Academic literature on the topic 'Random time change of Brownian motion and symmetry'

Chapter 22 studies stochastic dynamical equations, consistent with the detailed balance condition, which are generalized Langevin equations which describe a wide range of phenomena from Brownian motion to critical dynamics in continuous phase transitions. In the latter case, a dynamic action can be associated to the Langevin equation, which can be renormalized with the help of BRST symmetry. Dynamic renormalization group equations, describing critical dynamics, are then derived. Dynamic scaling follows, with a correlation time that exhibits critical slowing down governed by a dynamic exponent. In addition, Jarzinsky’s relation is derived in the case of a time–dependent driving force.

This chapter is devoted to the study of Langevin equations, first order in time differential equations, which depend on a random noise, and which belong to a class of stochastic differential equations that describe diffusion processes, or random motion. From a Langevin equation, a Fokker–Planck (FP) equation for the probability distribution of the solutions, at given time, of the Langevin equation can be derived. It is shown that observables averaged over the noise can also be calculated from path integrals, whose integrands define automatically positive measures. The path integrals involve dynamic actions that have automatically a Becchi–Rouet–Stora–Tyutin (BRST) symmetry and, when the driving force derives from a potential, exhibit the simplest form of supersymmetry. In some cases, like Brownian motion on Riemannian manifolds, difficulties appear in the precise definition of stochastic equations, quite similar to the quantization problem encountered in quantum mechanics (QM). Time discretization provides one possible solution to the problem.

Power laws and scaling are two features that have been known for some time in the distribution of returns (i.e., price fluctuations), and, more recently, in the distribution of volumes (i.e., numbers of shares traded) of financial assets. As in numerous examples in physics, these power laws can be understood as the asymptotic behavior of distributions that derive from nonextensive thermostatistics. Recent applications of the (Q-Gaussian distribution to returns of exchange rates and stock indices are extended here for individual U.S. stocks over very small time intervals and explained in terms of a feedback mechanism in the dynamics of price formation. In addition, we discuss some new empirical findings for the probability density of low volumes and show how the overall volume distribution is described by a function derived from q-exponentials. In March 1900 at the Sorbonne, a 30-year-old student—who had studied under Poincaré—submitted a doctoral thesis [2] that demonstrated an intimate knowledge of trading operations in the Paris Bourse. He proposed a probabilistic method to value some options on rentes, which were then the standard French government bonds. His work was based on the idea that rente prices evolved according to a random-walk process that resulted in a Gaussian distribution of price differences with a dispersion proportional to the square root of time. Although the importance of Louis Bachelier's accomplishment was not recognized by his contemporaries [24], it preceded by five years Einstein's famous independent, but mathematically equivalent, description of diffusion under Brownian motion. The idea of a Gaussian random-walk process (later preferably applied to logarithmic prices) eventually became one of the basic tenets of most twentieth-century quantitative works in finance, including the Black-Scholes [3] complete solution to the option-valuation problem—of which a special case had been solved by Bachelier in his thesis. In the times of the celebrated Black-Scholes solution, however, a change in perspective was already under way. Starting with the groundbreaking works of Mandelbrot [18] and Fama [11], it gradually became apparent that probability distribution functions of price changes of assets (including commodities, stocks, and bonds), indices, and exchange rates do not follow Bachelier's principle of Gaussian (or "normal") behavior.