sqrtdt

Sqrtdt is a shorthand expression encountered in stochastic calculus to denote the square root of an infinitesimal time increment dt, used to describe the stochastic increment of a Wiener process. In the standard model, a Wiener process W(t) has increments dW_t that are normally distributed with mean zero and variance dt. Equivalently, dW_t can be represented in discretized form as dW_t ≈ sqrt(dt) Z_t, where Z_t is a standard normal random variable with mean 0 and variance 1. The combination sqrt(dt) Z_t is sometimes referred to informally as sqrtdt, reflecting the idea that random fluctuations scale with the square root of the time step.

Usage and interpretation: In numerical schemes such as the Euler–Maruyama method, one uses ΔW = sqrt(Δt) ξ, with ξ ~ N(0,1). The symbol sqrtdt is a heuristic shorthand rather than a fixed quantity; it does not denote a deterministic differential but a stochastic increment. In Itô calculus, one typically writes dW_t, and the fundamental property dW_t^2 = dt justifies the square-root scaling of the stochastic term.

Limitations and context: The notation sqrtdt is not universal. Some texts prefer dW_t or ΔW to denote Wiener increments; sqrtdt appears mainly in informal derivations and teaching materials. Despite variegated notation, the underlying concept remains central to modeling stochastic differential equations, diffusion processes, and financial mathematics, where random fluctuations are modeled as proportional to the square root of the time step.

See also: Wiener process, Itô calculus, Itô integral, stochastic differential equation, Euler–Maruyama method.