Extensions of Black–Scholes processes and Benford’s law

Let Z be a stochastic process of the form Z ( t ) = Z ( 0 ) exp ( μ t + X ( t ) − 〈 X 〉 t / 2 ) where Z ( 0 ) > 0 , μ are constants, and X is a continuous local martingale having a deterministic quadratic variation 〈 X 〉 such that 〈 X 〉 t → ∞ as t → ∞ . We show that the mantissa (base b ) of Z ( t ) (denoted by M ( b ) ( Z ( t ) ) converges weakly to Benford’s law as t → ∞ . Supposing that 〈 X 〉 satisfies a certain growth condition, we obtain large deviation results for certain functionals (including occupation time) of ( M ( b ) ( Z ( t ) ) ) . Similar results are obtained in the discrete-time case. The latter are used to construct a non-parametric test for nonnegative processes ( Z ( t ) ) (based on the observation of significant digits of ( Z ( n ) ) ) of the null hypothesis H 0 ( σ 0 ) which says that Z is a general Black–Scholes process having a volatility σ ≥ σ 0 ( > 0 ) . Finally it is shown that the mantissa of Brownian motion is not even weakly convergent.