Write a note on ‘international actuarial notations’.

Note on international actuarial  notations is  the optional stopping theorem (or Doob's optional sampling theorem) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial expected value. Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional stopping theorem says that, on average, nothing can be gained by stopping play based on the information obtainable so far (i.e., without looking into the future). Certain conditions are necessary for this result to hold true. In particular, the theorem applies to doubling strategies.

The optional stopping theorem is an important tool of mathematical finance in the context of the fundamental theorem of asset pricing.

A discrete-time version of the theorem is given below:

Let X = (X_t)_{t∈â„•0} be a discrete-time martingale and τ a stopping time with values in â„•₀ ∪ {∞}, both with respect to a filtration (F_t)_{t∈â„•0}. Assume that one of the following three conditions holds:

(a) The stopping time τ is almost surely bounded, i.e., there exists a constant c ∈ â„• such that τ ≤ c a.s.

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