Stochastic differential

A random interval function $ dX $ defined by the formula

$$ ( dX) I = X _ {t} - X _ {s} ,\ \ I = ( s, t], $$

for every process $ X = ( X _ {t} , {\mathcal F} _ {t} , {\mathsf P}) $ in the class of semi-martingales $ S $, with respect to a stochastic basis $ ( \Omega , {\mathcal F} , ( {\mathcal F} _ {t} ) _ {t \geq 0 } , {\mathsf P}) $. In the family of stochastic differentials $ dS = \{ {dX } : {X \in S } \} $ one introduces addition $ ( A) $, multiplication by a process $ ( M) $ and the product operation $ ( P) $ according to the following formulas:

$ ( A) $ $ dX + dY = d( X+ Y) $;

$ ( M) $ $ ( \Phi dX) ( s, t] = \int _ {s} ^ {t} \Phi dX $( a stochastic integral, $ \Phi $ being a locally bounded predictable process which is adapted to the filtration $ ( {\mathcal F} _ {t} ) _ {t \geq 0 } $);

$ ( P) $ $ dX \cdot dY = d( XY) - X _ {-} dY - Y _ {-} dX $, where $ X _ {-} $ and $ Y _ {-} $ are the left-continuous versions of $ X $ and $ Y $.

It then turns out that

$$ ( dX \cdot dY) ( s, t] = \mathop{\rm l}.i.p. _ {| \Delta | \rightarrow 0 } \sum _ {i=1}^ { n } ( X _ {t _ {i} } - X _ {t _ {i-1} } )( Y _ {t _ {i} } - Y _ {t _ {i-1} } ), $$

where $ \Delta = ( s= t _ {0} < t _ {1} < \dots < t _ {n} = t) $ is an arbitrary decomposition of the interval $ ( s, t] $, l.i.p. is the limit in probability, and $ | \Delta | = \max | t _ {i} - t _ {i-1} | $.

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. In particular, it can be inferred from (1) that if $ X \in S $, then

$$ \tag{2 } f( X _ {t} ) = f( X _ {0} ) + \int\limits _ { 0 } ^ { t } f ^ { \prime } ( X _ {s - } ) dX _ {s} + $$

$$ + \frac{1}{2} \int\limits _ { 0 } ^ { t } f ^ { \prime\prime } ( X _ {s - } ) \ d\langle X\rangle ^ {c} + \sum _ {0< s\leq t } [ f( X _ {s} ) - f( X _ {s - } ) - f ^ { \prime } ( X _ {s - } ) \Delta X _ {s} ] , $$

where $ X ^ {c} $ is the continuous martingale part of $ X $, $ \Delta X _ {s} = X _ {s} - X _ {s - } $.

Formula (2) can be given the following form:

$$ f( X _ {t} ) = f( X _ {0} ) + \int\limits _ { 0 } ^ { t } f ^ { \prime } ( X _ {s - } ) dX _ {s} + \frac{1}{2} \int\limits _ { 0 } ^ { t } f ^ { \prime\prime } ( X _ {s - } ) d[ X, X] _ {s} + $$

$$ + \sum _ { 0< } s\leq t \left [ f( X _ {s)} - f ( X _ {s - } ) - f ^ { \prime } ( X _ {s - } ) \Delta X _ {s} - \frac{1}{2} f ^ { \prime\prime } ( X _ {s - } ) ( \Delta X _ {s} ) ^ {2} \right ] , $$

where $ [ X, X] $ is the quadratic variation of $ X $.

References

[1]	K. Itô, S. Watanabe, "Introduction to stochastic differential equations" K. Itô (ed.) , Proc. Int. Symp. Stochastic Differential Equations Kyoto, 1976 , Wiley (1978) pp. I-XXX
[2]	I.I. Gikhman, A.V. Skorokhod, "Stochastic differential equations and their applications" , Kiev (1982) (In Russian)

Comments

The product $ dX \cdot dY $ is more often written as $ d[ X, Y] $, where the so-called "square bracket" $ [ X, Y] $ is the process with finite variation such that $ [ X, Y] _ {t} = X _ {0} Y _ {0} + dX \cdot dY( 0, t] $. When $ X= Y $, one obtains the quadratic variation $ [ X, X] $ used at the end of the main article. Actually, it is a probabilistic quadratic variation: when $ X $ is a standard Brownian motion, $ d[ X, X] $ is the Lebesgue measure, but the true quadratic variation of the paths is almost surely infinite. See also Semi-martingale; Stochastic integral; Stochastic differential equation.

For the study of continuous-path processes evolving on non-flat manifolds the Itô stochastic differential is inconvenient, because the Itô formula (2) is incompatible with the ordinary rules of calculus relating different coordinate systems. A coordinate-free description can be obtained using the Stratonovich differential; see [a1], [a2], Chapt. 5, [a3], and Stratonovich integral.

References

[a1]	K.D. Elworthy, "Stochastic differential equations on manifolds" , Cambridge Univ. Press (1982)
[a2]	N. Ikeda, S. Watanabe, "Stochastic differential equations and diffusion processes" , North-Holland (1989) pp. 97ff
[a3]	P.A. Meyer, "Géométrie stochastique sans larmes" J. Azéma (ed.) M. Yor (ed.) , Sém. Probab. Strasbourg XV , Lect. notes in math. , 850 , Springer (1981) pp. 44–102
[a4]	I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus" , Springer (1988)
[a5]	L.C.G. Rogers, D. Williams, "Diffusion, Markov processes, and martingales" , 2. Itô calculus , Wiley (1987)
[a6]	S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, T. Lindstrøm, "Nonstandard methods in stochastic analysis and mathematical physics" , Acad. Press (1986)