In this case, SDE must be complemented by what is known as “interpretations of SDE” such as Itô or a Stratonovich interpretations of SDEs.
Numerical methods for solving stochastic differential equations include the Euler–Maruyama method, Milstein method and Runge–Kutta method (SDE).
This equation should be interpreted as an informal way of expressing the corresponding integral equation
The equation above characterizes the behavior of the continuous time stochastic process Xt as the sum of an ordinary Lebesgue integral and an Itô integral. Typically, SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process.

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An alternative view on SDEs is the stochastic flow of diffeomorphisms.
The most common form of SDEs in the literature is an ordinary differential equation with the right hand side perturbed by a term dependent on a white noise variable. citation needed
The notation used in probability theory (and in many applications of probability theory, for instance mathematical finance) is slightly different. In particular, it can be inferred from (1) that find this $ X \in S $,
then
$$ \tag{2 }
f( X _ {t} ) = f( X _ {0} ) + \int\limits _ { 0 } ^ { t } f ^ { \prime } ( X _ {s – } ) dX _ {s} +
$$
$$
+

\frac{1}{2}
\int\limits _ { 0 } ^ { t } f ^ { \prime\prime } ( X _ {s – } ) \
d\langle X\rangle ^ {c} + \sum _ {0 s\leq t } [ f( X _ {s} ) – f( X _ {s – } )
– f ^ continue reading this \prime } ( X _ {s – } ) \Delta X _ {s} ] ,
$$
where $ X ^ {c} $
is the continuous martingale part of $ X $,
$ \Delta X _ {s} = X _ {s} – X _ {s – } $.
In stochastic analysis, the principle of “differentiation” of random functions, or Itô formula, is of importance: If $ X ^ {1} \dots X ^ {n} \in S $
and the function $ f = f( x _ {1} \dots x _ {n} ) \in C ^ {2} $,
then
$$
Y = f( X ^ {1} \dots X ^ {n} ) \in S ,
$$
and
$$ \tag{1 }
df( X ^ {1} \dots X ^ {n} ) = \sum _ { i= } 1 ^ { n } \partial _ {i} f \cdot dX ^ {i} +

\frac{1}{2}
\sum _ { i,j= } 1 ^ { n } \partial _ {i} \partial _ {j} f \cdot dX ^ {i} dX
^ {j} ,
$$
where $ \partial _ {i} $
is the partial derivative with respect to the $ i $-
th coordinate. The main use of stochastic calculus in finance is through modeling the random motion of an asset price in the Black-Scholes model.

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The sense of randomly determined is from 1934, from German stochastik (1917). . Its known that no short-rate model can lead to Blacks pricing formula for caplets. Ito’s Lemma is a stochastic analogue of the chain rule of ordinary calculus. This notation makes the exotic nature of the random function of time

m

{\displaystyle \eta _{m}}

in the physics formulation more explicit. .

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