Bernstein Function

A Bernstein function, on the positive half-line [0,∞), is an absolutely continuous function $f: [0,∞) → [0,∞)$ that has a completely monotone derivative. This means that the nth derivative of the function, $f^n(x)$, satisfies the condition $(-1)^n * f^n(x)$ is non-decreasing for all $n$ in the set of natural numbers (including zero). Moreover, $f(0)$ should be zero and $f'(0)$ should exist and be non-negative. See BernsteinsTheorem for more details.

Representation

A Bernstein function can be represented in the form:

$$ f(x) = a + bx + \int_{0}^{\infty} (1 - e^{-sx})d\mu(s) $$

where $a$ and $b$ are non-negative constants, $\mu$ is a measure on $(0,∞)$ satisfying:

$$ \int_{0}^{\infty} min(1, s)d\mu(s) < \infty $$

This condition ensures that the integral is well-defined. The constants $a$, $b$ and the measure $\mu$ are uniquely determined by $f$.

Applications

Bernstein functions and their corresponding Bernstein processes are crucial in various mathematical domains including stochastic processes, operator theory, potential theory, risk theory, insurance mathematics, and financial mathematics.

Further Properties

The nth derivative $f^n(x)$ exists for all $n$ in the set of natural numbers and is given by:

$$ f^n(x) = n! * \int_{0}^{\infty} s^n * e^{-sx} d\mu(s) \quad \text{for all} \quad x > 0. $$

This formula shows that $f^n$ is a completely monotone function for all $n$ in the set of natural numbers.