Parameter estimation for the subcritical Heston model based on discrete time observations

where a > 0, b, α, β ∈ R, σ1 > 0, σ2 > 0, ̺ ∈ (−1, 1), and (Wt, Bt)t>0 is a 2-dimensional standard Wiener process, see Heston [7]. We investigate only the so-called subcritical case, i.e., when b > 0, see Definition 2.3, and we introduce some parameter estimator of (a, b, α, β) based on discrete time observations and derived from conditional least squares estimators (CLSEs) of some modified parameters starting the process (Y,X) from some known non-random initial value (y0, x0) ∈ (0,∞) × R. We do not estimate the parameters σ1, σ2 and ̺, since these parameters could—in principle, at least—be determined (rather than estimated) using an arbitrarily short continuous time observation (Xt)t∈[0,T ] of X, where T > 0, see, e.g., Barczy and Pap [1, Remark 2.6]. In Overbeck and Rydén [15, Theorems 3.2 and 3.3] one can find a strongly consistent and asymptotically normal estimator of σ1 based on discrete time observations for the process Y , and for another estimator of σ1, see Dokuchaev [5]. Eventually, it turns out that for the calculation of the estimator of (a, b, α, β), one does not need to know the values of the parameters σ1, σ2 and ̺. For interpretations of Y and X in financial mathematics, see, e.g., Hurn et al. [8, Section 4].