Stochastic differential equation

The concept of stochastic differential equation ( SDE or English abbreviation for stochastic differential equation SDE ) is in mathematics, a generalization of the concept of ordinary differential equation to stochastic processes. Stochastic differential equations are used in many applications to model time-dependent processes, in addition to deterministic influences in addition stochastic disturbances (noise) are exposed.

The mathematical formulation of the problem presented mathematicians with major problems, and so the formal theory of stochastic differential equations was formulated only in the 1940s by the Japanese mathematician Kiyoshi Itō. Together with the stochastic integration justified the theory of stochastic differential equations stochastic analysis.

From the differential to the integral equation

Just as with deterministic functions you want to formulate the relationship between the value of the function and its instantaneous change ( its derivative ) in an equation even in stochastic processes. What in one case leads to an ordinary differential equation, is problematic in the other case, since many stochastic processes, such as the Wiener process are nowhere differentiable.

Nonetheless, it is an ordinary differential equation

Always equivalent integral equation as

Write that requires no explicit mention of the derivation. For stochastic differential equations, is now the opposite way, ie, we define the concept with the help of the corresponding integral equation.

The formulation

Be given two functions, as well as a Brownian motion. The corresponding stochastic integral equation

Is by introducing the differential notation

To the stochastic differential equation. The first integral is to be read as a Lebesgue integral and the second as Itō integral. For given functions and ( as drift and diffusion coefficient referred ) and a Brownian motion so here is one process is sought that satisfies the above integral equation. This process is then a solution of the above SDE.

Existence and uniqueness

If any, on the same probability space as defined random variable, then from the above SDE by adding the condition almost certainly a stochastic initial value problem as a counterpart to the initial value problem for ordinary differential equations.

Also the existence and uniqueness theorem of Picard and Lindelöf found here a correlation: if the following three properties are satisfied:

• , I.e., has finite variance.
• Lipschitz condition: It is a constant, and so that all is applicable to all

• Linear boundedness: There is a constant, so all and applies to all

Then the initial value problem has a unique (up to almost sure equality) solution, which also at all times has finite variance.

Examples

• The SDE for the geometric Brownian motion reads. It is for example used in the Black- Scholes model for the description of stock prices.
• The SDE for an Ornstein - Uhlenbeck process. It is among other things used in the Vasicek model for mathematical modeling of interest rates.
• The SDE for the root diffusion process according to William Feller is

Solving stochastic differential equations and simulation of solutions

Just as with deterministic, there are also stochastic differential equations with no general approach for identifying the solution. In some cases ( as in the above -mentioned Black-Scholes SDE, the solution of a geometric Brownian motion ), it is also possible here, the solution to "guess " and verified by deriving (where the differentiation here by using the lemma Ito takes place ).

In most cases, which arise in practice, for example in the case of the root - diffusion process, however, no closed form solution can be reached. However, one is usually only interested in simulating random paths of the corresponding solution. This can be achieved by approximate numerical discretization, such as the Euler - Maruyama scheme (which is modeled on the explicit Euler method for ordinary differential equations) or the Milstein method.

Stochastic delay differential equations

In a stochastic delay equation ( SDDE, stochastic delay differential equation ) of the future increase depends not only on the present state, but also on the conditions in front of it a limited time interval. Existence and uniqueness are given under similar conditions as in "normal" SDEs. Be steadily, and is an m-dimensional Brownian motion. Then a stochastic delay differential equation is an equation of the form

In which

The corresponding differential notation is then