Analytic solution of geometric Brownian motion

Geometric Brownian motion

I. Description of the probability density function of a normal random variable:

(μ is the overall mean, σ is the standard deviation)

II. Mathematical description of the Brownian motion:

Price time function P(x), the difference between the price at the moment of T+t, P(T+t), and the price at the moment of T, P(T): P( T+t)-P(T) is a normal random variable distributed with mean expectation µt and standard deviation. (T>0, t>0)

Major flaws:

1, according to this price can theoretically have a negative value, but in practice the price is unlikely to have a negative value.

2, regardless of the initial value of the price of what value, a fixed length of time the price difference has the same normal distribution, does not conform to common sense.

Three, geometric Brownian motion:

Change the price difference to the price of the rise and fall: can avoid the defects of the direct use of Brownian motion to describe the price of geometric Brownian motion.

is a normal random variable, the distribution of the mean expected value μt, the standard deviation is. (T>0, t>0)


Geometric Brownian Motion

Geometric Brownian Motion is useful for modeling movements in stock prices. It has the advantage that whereas in a normal formal Brownian motion the value taken may be negative, a geometric Brownian motion takes a value that is never less than 0, which is consistent with the fact that the stock price is never negative.

The expression for the differential form of geometric Brownian motion. Or SDE (Stochastic Differential Equation) form:

Where S(t) can be interpreted as the stock price.

Geometric Brownian motion functional form formulation:

The above equation tells us that the geometric Brownian motion can be obtained by first generating a general form Brownian motion, then solving for its exponential function, and finally multiplying it by S(0), the stock price at the beginning of the period.

Added: why here t coefficient of one more item? You can refer to Ito’s formula for details.

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What was Einstein’s explanation of Brownian motion?

Brown discovered Brownian motion during his lifetime and failed to explain its root cause


Brown (1773 – 1858) was a British botanist, in June 1827 Brown used a microscope to observe the pollen of Clara flowers suspended in water, and found that the pollen particles kept making irregular movements, he wrongly believed that although the pollen was dead, but it had a kind of potential for life left behind, which prompted the pollen to keep moving. Brown and extend this study to a variety of movements, observed all the fresh pollen he collected, and later on coal dust, rock dust, metal powder and other inanimate material particles were observed, are similar to see the phenomenon of non-stop movement, that a variety of powders are present in some kind of activity. And in June and August 1828 published two papers in succession, announced his major discoveries, and later people called this phenomenon “Brownian motion”. But it was not explained correctly for a while.

Two years after Brown’s death, in 1860, the British physicist Maxwell made a preliminary explanation of “Brownian motion” based on his own theory of molecular motion. The discovery of Brownian motion provided the first direct evidence for the theory that matter is composed of molecules, and was a major contribution by Brown to physics.

Brown also made a lot of achievements in the study of botany, he first discovered and named the “cell nucleus” has been used until now.

The Tyndall effect, Brownian motion and electrophoresis of a simple understanding

The Tyndall effect is a colloidal solution of the unique phenomenon, specifically refers to a ray of light into the colloidal solution, available to the naked eye to see a light path, Tyndall phenomenon in fact, is the size of the particles and the scattering of the light caused by:

The particles are small is the real solution, such as saline is, because the particles are too small, the light is scattered is not obvious

The particle size is the suspension, the particle is too large, blocking the passage of light, because the light can not go through, not to mention the pathway.

Brownian motion is even simpler, when it is to observe the pollen is found, in fact, is the molecules due to heat and make irregular movement, because the macro-objects are larger, the force in all directions almost offset, so it can not appear, but small particles such as pollen, the force in all directions is not uniform, so there is a movement, in fact, it is the water molecules caused by the collision.

Electrophoresis is related to the nature of the colloid, the colloid is usually electrically charged, then the positively charged part will move to the cathode, and the negatively charged part will move to the anode, and in this way electrophoresis is produced.