Solving approximately a prediction problem for stochastic jump-diffusion systems

Conditional Optimization of Algorithms for Estimating Distributions of Solutions to Stochastic Differential Equations

“…The last equation together with (7) leads to (9). The randomization of the projection estimate ( 5) is obtained by calculating the linear functional…”

“…, n z and n z is the number of nodes of the interval [a,b]. The recurrence Relation (9) for Legendre polynomials avoids computer arithmetic errors for coefficients in the explicit Formula (8) for large m. Using the recurrence Relation ( 9) is recommended under condition m > 20 since in this case, computer arithmetic errors arise for the explicit Formula (8). However, in this case, it is more convenient to estimate the mean of Legendre polynomials to calculate the coefficients in the Expansion (5).…”

“…For analyzing, filtering, predicting, or smoothing random processes in various dynamical systems described by SDEs, the probability density is often estimated [8,9] because it provides maximum information about the random process.…”