A STOCHASTIC ALGEBRAIC-DIFFERENTIAL EQUATION OF THE NEWTON-NELSON TYPE

We investigate a stochastic algebraic-differential equation, modelling dynamically distorted signals in an electronic device with the presence of noise in the ingoing signal, on the basis of the theory of mean derivatives of stochastic processes. A new point here is that it is a second-order equation with mean derivatives where the second-order derivative is taken from the so-called Newton-Nelson equation of Nelson’s stochastic mechanics (a version of quantum mechanics). This is important since it allows us to take into account not only the noise in the ingoing signal, but also the noise generated by quantum effects in the device. The equation is ill-defined at time \(t=0\) $t=0$ . Fixing an arbitrary time instant \(t_0>0\) $t_0>0$ , we prove the existence of the process that satisfies the equation under consideration for \(t\ge t_0\) $t\ge t_0$ .