By H. Jerome Keisler
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Extra info for An Infinitesimal Approach to Stochastic Analysis
P). We assume that we can observe sequentially Y1 , Y2 , • •• and we denote by X 1 , X2 , •• • the sequence of rewards. If we stop at the n th stage, Xn = [,, (Y1 , Y2 , ••• , Y,). e. X, is measurable � for n = I , 2 , . . �, P) with target space the positive integers 1 , 2 , . whlch satisfies two conditions. First, . P[w: T(w) < oo ) = 1 , (8. 12) for each n . Eq. (8. 12) indicates that no future information is available to influence the decision to stop at time n. � = a(Y1 , Y2 , ... , Y, ).
12) Eq. 5. q;;; ] Y(T, t). Thus, futures pricing under the assumptions stated is a martingale. The theoretical paper of Samuelson ( 1 965) summarized in tllis application and the work of Mandelbrot (I 966) generated great interest in econometric test ing of the properties of stock prices. Although Samuelson's paper establishes the martingale property for futures pricing rather than for an equity asset, a share of a stock may be regarded as a sequence of futures claims due to mature at succes sive intervals.
2 1 ) is the stochastic generalization of (6. 1 7) and it can be used to help us decide under what conditions the sequence v1, , vt+ T is a martingale. ¥, ] = v1? � ] . This means that the martingale condition holds when the discount rate is equal to the conditional expected return of the stock. We conclude therefore that the sequence vr vt+ T ' T = l , 2 , ... 22) From this last equation we at once decide that v1, v1+ T ' T = 1 , 2 , ... 22) having � instead of= yields The submartingale property says that the conditional expected value of the stock next period is greater or equal to its current value.