How can you develop a continuous version of the recursive relations of reserves.

To develop a continuous version of the recursive relations of reserves, let's first establish the continuous version of the reserve at time

$�$, denoted as $�{�}^{-}(�;�)$. Given:

• $�(�)$: Represents the cash flow at time $�$ over the interval $[0,�]$.
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• $�(�,�)$: Represents the general discount function where $�$ represents the present time and $�$ represents the future time.
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• $�$: Represents the force of discount.
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• $�{�}^{-}(�;�)$: Represents the reserve at time $�$.
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The continuous version of the reserve at time $�$ can be expressed as follows:

$�{�}^{-}(�;�)=-{\int }_{�}^{�}�(�)\cdot �(�,�)��$

This represents the negative of the value at time $�$ of all future cash flows, discounted using the continuous discount function $�(�,�)$.

To derive the recursive relation, let's differentiate $�{�}^{-}(�;�)$ with respect to $�$:

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

Explanation:

• $��(�)\cdot �{�}^{-}(�;�)$: Represents the effect of discounting on the reserve at time $�$, where $�$ is the force of discount.
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• $�(�)$: Represents the cash flow at time $�$.

This recursive relation describes how changes in the reserve at time $�$ are influenced by both discounting and the cash flow at that time.

The continuous version of the recursive relations of reserves involves calculating the reserve at time $�$ by discounting all future cash flows and then differentiating this reserve with respect to $�$ to account for changes over time due to discounting and cash flows.