Optimal Allocation of Investments, Instantaneous Frequency Estimation and Adaptive Filters Using Malliavin Calculus

Stochastic calculus of variations (Malliavin Calculus) is used to estimate the sinusoids when the amplitudes are described as known stochastic processes (for example Ornstein-Uhlenbeck processes). The observations are a sum of more than one sinusoid. It is assumed that the unknown sinusoids are slowly varying over time. The problem is similar to the optimal allocation of resources for option pricing in the area of financial engineering. It is also similar to the estimation of time-varying parameters in the adaptive filters. The generalized Clark-Ocone formula is used to derive a closed form expression for the estimates of the unknown sinusoids. It is shown that the resulting equations are similar to the adaptive filter equation.