{{ - \frac{1}{2}}} } & { - 1} \\ 0 &{{\mathbb{E}}(Y_{\infty } )} \\ \end{array} } \right]} \right) \\ & = \frac{b}{a}\varvec{S} \otimes \left( {\left[ {\begin{array}{ll} {\sigma _{1}(2a + \sigma _{1}^{2} )^{{ - \frac{1}{2}}} } & 0 \\ { - 1} & {\frac{a}{b}} \\ \end{array} } \right]\left[ {\begin{array}{ll} {\frac{{(2a + \sigma _{1}^{2} )a}}{{\sigma _{1}^{2} b}}} & {\frac{{2a + \sigma _{1}^{2} }}{{\sigma _{1}^{2} }}} \\ {\frac{{2a + \sigma _{1}^{2} }}{{\sigma _{1}^{2} }}} & {\frac{{2b(a + \sigma _{1}^{2} )}}{{\sigma _{1}^{2} a}}} \\ \end{array} } \right]\left[ {\begin{array}{ll} {\sigma _{1} (2a + \sigma _{1}^{2} )^{{ - \frac{1}{2}}} } & { - 1} \\ 0 & {\frac{a}{b}} \\ \end{array} } \right]} \right) \\ & = \varvec{S} \otimes \varvec{I}_{2} . \\ \end{aligned} $$

Thus we obtain (4.2).□

Next, we formulate a corollary of Theorem 4.2 presenting separately the asymptotic behavior of the LSE of (a, b) based on continuous-time observations $(Y_t)_{t\in [0,T]}$, $T>0$. We call the attention that Overbeck and Rydén [27, Theorem 3.6] already derived this asymptotic behavior (for more details on the role of the initial distribution, see the Introduction); however, the covariance matrix of the limit normal distribution in their Theorem 3.6 is somewhat complicated. It turns out that it can be written in a much simpler form by making a simple reparametrization of the SDE (1) in Overbeck and Rydén [27], estimating $-b$ instead of b (with the notations of Overbeck and Rydén [27]), i.e., considering the SDE (1.1) and estimating b (with our notations).

Corollary 4.3

If $a, b, \sigma _1 \in {\mathbb {R}}_{++}$, and ${\mathbb {P}}(Y_0\in {\mathbb {R}}_{++})=1$, then the LSE of (a, b) given in (3.4) based on continuous-time observations $(Y_t)_{t\in [0,T]}$, $T>0$, is strongly consistent and asymptotically normal, i.e., $\bigl (\widehat{a}_T^{\mathrm {LSE}}, \widehat{b}_T^{\mathrm {LSE}}\bigr )\,{\mathop {\longrightarrow }\limits ^{{\mathrm {a.s.}}}}(a, b)$ as $T \rightarrow \infty$, and

$$ T^{{\frac{1}{2}}} \left[ {\begin{array}{l} {\hat{a}_{T}^{{{\text{LSE}}}} - a} \\ {\hat{b}_{T}^{{{\text{LSE}}}} - b} \\ \end{array} } \right]\mathop \to \limits^{{\mathcal{L}}} {\mathcal{N}}_{2} \left( {{\mathbf{0}},\left[ {\begin{array}{ll} {\frac{{(2a + \sigma _{1}^{2} )a}}{b}} \hfill & {2a + \sigma _{1}^{2} } \hfill \\ {2a + \sigma _{1}^{2} } \hfill & {\frac{{2b(a + \sigma _{1}^{2} )}}{a}} \hfill \\ \end{array} } \right]} \right)\qquad {\text{as }}T \to \infty. $$

5 Numerical Illustrations

In this section, first, we demonstrate some methods for the simulation of the Heston model (1.1), and then we illustrate Theorem 4.1 and convergence (4.1) in Theorem 4.2 using generated sample paths of the Heston model (1.1). We will consider a subcritical Heston model (1.1) (i.e., $b \in {\mathbb {R}}_{++}$) with a known non-random initial value $(y_0, x_0) \in {\mathbb {R}}_{++} \times {\mathbb {R}}$. Note that in this case, the augmented filtration $({\mathcal F}_t)_{t\in {\mathbb {R}}_+}$ corresponding to $(W_t,B_t)_{t\in {\mathbb {R}}_+}$ and the initial value $(y_0,x_0)\in {\mathbb {R}}_{++}\times {\mathbb {R}}$, in fact, does not depend on $(y_0,x_0)$. We recall five simulation methods which differ from each other in how the CIR process in the Heston model (1.1) is simulated.

