  Intrinsically noisy mechanisms drive most physical, biological and economic
phenomena, from stock pricing to phenotypic variability. Frequently, the
system's state influences the driving noise intensity, as, for example, the
actual value of a commodity may alter its volatility or the concentration of
gene products may regulate their expression. All these phenomena are often
modeled using stochastic differential equations (SDEs). However, an SDE is not
sufficient to fully describe a noisy system with a multiplicative feedback,
because it can be interpreted according to various conventions -- in
particular, It\^{o} calculus and Stratonovitch calculus --, each of which leads
to a qualitatively different solution. Which convention to adopt must be
determined case by case on the basis of the available experimental data; for
example, the SDE describing electrical circuits driven by a noise are known to
obey Stratonovich calculus. Once such an SDE-convention pair is determined, it
c an be employed to predict the system's behavior under new conditions. Here,
we experimentally demonstrate that the convention for a given physical system
may actually vary under varying operational conditions. We show that, under
certain conditions, a noisy electric circuit shifts to obey It\^o calculus,
which may dramatically alter the system's long term stability. We track such
Stratonovich-to-It\^o transition to the underlying dynamics of the system and,
in particular, to the ratio between the driving noise correlation time and the
feedback delay time. We briefly discuss ramifications of our conclusions for
biology and economics: the possibility of similar transitions and their
dramatic consequences should be recognized and accounted for where SDEs are
employed to predict the evolution of complex phenomena.
