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WIP: Covid sde #867

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Comment on heuristics in the CIS process
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jlperla committed Jul 13, 2020
commit 137ca39c68b77db3d5313be73f6fabd5f5158920
2 changes: 1 addition & 1 deletion source/rst/continuous_time/seir_model_sde.rst
Original file line number Diff line number Diff line change
Expand Up @@ -533,7 +533,7 @@ Consider that the effective transmission rate :math:`R(t)` could depend on degre
* Deviations in the implementation and timing of lockdown policy within demographics, locations, or businesses within the system.
* Aggregate shocks in opening/closing industries

To implement this, we will add on a diffusion term to :eq:`Rode` with an instantaneous volatility of :math:`\zeta \sqrt{R}`. The scaling by the :math:`\sqrt{R}` ensures that the process can never go negative since the variance converges to zero as R goes to zero.
To implement this, we will add on a diffusion term to :eq:`Rode` with an instantaneous volatility of :math:`\zeta \sqrt{R}`. The scaling by the :math:`\sqrt{R}` ensure that the process (used in finance as the `CIR <https://en.wikipedia.org/wiki/Cox%E2%80%93Ingersoll%E2%80%93Ross_model>`__ model of interest rates) stays weakly positive. The heuristic explanation is that the variance of the shocks converges to zero as R goes to zero, enabling the upwards drift to dominate. See `here <https://en.wikipedia.org/wiki/Cox%E2%80%93Ingersoll%E2%80%93Ross_model#Distribution>`__ for a heuristic description of when the process is weakly and strictly positive.

The notation for this `SDE <https://en.wikipedia.org/wiki/Stochastic_differential_equation#Use_in_probability_and_mathematical_finance>`__ is then

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