In what follows, let $\eta _{k}$, $k \in \{1, \ldots , N\}$, be independent standard normally distributed random variables with some $N \in \mathbb {N}$, and put $t_k := k \frac{T}{N}$, $k \in \{0, 1, \ldots , N\}$, with some $T \in {\mathbb {R}}_{++}$.

Higham and Mao [15] introduced the Absolute Value Euler (AVE) method

$$\begin{aligned} Y^{(N)}_{t_{k}}=Y^{(N)}_{t_{k-1}}+(a-bY^{(N)}_{t_{k-1}})(t_{k}-t_{k-1})+\sigma _{1}\sqrt{|Y^{(N)}_{t_{k-1}}|}\sqrt{t_{k}-t_{k-1}}\,\eta _{k} , \qquad k \in \{1, \ldots , N\} , \end{aligned}$$

with $Y^{(N)}_0=y_0$ for the approximation of the CIR process, where $a,b,\sigma _1 \in {\mathbb {R}}_{++}$. This scheme does not preserve non-negativity of the CIR process.

The Truncated Euler (TE) scheme uses the discretization

$$\begin{aligned} Y^{(N)}_{t_{k}}=Y^{(N)}_{t_{k-1}}+(a-bY^{(N)}_{t_{k-1}})(t_{k}-t_{k-1})+\sigma _{1}\sqrt{\max (Y^{(N)}_{t_{k-1}},0)}\sqrt{t_{k}-t_{k-1}}\,\eta _{k}, \qquad k \in \{1, \ldots , N\}, \end{aligned}$$

with $Y^{(N)}_0=y_0$, where $a,b,\sigma _1 \in {\mathbb {R}}_{++}$, for approximation of the CIR process Y, see, e.g., Deelstra and Delbaen [10]. This scheme does not preserve non-negativity of the CIR process.

The Symmetrized Euler (SE) method gives an approximation of the CIR process Y via the recursion

$$\begin{aligned} Y^{(N)}_{t_{k}}=\left| Y^{(N)}_{t_{k-1}}+\left( a-bY^{(N)}_{t_{k-1}}\right) (t_{k}-t_{k-1}) +\sigma _{1}\sqrt{Y^{(N)}_{t_{k-1}}}\sqrt{t_{k}-t_{k-1}}\,\eta _{k}\right| , \qquad k \in \{1, \ldots , N\}, \end{aligned}$$

with $Y^{(N)}_0=y_0$, where $a,b,\sigma _1 \in {\mathbb {R}}_{++}$, see, Diop [12] or Berkaoui et al. [7] (where the method is analyzed for more general SDEs including so-called alpha-root processes as well with diffusion coefficient $\root \alpha \of {x}$ with $\alpha \in (1,2]$ instead of $\sqrt{x}$). This scheme gives a non-negative approximation of the CIR process Y.

The following two methods do not directly simulate the CIR process Y, but its square root $Z=(Z_t:=\sqrt{Y_t})_{t\in {\mathbb {R}}_+}$. If $a>\frac{\sigma _1^2}{2}$, then ${\mathbb {P}}(Y_t\in {\mathbb {R}}_{++}, \;\forall \, t\in {\mathbb {R}}_+)=1$, and, by Itô’s formula,

$$\begin{aligned} {\mathrm {d}}Z_t = \left( \left( \frac{a}{2} - \frac{\sigma _1^2}{8}\right) \frac{1}{Z_t} - \frac{b}{2}Z_t\right) {\mathrm {d}}t +\frac{\sigma _1}{2}\,{\mathrm {d}}W_t,\qquad t\in {\mathbb {R}}_+. \end{aligned}$$

The Drift Explicit Square Root Euler (DESRE) method (see, e.g., Kloeden and Platen [23, Section 10.2] or Hutzenthaler et al. [19, equation (4)] for general SDEs) simulates Z by

$$\begin{aligned} Z^{(N)}_{t_k}=Z^{(N)}_{t_{k-1}}+\left( \left( \frac{a}{2}-\frac{\sigma _{1}^2}{8}\right) \frac{1}{Z^{(N)}_{t_{k-1}}} -\frac{b}{2}Z^{(N)}_{t_{k-1}}\right) (t_{k}-t_{k-1})+\frac{\sigma _{1}}{2}\sqrt{t_{k}-t_{k-1}}\,\eta _{k}, \qquad k \in \{1, \ldots , N\}, \end{aligned}$$

