Solving stochastic differentials for time series forecasting

jjhyun90

I am trying to reproduce results of a paper. The model is:

[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.

Mute

Are you trying to solve the equations numerically? I think people typically interpret the dt as a small timestep Δt. I guess the tricky part is the stochastic terms - it looks like they should be Gaussian distributed random variables with covariances [itex]\rho_{12}[/itex], etc? Generating those might be tricky? Otherwise everything else looks straightforward.

The equations are simple enough to solve analytically, but if you're not familiar with differential equations that might not be the best avenue for you. But, if you want to try, you can solve the y and v equations using integrating factors, and then plug the solutions into the S equation, which you can solve by dividing out the S and writing [itex]dS/S = d(\ln S)[/itex], then it's a simple matter of integration. (I would write [itex]dz_1/dt = \eta_1(t)[/itex], etc.)