  We study the temporal fluctuations in time-dependent stock prices (both
individual and composite) as a stochastic phenomenon using general techniques
and methods of nonequilibrium statistical mechanics. In particular, we analyze
stock price fluctuations as a non-Markovian stochastic process using the
first-passage statistical concepts of persistence and survival. We report the
results of empirical measurements of the normalized $q$-order correlation
functions $f_q(t)$, survival probability $S(t)$, and persistence probability
$P(t)$ for several stock market dynamical sets. We analyze both
minute-to-minute and higher frequency stock market recordings (i.e., with the
sampling time $\delta t$ of the order of days). We find that the fluctuating
stock price is multifractal and the choice of $\delta t$ has no effect on the
qualitative multifractal behavior displayed by the $1/q$-dependence of the
generalized Hurst exponent $H_q$ associated with the power-law evolution of the
correlation function $f_q(t)\sim t^{H_q}$. The probability $S(t)$ of the stock
price remaining above the average up to time $t$ is very sensitive to the total
measurement time $t_m$ and the sampling time. The probability $P(t)$ of the
stock not returning to the initial value within an interval $t$ has a universal
power-law behavior, $P(t)\sim t^{-\theta}$, with a persistence exponent
$\theta$ close to 0.5 that agrees with the prediction $\theta=1-H_2$. The
empirical financial stocks also present an interesting feature found in
turbulent fluids, the extended self-similarity.
