What is Brownian Motion?

Brownian Motion refers to the irregular movement exhibited by tiny particles. In 1827, British botanist R. Brown observed the pollen's non-stop irregular movement in an aqueous solution of pollen particles. Further experiments have confirmed that not only pollen particles but also other particles suspended in fluids also exhibit such irregular movements, such as dust suspended in air. Later generations called this particle motion Brownian motion. Taking garcinia cambogia particles suspended in water as an example, a garcinia cambogia particle with a radius of 2x10 ^ -7 meters has a mass of about 3x10 ^ -17 kilograms, and its movement rate is close to 0.02 m / s at 27 ° C. At first people did not understand the cause of this movement. In 1877, J. Delsoe first pointed out that Brownian motion was caused by the imbalance force of particles colliding with liquid molecules. Subsequently, in 1904, the French scientist H. Pancaré made a further explanation: large objects (such as 0.1 mm linearity) will be impacted by moving atoms from various aspects, and the blows are very frequent. The law of probability makes them compensate each other, so they do not mobile. Tiny particles are hit too little to compensate. In other words, Brownian motion is the macroscopic expression of liquid molecules in non-stop and irregular thermal motion. 1905-1906 A. Einstein and M.von Smolukhovsky respectively published articles on theoretical analysis of Brownian motion. In 1908 Piran experimentally verified Einstein's theory, so that the physical images of molecular kinetic theory were widely accepted.

{B (t)} Brownian motion is also known as

Linking the Brownian motion with stock price behavior, and then establishing a mathematical model of the Wiener process is an important financial innovation of this century and occupies an important position in modern financial mathematics. So far, the general view still holds that the stock market is fluctuating randomly. Random fluctuations are the most fundamental characteristic of the stock market and the normal state of the stock market.

The Brownian motion assumption is the core assumption of modern capital market theory. Modern capital market theory believes that securities and futures prices have random characteristics. The so-called randomness here refers to the memorylessness of the data, that is, past data does not constitute a basis for predicting future data. No surprisingly similar iterations will occur. The mathematical definition of a random phenomenon is: the results of individual experiments show uncertainty; the results of a large number of repeated experiments have statistical regularity. The Wiener process of Brownian motion, which describes one of the stock price behavior models, is a special form of Markov stochastic process;

One-dimensional brownian motion

The husband process is a special type of random process. Stochastic process is a probability model based on probability space, which is considered to be the dynamics of probability theory, that is, its research object is a random phenomenon that evolves with time. So random behavior is a statistically regular behavior. Stock price behavior models are usually expressed using the well-known Wiener process. It is tempting to assume that the stock price follows a generalized Wiener process, that is, it has a constant rate of expected drift and variance. The Wiener process shows that only the current value of the variable is related to future predictions, and the past history of the variables and how the variables have evolved from the past to the present are not related to future predictions. The Markovian nature of stock prices is consistent with the weak form of market efficiency, that is, the current price of a stock already contains all the information, including of course all past price records. But when people started to use fractal theory to study financial markets, they found that its operation does not follow Brownian motion, but obeys the more general geometrical Browmrian motion. ^[11]