with $Z^{(N)}_0=\sqrt{y_0}$, where $a > \frac{\sigma _{1}^2}{2}$ and $b,\sigma _1\in {\mathbb {R}}_{++}$. Here note that ${\mathbb {P}}(Z^{(N)}_{t_k} =0)=0$, $k\in \{1,\ldots ,N\}$, since $Z^{(N)}_{t_k}$ is absolutely continuous. Transforming back, i.e., $Y^{(N)}_{t_k}=(Z^{(N)}_{t_k})^2$, $k\in \{0,1,\ldots ,N\}$, gives a non-negative approximation of the CIR process Y.

The Drift Implicit Square Root Euler (DISRE) method (see, Alfonsi [1] or Dereich et al. [11]) simulates Z by

$$\begin{aligned} Z^{(N)}_{t_k}=Z^{(N)}_{t_{k-1}}+\left( \left( \frac{a}{2}-\frac{\sigma _{1}^2}{8}\right) \frac{1}{Z^{(N)}_{t_k}} -\frac{b}{2}Z^{(N)}_{t_k}\right) (t_{k}-t_{k-1})+\frac{\sigma _{1}}{2}\sqrt{t_{k}-t_{k-1}}\,\eta _{k}, \qquad k \in \{1, \ldots , N\}, \end{aligned}$$

with $Z^{(N)}_0=\sqrt{y_0}$, where $a > \frac{\sigma _{1}^2}{2}$ and $b,\sigma _1\in {\mathbb {R}}_{++}$. This recursion has a unique positive solution given by

$$\begin{aligned} Z^{(N)}_{t_{k}}=\frac{Z^{(N)}_{t_{k-1}}+\frac{\sigma _{1}}{2}\sqrt{t_{k}-t_{k-1}}\,\eta _{k}}{2+b(t_{k}-t_{k-1})} +\sqrt{\frac{\left( Z^{(N)}_{t_{k-1}}+\frac{\sigma _{1}}{2}\sqrt{t_{k}-t_{k-1}}\,\eta _{k}\right) ^{2}}{(2+b(t_{k}-t_{k-1}))^{2}} +\frac{\left( a-\frac{\sigma _{1}^2}{4}\right) (t_{k}-t_{k-1})}{2+ b(t_{k}-t_{k-1})}} \end{aligned}$$

for $k \in \{1, \ldots , N\}$ with $Z^{(N)}_0=\sqrt{y_0}$. Transforming again back, i.e., $Y^{(N)}_{t_k}=(Z^{(N)}_{t_k})^2$, $k\in \{0,1,\ldots ,N\}$, gives a strictly positive approximation of the CIR process Y.

We mention that there exist so-called exact simulation methods for the CIR process, see, e.g., Alfonsi [2, Section 3.1]. In our simulations, we will use the SE, DESRE and DISRE methods for approximating the CIR process which preserve non-negativity of the CIR process.

The second coordinate process X of the Heston process (1.1) will be approximated via the usual Euler–Maruyama scheme given by

$$\begin{aligned} X^{(N)}_{t_{k}} = X^{(N)}_{t_{k-1}}+(\alpha -\beta Y^{(N)}_{t_{k-1}})(t_{k}-t_{k-1}) + \sigma _2\sqrt{Y^{(N)}_{t_{k-1}}}\sqrt{t_{k}-t_{k-1}}\big (\varrho \,\eta _{k}+\sqrt{1-\varrho ^2}\,\zeta _k\big ) \end{aligned}$$

(5.1)

for $k \in \{1, \ldots , N\}$ with $X^{(N)}_0=x_0$, where $\alpha ,\beta \in {\mathbb {R}}$, $\sigma _2\in {\mathbb {R}}_{++}$, $\varrho \in (-1,1)$, and $\zeta _{k}$, $k \in \{1, \ldots , N\}$, be independent standard normally distributed random variables independent of $\eta _k$, $k\in \{1,\ldots ,N\}$. Note that in (5.1) the factor $\sqrt{Y^{(N)}_{t_{k-1}}}$ appears, which is well-defined in case of the CIR process Y is approximated by the SE, DESRE or DISRE methods, that we will consider.

We also mention that there exist exact simulation methods for the Heston process (1.1), see, e.g., Broadie and Kaya [8] or Alfonsi [2, Section 4.2.6].

