Stratonovich integral

Let $ ( X, Y)=( X( t), Y( t)) _ {t \geq 0 } $ be continuous semi-martingales (cf. Semi-martingale) defined on a filtered probability space $ ( \Omega , {\mathcal F} , {\mathcal F} _ {t} , P) $. The Stratonovich integral of $ Y $ with respect to $ X $ on the interval $ [ 0, t] $ is defined as

$$ \tag{a1 } \int\limits _ { 0 } ^ { t } Y( s) \circ dX( s) = \int\limits Y( s) dX( s)+ \frac{1 }{2 } \langle X, Y \rangle _ {t} , $$

where the integral on the right-hand side is the Itô stochastic integral and $ \langle X, Y \rangle _ {t} $ denotes the quadratic cross-variation process of $ X $ and $ Y $. There is no universally agreed notation for the integral, but the above is the most common (see [a1], [a2], for example). It is also known as the Fisk, Fisk–Stratonovich or symmetrized stochastic integral, the latter in view of the property that

$$ \tag{a2 } \int\limits _ { 0 } ^ { t } Y( s) \circ dX( s) = $$

$$ = \ \lim\limits _ {| \Delta | \rightarrow 0 } \sum _ {i= 1 } ^ { n } \frac{Y( t _ {i} )+ Y( t _ {i- 1 } ) }{2} ( X( t _ {i} )- X( t _ {i- 1 } )), $$

where $ \Delta $ denotes a partition $ 0= t _ {0} < t _ {1} < {} \dots < t _ {n} = t $, $ | \Delta | = \max _ {i} ( t _ {i} - t _ {i- 1 } ) $, and the limit is in probability. Indeed, this property was the original definition of the integral, [a3], [a4]. As an immediate application of (a2) one sees that $ \int _ {[ 0,t] } X \circ dX =( 1/2) X ^ {2} ( t) $, and this points to the main feature of the Stratonovich integral, namely that the Itô formula expressed in terms of Stratonovich integrals coincides with the "ordinary" (Newton–Leibniz) formula. Let $ X ^ {1} ( t) \dots X ^ {d} ( t) $ be continuous semi-martingales, defining $ X( t)=( X ^ {1} ( t) \dots X ^ {d} ( t)) $, and let $ f: \mathbf R ^ {d} \rightarrow \mathbf R $ be a $ C ^ {2} $- function. The Itô formula is

$$ \tag{a3 } f( X( t))- f( X( 0)) = $$

$$ = \ \sum _ {i= 1 } ^ { d } \int\limits _ { 0 } ^ { t } D _ {i} f( X( s)) dX ^ {i} ( s) + \frac{1}{2} \sum _ {i,j= 1 } ^ { d } \int\limits _ { 0 } ^ { t } D _ {i} D _ {j} f( X( s)) d\langle X ^ {i} , X ^ {j} \rangle _ {s} , $$

where $ D _ {i} = \partial / \partial x ^ {i} $. If one writes $ Y( t)= D _ {i} f( X( t)) $, then for $ f \in C ^ {3} $ it is readily verified that

$$ \tag{a4 } \langle Y, X ^ {i} \rangle _ {s} = \sum _ {j= 1 } ^ { d } \int\limits _ { 0 } ^ { t } D _ {i} D _ {j} f( X( s)) d\langle X ^ {i} , X ^ {j} \rangle _ {s} , $$

and hence (a3) becomes simply

$$ \tag{a5 } f( X( t))- f( X( 0)) = \sum _ {i= 1 } ^ { d } \int\limits _ { 0 } ^ { t } D _ {i} f( X( s)) \circ dX ^ {i} ( s) $$

— the formula of "ordinary calculus" . Equations (a4), (a5) remain valid for $ f \in C ^ {2} $[a2], Thm. V20.

At first sight the Stratonovich integral appears to be simply a notational trick to obtain (a5). However, it plays an important role in several areas of stochastic analysis, including:

a) Approximation of stochastic differential equations, [a1], Sect. V1.7. Let $ W ^ {1} ( t) \dots W ^ {p} ( t) $ be independent Brownian motions (cf. Brownian motion) and consider the stochastic differential equation

$$ \tag{a6 } dX( t) = b( X( t)) dt+ \sigma ( X( t)) dW( t), $$

where $ b: \mathbf R ^ {d} \rightarrow \mathbf R ^ {d} $ and $ \sigma : \mathbf R ^ {d} \rightarrow \mathbf R ^ {d \times p } $ are bounded functions with $ b _ {i} \in C ^ {1} ( \mathbf R ^ {d} ) $ and $ \sigma _ {ij } \in C ^ {2} ( \mathbf R ^ {d} ) $. Written in terms of Stratonovich integrals this becomes

$$ \tag{a7 } dX( t) = \widetilde{b} ( X( t)) dt+ \sigma ( X( t)) \circ dW( t), $$

where $ \widetilde{b} ( x) = b( x)- \widehat{b} ( x) $ and

$$ \widehat{b} _ {i} ( x) = \frac{1}{2} \sum _ {i,j } \sigma _ {jk } ( x) D _ {k} \sigma _ {ij} ( x) . $$

Now, let $ W ^ {(} n) ( s) $ be a piecewise-linear approximation to the Brownian path, i.e.

$$ W ^ {(} n) ( s) = \frac{( t _ {i} - s) W( t _ {i-} 1 ) +( s- t _ {i-} 1 ) W( t _ {i} ) }{t _ {i} - t _ {i-} 1 } , $$

