For a strictly positive continuous semimartingale , its stochastic logarithm is . When is a continuous local martingale starting at one, this is a local martingale and the Itô formula gives . Unlike the ordinary logarithm, it has no compensating finite-variation term.
Write a positive continuous local martingale as using its stochastic logarithm. If , its logarithmic bracket must diverge. Otherwise the finite-bracket convergence lemma makes converge finitely and the exponential has a positive limit.
Articles by others on the same topic
The stochastic logarithm is a mathematical concept that arises in the field of stochastic calculus, specifically in the study of stochastic processes. It is used to analyze the logarithmic transformation of stochastic processes, especially when these processes are modeled as continuous-time martingales or processes with some form of randomness, such as Brownian motion. In a more formal sense, the stochastic logarithm refers to the logarithmic transformation applied to stochastic processes, particularly in the context of Itô's calculus.