$$\frac{d}{dt} (_tV)=\delta (_tV)+(P_t-e_t)-\mu_{x+t}(S_t+E_t-_{t}V)$$

This formula makes sense to me, except for the last term. The book says we need to increase our policy value by \(_tV\) if the life dies. I understand that when the life dies, we no longer need to hold that amount in reserve for that policyholder. But \(\frac{d}{dt} (_tV)\) represents the change in the reserve fund value over a very short time. So it makes sense that the fund increases by the interest earned, the premiums received (less the expenses) and decreases by the expected payout of \(\mu_{x+t}(S_t+E_t)\).

But shouldn't the fund decrease by the expected amount of \(\mu_{x+t}(_{t}V)\) since when the individual dies, we can release the money which was held as reserve?

I think perhaps where my knowledge is faulty is in the definition of \(_tV\). Is this the amount required that the insurer needs to hold for future claims or is it the value of the fund (after premiums, expenses and expected payout of benefits)