Give Thiele's differential equation.

 Thiele's differential equation describes the rate of change of the reserve

${�}^{-}(�)$ over time in a continuous payment insurance policy.

Given the conditions provided, where ${�}^{-}(�)$ represents the reserve at time $�$, $�$ represents the force of discount, ${�}_{�}(�)$ represents the force of mortality, $�(�)$ represents the death benefit payable at the moment of death, and $�(�)$ represents the continuous premiums paid at time $�$, Thiele's differential equation can be written as follows:

$\frac{�}{��}�{�}^{-}(�)=�{�}_{�}(�)\cdot �{�}^{-}(�;�;{�}_{�})+�(�)-�(�)\cdot {�}_{�}(�)$

Here's the breakdown of the equation:

• $\frac{�}{��}�{�}^{-}(�)$: Rate of change of the reserve at time $�$.
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• $�$: Force of discount.
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• ${�}_{�}(�)$: The present value of the contingent payment at time $�$.
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• $�{�}^{-}(�;�;{�}_{�})$: The reserve at time $�$ for the cash flows $�(�)=�(�)-�(�)\cdot {�}_{�}(�)$.
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• $�(�)-�(�)\cdot {�}_{�}(�)$: Represents the net cash flow at time $�$, which is the difference between premiums paid continuously and the expected death benefit liability.
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This equation describes how the reserve changes over time due to the force of discount and the net cash flows, considering both the premiums paid and the expected death benefit liability.

In addition, $�{�}_{�}(�)$ is defined as:

$�{�}_{�}(�)=�(�)+{�}_{�}(�)$

Where $�(�)$ is the force of interest, representing the rate at which reserves grow due to interest.