expectationlike

Expectationlike refers to a class of functionals defined on a space of random variables that play a role similar to the mathematical expectation but without asserting all axioms of standard probability expectation. The phrase is used informally in probability, statistics, and risk assessment to describe operators that assign a real number to each random variable and that retain some intuitive features of expectation, such as translating constants to themselves and reacting to linear combinations in a predictable way. In different contexts, expectationlike functionals may be linear, sublinear, or otherwise convex.

A common linear variant satisfies E'[aX + bY] = aE'[X] + bE'[Y], E'[c] = c for constants c, and monotonicity: if X ≥ Y almost surely then E'[X] ≥ E'[Y]. A broader and more widely used family is the sublinear (or nonlinear) expectation, where E'[X + Y] ≤ E'[X] + E'[Y] and E'[aX] = aE'[X] for a ≥ 0, with E'[0] = 0. Sublinear expectation captures ambiguity or model uncertainty and underpins coherent risk measures and robust pricing. The theory includes specific realizations such as G-expectation, introduced by Shige Peng, and other capacity-based or upper-lower expectation frameworks.

Applications of expectationlike functionals appear in risk management, finance under model uncertainty, stochastic control, and imprecise probability. They provide a flexible toolbox for evaluating uncertain outcomes when a single probability model is insufficient or when monotone/convex risk assessment is preferred. While not a standardized term, expectationlike serves as a useful shorthand for operators that resemble expectation in essential ways without committing to full probabilistic structure.