I am trying to reproduce results of a paper. The model is:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]

dS = (v-y-\lambda_1)Sdt + \sigma_1Sdz_1 \\

dy = (-\kappa y - \lambda_2)dt + \sigma_2 dz_2 \\

dv = a((\bar{v}-v)-\lambda_3)dt + \sigma_3 dz_3 \\

dz_1dz_2 = \rho_{12}dt \\

dz_1dz_3 = \rho_{13}dt \\

dz_2dz_3 = \rho_{23}dt \\

[/itex]

where S, y, and v are state variables and [itex] z_1, z_2, z_3 [/itex] are Wiener processes.

The parameters [itex] \kappa, a, \bar{v}, \lambda_1, \lambda_2, \lambda_3, \sigma_1, \sigma_2, \sigma_3, \rho_{12}, \rho_{13}, \rho_{23} [/itex] are given. There is time series data such for each day I can estimate S, y, and v.

The goal is to estimate the next day's S, y, and v. It feels like an easy task, but am not familiar with differential equations or Wiener processes (let alone stochastic calculus). Without understanding the model, I was naively trying to "substitute" dt with Δt, etc, but the given parameters are such that [itex]\rho_{12} > 0, \, \rho_{23} > 0, \, \rho_{13} < 0 [/itex], so I realized I should not do that.

Any help would be greatly appreciated.

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# Solving stochastic differentials for time series forecasting

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