One calls method of Monte Carlo any method aiming at calculating a numerical value, and using random processes, i.e. probabilistic techniques. The name of these methods refers to the games of chance practiced with Monte Carlo.

The methods of Monte Carlo are particularly used to calculate integrals in dimensions larger than 1 (in particular, to calculate surfaces, volumes, etc)

The method of simulation of Monte Carlo also makes it possible to introduce a statistical approach of the risk into a financial decision. It consists in isolating a certain number of variable-keys from the project such as the turnover or the margin… and affecting a probability distribution to them. For each one of these factors, one carries out a great number of random pullings in the probability distributions determined previously, in order to determine the probability of occurrence of each result.

The true development of the methods of Monte Carlo was carried out, under the impulse of John von Neumann and Stanislas Ulam in particular, at the time of the Second world war and research on the manufacture of the atomic bomb. In particular, they used these probabilistic methods to solve partial derivative equations.

Theory

We lay out of the definition of the mathematical Espérance of a function $g$ of random variable $X$, according to which $E\; (G\; (X))=\; \backslash \; int\_\; \{\backslash \; Omega\}\; G\; (X)\; f\_X\; (X)\; dx$

$f\_X$ is the function of density. This can be wide with the probabilities discréte while summoning thanks to a measurement $\backslash \; nu$ discréte, of type Dirac.

The idea is to produce a sample $(x\_1,\; x\_2,\dots ,\; x\_N)$ of the law $X$, and to calculate a new estimator known as of Monte Carlo, starting from this sample. This estimator is built starting from the empirical average, which is an unbiassed estimate of the hope: $\backslash \; tilde\; \{g\_n\}\; (X)\; =\; \backslash \; frac\; \{1\}\; \{N\}\; \backslash \; sum\_\; \{i=1\}\; ^\; \{NR\}\; G\; (x\_i)$

This is the estimator of Monte Carlo. We see well that by replacing the sample by a whole of values taken in the support of a Intégrale, and function has to integrate, we can thus build an approximation of its value, built statistically.

Examples

Resolution of the Problem of the sales representative

The resolution of the Problem of the sales representative request of time, and the complicated algorithms. The method of Monte Carlo can provide within this framework a method of resolution effective.

Determination of the value of π (pi)

This method is close to that of Buffon.

That is to say a point M of coordinates (X, there) 0 One draws by chance the values from X and Y.

If then the point M belongs to the disc of center (0,0) of 1.

The probability that the point M belongs to the disc is π/4.

By submitting the report/ratio of the number of points in the disc compared to the number of pulling one obtains an approximation of the π/4 number if the number of pulling is large.

Determination of the surface of a lake

This example is traditional in popularization of the method of Monte Carlo. That is to say a rectangular or square zone of which the length on the sides are known. Within this surface a lake is whose surface is unknown. Thanks to measurements on the sides of the zone, one knows the surface of the rectangle. To find the surface of the lake, one asks an army to draw X blows from gun in a random way on this zone. One counts then the number NR of balls which remained on the ground, one can thus determine the number of balls which fell into the lake: X-N. It is then enough to draw up a relationship between the values:

$\backslash \; frac\; \{\backslash \; mathrm\; \{surface\}\; \_\; \{~\; \backslash \; mathrm\; \{ground\}\}\}\; \{\backslash \; mathrm\; \{surface\}\; \_\; \{~\; \backslash \; mathrm\; \{lake\}\}\}\; =\; \backslash \; frac\; \{X\}\; \{X-N\}$

$\backslash \; Longrightarrow\; \backslash \; qquad\; \backslash \; mathrm\; \{surface\}\; \_\; \{~\; \backslash \; mathrm\; \{lake\}\}\; =\; \backslash \; frac\; \{(X-N)\}\{X\}\; \backslash \; \backslash \; times\; \backslash \; \backslash \; mathrm\; \{surface\}\; \_\; \{~\; \backslash \; mathrm\; \{ground\}\}$

For example, if the ground makes 1000 m, that the army draws 500 balls and that 100 projectiles fell into the lake then the surface from the water level is of: 100*1000/500 = 200 Mr.

Of course, the quality of the estimate improves by increasing the number of shootings and by making sure that the artillerists always do not aim the same place but cover the zone well. This last remark is to be put in parallel with the quality of the random generating which is paramount to have good performances in the method of Monte Carlo. A skewed generator is like a gun which always draws at the same place: information which it brings is reduced.

Application to the model of Ising

See also: Model of Ising

August 1st

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