A probabilistic note on the Cauchy functional equation

Martingale Transformations of Brownian Motion with Application to Functional Equations

“…Lemma 2 implies that any measurable solution of ( 21) is locally integrable. Similar assertion for Cauchy's additive functional equation was proved in [10] using the Bernstein theorem on characterization of the normal distributions.…”

“…where λ is the Lebesgue measure and by definition of u(t, x) the function f (x) is continuous (moreover, it is two-times differentiable). It follows from (10) that P (f (W t ) = f (W t )) = 1 for any t ≥ 0 and since Ef (W t ) = E f (W t ), we obtain that for any t ≥ 0…”

Functional Equations for the Stochastic Exponential

“…Proof. To show that f (t, W t ) is integrable for any t ≥ 0 we shall use the idea from [13] on application of the Bernstein theorem.…”

Functional equations and martingales

“…To show that f (W t ) is integrable for any t ≥ 0 we shall use the idea from [21] on application of the Bernstein theorem.…”

“…The present paper was motivated by a note of S. Smirnov [21], where an application of Bernstein's characterization of the normal distribution is given to show that any measurable solution of the Cauchy functional equation ( 1) is locally integrable. We use this idea to show the integrability of the transformed processes f (W t ), where W = (W t , t ≥ 0) is a Brownian Motion.…”