A sequence of functions (f_n) converges pointwise to a function f on an interval I if, for every point x in I, f_n(x) converges to f(x) as n tends to infinity.
For every x in I, the limit of f_n(x) as n tends to infinity is equal to f(x).
The convergence depends on each point x individually, which implies that the speed of convergence can vary from one
point to another.
A sequence of functions (f_n) converges uniformly to a function f on an interval I if, regardless of the point x, the
functions f_n get closer to f at a rate that does not depend on x.
The limit as n tends to infinity of the supremum (*) of |f_n(x) - f(x)| over the interval I is equal to 0.
This means that for every ε > 0, there exists an integer N such that for all n ≥ N and for every x in
I, |f_n(x) - f(x)| < ε.
The convergence is uniform over the interval I, meaning that the quality of the approximation does not depend on the
chosen point.
Financial asset prices are often modeled using sequences of functions or stochastic processes.
Pointwise Convergence ensures that for every possible market state (each point), the model approximation converges to
the real asset price. This is useful in scenarios where we are interested in specific values, such as simulating a particular market scenario.
For example, when pricing an option, we want to ensure that the estimated price is accurate for all possible levels of
the underlying asset's price.
Integrals and conditional expectations play also a central role in valuing financial assets. Uniform convergence allows
passing to the limit under the integral, which is crucial for correctly evaluating prices and expectations.
(*) The supremum (« sup ») is the least upper bound of the
values that |f_n(x) - f(x)| can reach. In the context of uniform convergence, the supremum helps us measure the maximum deviation between the functions f_n and f across the entire interval,
ensuring that this maximum deviation can be made arbitrarily small as n increases.
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