On a certain exponential inequality for Gaussian processes

If X = (Xj) j=1m is a zero-mean Gaussian stochastic process and σ j = (E[Xj2]1/2, j = 1, ..., m, Tsirel'son (Theory Probab. Appl., 30, 820-828, 1985) and more explicitly Vitale (Ann. Probab., 24, 2172-2178, 1996 and A log-concavity proof for a Gaussian exponential bound. In: Hill, T.P., Houdré, C. (eds.) Advances in Stochastic Inequalities, Contemporary Mathematics, vol. 234, pp. 209-212. AMS, Providence, RI, 1999) applied results from Brunn-Minkowski theory to show that X satisfies the following inequality: E [exp(max 1≤j≤m(Xj - jσ2/2))] ≤ exp (E[max1≤j≤m Xj]). In this paper a more general inequality will be derived using a known formula for Gaussian integrals. In particular, it also follows that E[exp(min1≤j≤m(Xj - jσ2/2))] ≤ exp(E[min1≤j≤m Xj]). In the last section of this article the above exponential inequalities are combined with a well known variant of the Slepian lemma to compare certain option prices in the Black-Scholes and Bachelier models. © 2007 Springer Science+Business Media, LLC.