We will approximate the estimator $\bigl (\widehat{a}_T^{\mathrm {LSE}}, \widehat{b}_T^{\mathrm {LSE}}, \widehat{\alpha }_T^{\mathrm {LSE}}, \widehat{\beta }_T^{\mathrm {LSE}}\bigr )$ given in (3.4) and (3.5) using the generated sample paths of (Y, X). For this, we need to simulate, for a large time $T\in {\mathbb {R}}_{++}$, the random variables

$$\begin{aligned} Y_T,\quad X_T, I_{1,T}:=\int _0^T Y_s\,{\mathrm {d}}s, I_{2,T}:=\int _0^TY_s^2\,{\mathrm {d}}s, I_{3,T}:=\int _0^T Y_s\,{\mathrm {d}}Y_s, I_{4,T}:=\int _0^T Y_s\,{\mathrm {d}}X_s. \end{aligned}$$

We can easily approximate the $I_{i,T}$, $i\in \{1,2,3,4\}$, respectively, by

$$\begin{aligned}&I^N_{1,T}:= \sum _{k=1}^N Y^{(N)}_{t_{k-1}} (t_k-t_{k-1}) = \frac{T}{N} \sum _{k=1}^N Y^{(N)}_{t_{k-1}}, \qquad I^N_{2,T}:= \sum _{k=1}^N (Y^{(N)}_{t_{k-1}})^2 (t_k-t_{k-1}) = \frac{T}{N} \sum _{k=1}^N (Y^{(N)}_{t_{k-1}})^2 , \\&I^N_{3,T}:= \sum _{k=1}^N Y^{(N)}_{t_{k-1}} (Y^{(N)}_{t_k}-Y^{(N)}_{t_{k-1}}), \qquad I^N_{4,T}:= \sum _{k=1}^N Y^{(N)}_{t_{k-1}} (X^{(N)}_{t_k}-X^{(N)}_{t_{k-1}}). \end{aligned}$$

Hence, we can approximate $\widehat{a}_T^{\mathrm {LSE}}$, $\widehat{b}_T^{\mathrm {LSE}}$, $\widehat{\alpha }_T^{\mathrm {LSE}}$, and $\widehat{\beta }_T^{\mathrm {LSE}}$ by

$$\begin{aligned} \widehat{a}_T^{(N)}:=\frac{(Y^{(N)}_T - y_0) I^N_{2,T} - I^N_{1,T} I^N_{3,T}}{TI^N_{2,T} - (I^N_{1,T})^2},\qquad \qquad \widehat{b}_T^{(N)}:=\frac{(Y^{(N)}_T - y_0) I^N_{1,T} - T I^N_{3,T}}{TI^N_{2,T} - (I^N_{1,T})^2},\\ \widehat{\alpha }_T^{(N)}:= \frac{(X^{(N)}_T - x_0) I^N_{2,T} - I^N_{1,T} I^N_{4,T}}{TI^N_{2,T} - (I^N_{1,T})^2},\qquad \qquad \widehat{\beta }_T^{(N)}:= \frac{(X^{(N)}_T - x_0) I^N_{1,T} - T I^N_{4,T}}{TI^N_{2,T} - (I^N_{1,T})^2}. \end{aligned}$$

We point out that $\widehat{a}_T^{(N)}$, $\widehat{b}_T^{(N)}$, $\widehat{\alpha }_T^{(N)}$ and $\widehat{\beta }_T^{(N)}$ are well-defined, since

$$\begin{aligned} TI^N_{2,T} - (I^N_{1,T})^2 = \frac{T^2}{N} \sum _{k=1}^N \left( Y^{(N)}_{t_k} - \frac{1}{N} \sum _{k=1}^N Y^{(N)}_{t_{k-1}} \right) ^2 \geqslant 0, \end{aligned}$$

and

$$\begin{aligned} TI^N_{2,T} - (I^N_{1,T})^2 = 0&\qquad \Longleftrightarrow \qquad Y^{(N)}_{t_k} = \frac{1}{N} \sum _{\ell =1}^N Y^{(N)}_{t_{\ell -1}}, \qquad k\in \{1,\ldots ,N\}\\&\qquad \Longleftrightarrow \qquad Y^{(N)}_0 = Y^{(N)}_{t_1} = \cdots = Y^{(N)}_{t_{N-1}}. \end{aligned}$$

Consequently, using that $Y^{(N)}_{t_1}$ is absolutely continuous together with the law of total probability, we have ${\mathbb {P}}(TI^N_{2,T} - (I^N_{1,T})^2 \in {\mathbb {R}}_{++})=1$.

