Stochastic Calculation

The stochastic calculation is the study of the random phenomena depending on time. For this reason, it is an extension of the theory of the Probabilité S.

Applications

The scope of application of stochastic calculation includes/understands:

the quantum Mechanical ,
the Treatment of the signal,
the Chemistry,
financial Mathematical ,
and even the music.

It is also used in the forecasts of behavior of the wind and the airstreams.

Random processes

A random process $X$ is a family of random variable indexed by a subset of $\ mathbb R$ or $\ mathbb N$ , often compared to time (see also stochastic Processus). It is thus a function of two variables: the time and the state of the world $\ omega$ . The whole of the states of the world is traditionally noted $\ Omega$ . The application which with $t$ associates $X (\ Omega, T)$ is called trajectory process. The Brownian Movement is a particularly simple example of random process indexed by $\ mathbb R$ . It can be defined like the single process $W_t$ with Gaussian increase such as the correlation between $W_t$ and $W_s$ is $min (T, S)$ . One can also see it like the limit of a random walk when the step of time tends towards 0.

Filtrations

A filtration

F_t

t \ in \ mathbb {NR}

is a family of encased subgroups of

\ Omega

, who can be interpreted as the information available which evolves/moves during time. to supplement

Conditional hope according to a filtration

Process of Itō

Integral of Itô

Definition of a process of Itô

Once specified the definition chosen for a stochastic integral, one defines then a process of Itô as being a stochastic process

X_t

form

$X_t = X_0 + \ int_0^t U (S, \ Omega) {\ rm D} S + \ int_0^t v (S, \ Omega) {\ rm D} B_s$

with $u$ and $v$ two random functions satisfying some technical assumptions of adaptation to the process $B_t$ and $\ omega$ is a realization in the subjacent space of probability.

In the formalism of differential calculus with the regulation of Itô one notes in an equivalent way the preceding relation like

Another regulation

There exists another notable regulation to define a stochastic integral, it is the prescritpion of Stratonovich. The Intégrale of Stratonovich is defined like the limit of the discrete sums

$\ sum X_ {\ frac {t_i+t_ {i+1}} {2}} (B_ {t_ {i+1}} - B_ {t_i}).$

The notable difference with the regulation of Itô is that the quantity $X_ {\ frac {t_i+t_ {i+1}} {2}}$ is not independent within the meaning of the probabilities of the variable $B_ {t_ {i+1}} - B_ {t_i}$ . Thus, contrary to the regulation of Ito, in the regulation of Stratonovich one has

$E \ left X_t \, dB_t \ right \ neq 0$

what complicates, from this point of view, certain calculations. However the use of the regulation of Stratonovich does not choose a direction of time privileged contrary to that of Itô what implies that the stochastic processes defined in by the integral of Stratonovich satisfy invariant stochastic differential equations by inversion of time. For this reason, this regulation is often used in Physique statistics.

It should be noted however that it is possible to pass from the one to the other of the regulations by carrying out simple changes of variables what make them equivalent. The choice of regulation is thus a question of suitability.

Usual processes

Exponential martingales

Integral of Wiener and stochastic integral

to supplement

Either $Z$ the definite standard Brownian movement on space probabilized $(\ Omega, has, F, P)$ and σ a process adapted to F. One supposes in addition that σ checks:
$E \ left (\ int_0^T \ sigma_s^2 ds \ right) < + \ infty$ .
Then, the stochastic integral of σ compared to Z is the random variable:
$\ left (\ int_0^T \ sigma_s dZ_s \ right) = \ lim_ {NR \ to + \ infty} \ sum_ {n=1} ^N \ sigma_ {n-1} \ left (Z_n - Z_ {n-1} \ right)$ .

Lemma of Itô

Either X a stochastic process such as one has dx = a*dt + B *dz where Z is a process of standard Wiener.

Then according to the Lemma of Ito, one has for a function G = G (X, T)

$dG = \ frac {dG} {dt} dt + \ frac {dG} {dx} dx + \ frac {1} {2} b^2 \ frac {d^2 G} {dx^2} dt$

Stochastic differential equations

A stochastic differential equation (EDS) is the data of an equation of the

dX type = \ driven (X, T) dt + \ sigma (X, T) dW_t

, where

X

is an unknown random process, that one calls equation of diffusion commonly. To integrate the EDS, it is to find the whole of the process checking the diffusion entiere.

Process of Orstein-Uhlenbeck

The process of Ornstein-Uhlenbeck is a stochastic process describing (inter alia) the speed of a particle in a fluid, in dimension 1.

One defines it as being the solution $X_t$ of the following stochastic differential equation: $dX_t= \ sqrt2dB_t-X_tdt$ , where $B_t$ is a standard Brownian movement, and with $X_0$ a given random variable. The $dB_t$ term translates the many random shocks undergone by the particle, whereas the term $- {X_t} dt$ represents the force of friction undergone by the particle.

The formula of Itô applied to the process ${e^t} X_t$ gives us: $d ({e^t} X_t) = {e^t} {X_t} dt+ {e^t} (\ sqrt {2} {dB_t} - {X_t} dt) + {e^t} dt= {e^t} \ sqrt {2} {dB_t} + {e^t} dt$ , maybe, in integral form: $X_t= {X_0} e^ {- T} + \ sqrt {2} e^ {- T} \ int_0^t {e^s} dB_s$

For example, if $X_0$ is worth Presque surely $x$ , the law of $X_t$ is a Gaussian law of average $xe^ {- T}$ and of variance $1-e^ {- 2t}$ , what converges in law when $t$ tends towards the infinite one towards the reduced centered Gaussian law.

Problems of optimal control

Methods of simulation

Method of Monte Carlo

The methods of Monte Carlo rest on the law of the great numbers: by repeating a great number of times an experiment, a way (theoretically) independent, one obtains an increasingly reliable approximation of the true value of the hope of the phenomenon observed.

Such methods are in particular used finances some for the valorization of Options for which there does not exist closed formula, but only of the numerical approximations.

Simulation by recombining trees

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