$ s \in [ t _ {i-} 1 , t _ {i} ] $, for a partition $ ( t _ {i} ) $ of $ [ 0, t] $ as above, and consider the approximating sequence of random ordinary differential equations

$$ \tag{a8 } \dot{X} ^ {(} n) ( s) = \widetilde{b} ( X ^ {(} n) ( s))+ \sigma ( X ^ {(} n) ( s)) \dot{W} ^ {(} n) ( s), $$

where the dot denotes $ d/ds $. Then

$$ \lim\limits _ {| \Delta | \rightarrow 0 } {\mathsf E} \left ( \sup _ {0 \leq s \leq t } | X( s)- X ^ {(} n) ( s) | ^ {2} \right ) = 0. $$

Thus, the approximation scheme (a8) derives naturally from the Stratonovich equation (a7), and not from the Itô equation (a6). In general, the limit obtained depends on the specific approximation to the Brownian path chosen, unless the $ p $ columns of $ \sigma $, considered as vector fields (cf. Vector field), commute, i.e. $ \sum _ {mk} ( \sigma _ {ml} D _ {m} \sigma _ {kj} - \sigma _ {mj} D _ {m} \sigma _ {kl} ) = 0 $, $ l, j= 1 \dots p $, in which case the limit (a6) or (a7) is obtained for any "reasonable" approximation to the Brownian path; in particular, this is true when $ p= 1 $. Questions of this sort were first investigated by E. Wong and M. Zakai [a5], and $ \widehat{b} ( x) $ is sometimes known as the Wong–Zakai correction term.

b) Support of diffusion processes, [a1], Sect. V1.8, [a6]. Consider the solution of (a6) or (a7) starting from a fixed point $ x \in \mathbf R ^ {d} $. It defines a measure $ {\mathcal P} $ on the sample space $ {\mathcal C} ^ {d} = C([ 0, \infty ) ; \mathbf R ^ {d} ) $. Let $ {\mathcal S} $ be the support of this measure, i.e. the smallest closed set in $ {\mathcal C} ^ {d} $ with $ {\mathcal P} $- measure $ 1 $. Let $ \Phi $ be the set of $ C ^ \infty $- functions in $ {\mathcal C} ^ {p} $, and, for $ \phi \in \Phi $, let $ \zeta ^ \phi \in {\mathcal C} ^ {d} $ be the solution of the ordinary differential equation

$$ \frac{d}{dt} \zeta _ \phi ( t) = \widetilde{b} ( \zeta _ \phi ( t))+ \sigma ( \zeta _ \phi ( t)) \dot \phi ( t) ,\ \zeta _ \phi ( 0) = x. $$

Then $ {\mathcal S}= \mathop{\rm cl} \{ {\zeta _ \phi } : {\phi \in \Phi } \} $. Thus $ {\mathcal S} $ is just the closure of the set of "outputs" $ X( \cdot ) $ of equation (a7) when the "input" $ W $ is replaced by smooth functions $ \phi $. As in a), the Stratonovich formulation is required to preserve consistency between systems with "smooth" and "Brownian" inputs.

c) Diffusion processes on manifolds. Let $ M $ be a $ C ^ \infty $- manifold (cf. Manifold), let $ A _ {0} \dots A _ {p} $ be smooth vector fields on $ M $( cf. Vector field on a manifold), and fix $ x _ {0} \in M $. Then there is a unique $ M $- valued process $ X( t) $ such that $ X( 0)= x _ {0} $ and for $ f \in C ^ \infty ( M) $,

$$ \tag{a9 } df( X( t)) = $$

$$ = \ A _ {0} f( X( t)) dt+ \sum _ { j= } 1 ^ { p } A _ {j} f( X( t)) \circ dW ^ {j} ( t). $$

It is essential to use the Stratonovich integral here; the same equation written with Itô integrals fails to provide a coordinate-independent description, since the rules of Itô calculus conflict with the ordinary rules of calculus relating different coordinate systems. It is immediate from (a9) that $ d\langle A _ {j} f( X( \cdot )), W ^ {j} \rangle _ {t} = A _ {j} ^ {2} f( X( t)) $ and hence that the Itô form of (a9) is

$$ df( X( t)) = $$

$$ = \ \left ( A _ {0} + \frac{1}{2} \sum _ { j= } 1 ^ { p } A _ {j} ^ {2} \right ) f( X( t)) dt + \sum _ { j= } 1 ^ { p } A _ {j} f( X( t)) dW ^ {j} ( t). $$

It follows readily that $ X( t) $ is a diffusion process with differential generator

$$ {\mathcal A} = A _ {0} + \frac{1}{2} \sum _ {1} ^ {p} A _ {j} ^ {2} . $$

A detailed account of the Stratonovich integral and its properties is contained in [a2], Sect. V.5, including an extension of the definition to possibly discontinuous semi-martingales.

References

[a1]	N. Ikeda, S. Watanabe, "Stochastic differential equations and diffusion processes" , North-Holland (1989)
[a2]	P. Protter, "Stochastic integrals and differential equations" , Springer (1990)
[a3]	R.L. Stratonovich, "A new representation for stochastic integrals and equations" SIAM J. Control , 4 (1966) pp. 362–371
[a4]	D.L. Fisk, "Quasi-martingales and stochastic integrals" Techn. Report Dept. Math. Michigan State Univ. , 1 (1963)
[a5]	E. Wong, M. Zakai, "On the relation between ordinary and stochastic differential equations" Int. J. Engin. Sci. , 3 (1965) pp. 213–229
[a6]	D.W. Stroock, S.R.S. Varadhan, "On the support of diffusion processes with applications to the strong maximum principle" , Proc. VI Berkeley Symp. Math. Statist. Probab. , III , Univ. California Press (1972) pp. 333–359