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Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility. (English) Zbl 1355.60071

Summary: We consider a stochastic differential equation of the form \[ dX_t = \theta a(t,X_t)\,dt + \sigma_1(t,X_t) \sigma_2(t,Y_t)\,dW_t \] with multiplicative stochastic volatility, where \(Y\) is some adapted stochastic process. We prove existence-uniqueness results for weak and strong solutions of this equation under various conditions on the process \(Y\) and the coefficients \(a\), \(\sigma_1\), and \(\sigma_2\). Also, we study the strong consistency of the maximum likelihood estimator for the unknown parameter \(\theta \). We suppose that \(Y\) is in turn a solution of some diffusion SDE. Several examples of the main equation and of the process \(Y\) are provided supplying the strong consistency.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
62F10 Point estimation
62F12 Asymptotic properties of parametric estimators
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