sqrtT

sqrtT is a notational shorthand commonly used in mathematics, statistics, and quantitative finance to denote the square root of a nonnegative quantity T. In many formulas, T represents a time horizon, duration, or maturity, and sqrtT = sqrt(T) is used to scale quantities with time. There is no separate universal function named sqrtT; it is simply shorthand for the square root of T.

Properties: The domain is T >= 0. At T = 0, sqrtT = 0. For T > 0, sqrtT is strictly increasing and concave, with derivative d/dT sqrtT = 1/(2 sqrtT). As T grows large, sqrtT increases sublinearly with T, reflecting the general property that square roots grow more slowly than linear functions.

Applications in probability and finance: In stochastic processes, the standard Brownian motion W_T has standard deviation sqrt(T); equivalently W_T = sqrt(T) Z with Z ~ N(0,1). In numerical simulations of stochastic differential equations, sqrt(T) scales random normal increments to reflect diffusion over a time horizon T. In finance, over a time to maturity T, the diffusion term scales as sigma sqrt(T); the distribution of log returns over the period has variance sigma^2 T, a key component in pricing models such as Black-Scholes where sqrt(T) appears in the diffusion term and in the variance expression.

Practical notes: When T = 0, sqrtT is 0; for T < 0, the real-valued square root is not defined. In software, sqrt is the standard function, and sqrtT may appear as a variable name or shorthand within formulas. See also square root, time to maturity, Brownian motion.