Short Rate - Growth & Discount

Let B(t) be the cash in the bank, which pays interest at the rate r(t). After a time , the money in the bank will be

Re-arranging

As the time interval becomes very small, in the limit where it tends to zero we can write

Integrating from t to T, for the continuous case

which gives

Applying the limits and taking the inverse of the log

In the continuous case, the growth of an asset at the rate r(t) is given as

If the interest rate r(t) is deterministic and a constant (say r), then the eight hand integrand can be completed as (T-t). However, if r(t) is stochastic, then equation (3) is a stochatic equation.

Similarly, the stochastic discount factor can be defined as