For the numerical implementation, we take $y_0=0.2$, $x_0=0.1$, $a=0.4$, $b=0.3$, $\alpha =0.1$, $\beta =0.15$, $\sigma _{1}=0.4$, $\sigma _{2}=0.3$, $\varrho =0.2$, $T=3000$, and $N=30000$ (consequently, $t_{k}-t_{k-1}=0.1$, $k\in \{1,\ldots ,N\}$). Note that $a>\frac{\sigma _1^2}{2}$ with this choice of parameters. We simulate 10,000 independent trajectories of $(Y_T,X_T)$ and the normalized error $T^{\frac{1}{2}}\bigl (\widehat{a}_T^{\mathrm {LSE}}-a, \widehat{b}_T^{\mathrm {LSE}}-b, \widehat{\alpha }_T^{\mathrm {LSE}}-\alpha , \widehat{\beta }_T^{\mathrm {LSE}}-\beta \bigr )$. Table 1 contains the empirical mean of $Y^{(N)}_T$ and $\frac{1}{T} X^{(N)}_T$, based on 10,000 independent trajectories of $(Y_T,X_T)$, and the (theoretical) limit $\lim _{t\rightarrow \infty }{\mathbb {E}}(Y_t)=\frac{a}{b}$ and $\lim _{t\rightarrow \infty }t^{-1}{\mathbb {E}}(X_t)=\alpha -\frac{\beta a}{b}$, respectively (following from Proposition 2.2), using the schemes SE, DESRE and DISRE for simulating the CIR process.

Table 1 Empirical mean of $Y^{(N)}_T$ (first row) and $\frac{1}{T} X^{(N)}_T$ (second row)

Henceforth, we will use the above choice of parameters except that $T=5000$ and $N=50,000$ (yielding $t_{k}-t_{k-1}=0.1$, $k\in \{1,\ldots ,N\}$).

In Table 2, we calculate the expected bias (${\mathbb {E}}(\widehat{\theta }^{\mathrm {LSE}}_T-\theta )$), the $L_{1}$-norm of error (${\mathbb {E}}|\widehat{\theta }^{\mathrm {LSE}}_T-\theta |$) and the $L_{2}$-norm of error $\big (\big ({\mathbb {E}}(\widehat{\theta }^{\mathrm {LSE}}_T-\theta )^{2}\big )^{1/2}\big )$, where $\theta \in \{a,b,\alpha ,\beta \}$, using the scheme DISRE for simulating the CIR process.

Table 2 Expected bias, $L_1$- and $L_2$-norm of error using DISRE scheme

In Table 3, we give the relative errors $(\widehat{\theta }^{(N)}_T - \theta )/\theta$, where $\theta \in \{a,b,\alpha ,\beta \}$, for $T=5000$ using the scheme DISRE for simulating the CIR process.

Table 3 Relative errors using DISRE scheme

In Fig. 1, we illustrate the limit law of each coordinate of the LSE $\bigl (\widehat{a}_T^{\mathrm {LSE}}, \widehat{b}_T^{\mathrm {LSE}}, \widehat{\alpha }_T^{\mathrm {LSE}}, \widehat{\beta }_T^{\mathrm {LSE}}\bigr )$ given in (4.1). To do so, we plot the obtained density histograms of each of its coordinates based on 10,000 independently generated trajectories using the scheme DISRE for simulating the CIR process, we also plotted the density functions of the corresponding normal limit distributions in red.

Fig. 1

In the first line from left to right, the density histograms of the normalized errors of $T^{1/2}(\widehat{a}_T^{(N)} - a)$ and $T^{1/2}(\widehat{b}_T^{(N)} - b)$, in the second line from left to right, the density histograms of the normalized errors of $T^{1/2}(\widehat{\alpha }_T^{(N)} - \alpha )$ and $T^{1/2}(\widehat{\beta }_T^{(N)} - \beta )$. In each case, the red line denotes the density function of the corresponding normal limit distribution

