Estimation of parameters in a system of stochastic differential equations from discrete observations (Q1975809)

The authors consider an \(N\)-dimensional random process \(Y(t)\) satisfying the following system of stochastic differential equations in Ito's sense: \[ dY(t)=F(t,Y)\Theta dt+G(t,Y)S dW(t),\quad Y(0)=0, \quad t\geq 0, \] where \(W(t)\) is an \(N\)-dimensional standard Wiener process, \(\Theta\) is a \(K\)-dimensional vector of constant parameters, \(S\) is an \((N\times M)\)-matrix of constant parameters, \(F(t,Y)\) and \(G(t,Y)\) are \((N\times K)\) - and \((N\times M)\) - matrices of functions satisfying a uniform Lipschitz condition in \(Y\). A maximum likelihood estimate for the unknown parameters \(\Theta\) and \(S\) is constructed. This estimate is based on the approximation, by the Euler scheme, of the observations of a trajectory of a solution \(Y_i=Y(t_i)\) to the system of discrete times \(t_i\), \(i=0,\dots,N_p\), where the number \(N_p\) of observations is such that \(N_p\gg\max(N,K,M)\).