Improved estimators of security price volatility are formulated. These estimators employ data of the type commonly found in the financial pages of a newspaper, namely the high, low, opening and closing prices, and the transaction volume. The new estimators have much higher relative efficiency than the standard estimators.

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We consider a financial market model with two assets. One has deterministic rate of growth, while the rate of growth of the second asset is governed by a Brownian motion with drift. We can shift money from one asset to another; however, there are losses of money (brokerage fees) involved in shifting money from the risky to the nonrisky asset. We want to maximize the expected rate of growth of funds. It is proved that an optimal policy keeps the ratio of funds in risky and nonrisky assets within a certain interval with minimal effort.

Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several exponential and moment inequalities, particularly those related to laws of the iterated logarithm, for self-normalized random variables including martingales. Tail probability bounds are also derived. For random variables Bt > 0 and At, let Yt(λ) = exp{λAt − λ 2 B 2 t /2}. We develop inequalities for the moments of At/Bt or sup t≥0 At/{Bt(log log Bt) 1/2 } and variants thereof, when EYt(λ) ≤ 1 or when Yt(λ) is a supermartingale, for all λ belonging to some interval. Our results are valid for a wide class of random processes including continuous martingales with At = Mt and Bt = M t, and sums of conditionally symmetric variables di with At = t i=1 di and Bt = 1. Introduction. A prototypical example of self-normalized random variables is Student's t-statistic which replaces the population standard devi-

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