With the above choice of parameters, as a consequence of (4.1), we have

$$ \begin{gathered} T^{{\frac{1}{2}}} (\hat{a}_{T}^{{{\text{LSE}}}} - a)\xrightarrow{{\mathcal{L}}}{\mathcal{N}}\left( {0,\frac{a}{b}(2a + \sigma _{1}^{2} )} \right) = {\mathcal{N}}(0,1.28)\qquad {\text{as }}T \to \infty , \hfill \\ T^{{\frac{1}{2}}} (\hat{b}_{T}^{{{\text{LSE}}}} - b)\xrightarrow{{\mathcal{L}}}{\mathcal{N}}\left( {0,\frac{{2b}}{a}(a + \sigma _{1}^{2} )} \right) = {\mathcal{N}}(0,0.84)\qquad {\text{as }}T \to \infty , \hfill \\ T^{{\frac{1}{2}}} (\hat{\alpha }_{T}^{{{\text{LSE}}}} - \alpha )\xrightarrow{{\mathcal{L}}}{\mathcal{N}}\left( {0,\frac{{a\sigma _{2}^{2} }}{{b\sigma _{1}^{2} }}(2a + \sigma _{1}^{2} )} \right) = {\mathcal{N}}(0,0.72)\qquad {\text{as }}T \to \infty , \hfill \\ T^{{\frac{1}{2}}} (\hat{\beta }_{T}^{{{\text{LSE}}}} - \beta )\xrightarrow{{\mathcal{L}}}{\mathcal{N}}\left( {0,\frac{{2b\sigma _{2}^{2} }}{{a\sigma _{1}^{2} }}(a + \sigma _{1}^{2} )} \right) = {\mathcal{N}}(0,0.4725)\qquad {\text{as }}T \to \infty . \hfill \\ \end{gathered} $$

In case of the parameters a and b, one can see a bias in Fig. 1, which, in our opinion, may be related with the different speeds of weak convergence for the LSE of (a, b) and that of $(\alpha ,\beta )$, and with the bad performance of the applied discretization scheme for Y.

Table 4 contains the skewness and excess kurtosis of $T^{\frac{1}{2}}(\widehat{\theta }^{(N)}_T - \theta )$, where $\theta \in \{a,b,\alpha ,\beta \}$, using the scheme DISRE for simulating the CIR process. This confirms our results in (4.1) as well.

Table 4 Skewness and excess kurtosis using the scheme DISRE for simulating the CIR process

Using the Anderson–Darling and Jarque–Bera tests, we test whether each of the coordinates of $T^{\frac{1}{2}}\bigl (\widehat{a}_T^{\mathrm {LSE}}-a, \widehat{b}_T^{\mathrm {LSE}}-b, \widehat{\alpha }_T^{\mathrm {LSE}} - \alpha , \widehat{\beta }_T^{\mathrm {LSE}}-\beta \bigr )$ follows a normal distribution or not for $T=5000$. In Table 5 we give the test values and (in parenthesis) the p-values of the Anderson–Darling and Jarque–Bera tests using the scheme DISRE for simulating the CIR process (the $*$ after a p-value denotes that the p-value in question is greater than any reasonable significance level). It turns out that, with this choice of parameters, at any reasonable significance level the Anderson–Darling test accepts that $T^{\frac{1}{2}}(\widehat{a}_T^{\mathrm {LSE}} - a)$, $T^{\frac{1}{2}}(\widehat{b}_T^{\mathrm {LSE}} - b)$, $T^{\frac{1}{2}}(\widehat{\alpha }_T^{\mathrm {LSE}} - \alpha )$, and $T^{\frac{1}{2}}(\widehat{\beta }_T^{\mathrm {LSE}} - \beta )$ follow normal laws. The Jarque–Bera test also accepts that $T^{\frac{1}{2}}(\widehat{b}_T^{\mathrm {LSE}} - b)$, $T^{\frac{1}{2}}(\widehat{\alpha }_T^{\mathrm {LSE}} - \alpha )$, and $T^{\frac{1}{2}}(\widehat{\beta }_T^{\mathrm {LSE}} - \beta )$ follow normal laws, but rejects that $T^{\frac{1}{2}}(\widehat{a}_T^{\mathrm {LSE}} - a)$ follows a normal law.

Table 5 Test of normality in case of $y_0=0.2$, $x_0=0.1$, $a=0.4$, $b=0.3$, $\alpha =0.1$, $\beta =0.15$, $\sigma _{1}=0.4$, $\sigma _{2}=0.3$, $\varrho =0.2$, $T=5000$, and $N=50,000$ generating 10,000 independent sample paths using the scheme DISRE for simulating the CIR process

All in all, our numerical illustrations are more or less in accordance with our theoretical results in (4.1). Finally, we note that we used the open source software R for making the simulations.