Please post here any mathematical result that traders would like to know. Deep, elegant, surprising, beautiful, easy to formulate but difficult to prove theorems from the theory of point processes, random walks, martingale theory, genetic algorithms, time series, wavelet, extreme value theory, records theory, mathematical gaming etc. are welcome.
Let me start with this one:
In a random walk with state space = Z and transition probabilities P(k --> k+1)=p, P(k --> k)=q, P( k --> k-1)=r
with p+q+r=1, the expected number of steps before moving up is either finite or infinite depending on p, q, r. The borderline between the finite and infinite case is when p=r. When p>r, the expectation is function of 1/(p-r). When p<=r, the expectation is infinite.
This means (applied to the stock market) that it is possible for a stock price to *never* surpass its present value, as well as to *never* go below the present price, depending on p, q, and r.
Vincent Granville
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datashaping.com : Advanced Trading Strategies |
