Let us consider a betting game as follows. Let be a sequence of i.i.d. random variables taking values in such that and . Suppose that at the initial time your capital is . We start a sequence of betting rounds. At time , you either win your stake if , or you loose if . Let and respectively stand for your capital and your stake at time . Then can be recursively defined by
Let be a fixed (small) number in (0,1). Assume that and is predictable, i.e. is -measurable with . You certainly wish to find an optimal strategy of stakes to maximize your fortune. This is equivalent to the maximization of the expected interest rate given by
where is the length of the game. For , set , which is a -measurable random variable. The sequence is called a strategy.
Proposition 1 Let
with . Then is a supermartingale.
Proof: We fisrt show that for all . Indeed, we notice that
We also have since . It immediately follows that
Hence, by the principle of induction, we obtain
and thus . On the other hand, we have
Applying Jensen’s inequality , we thus have
Hence is a supermartingale.
Proposition 2 We have that is an optimal strategy to maximize the expected interest rate.
Proof: Since is a supermartingale, we have . Therefore,
We note that is a martingale if
for all . This is equivalent to . Choosing for all , we must have that . In this case, we obtain
Hence, is an optimal strategy.