Instochastic processes, theStratonovich integral orFisk–Stratonovich integral (developed simultaneously byRuslan Stratonovich andDonald Fisk) is astochastic integral, the most common alternative to theItô integral. Although the Itô integral is the usual choice in applied mathematics, the Stratonovich integral is frequently used in physics.

In some circumstances, integrals in the Stratonovich definition are easier to manipulate. Unlike theItô calculus, Stratonovich integrals are defined such that thechain rule of ordinary calculus holds.

Perhaps the most common situation in which these are encountered is as the solution to Stratonovichstochastic differential equations (SDEs). These are equivalent to Itô SDEs and it is possible to convert between the two whenever one definition is more convenient.

The Stratonovich integral can be defined in a manner similar to theRiemann integral, that is as alimit ofRiemann sums. Suppose that${\displaystyle W:[0,T]\times \mathrm{\Omega }\to \mathbb{R}}$ is aWiener process and${\displaystyle X:[0,T]\times \mathrm{\Omega }\to \mathbb{R}}$ is asemimartingaleadapted to thenatural filtration of the Wiener process. Then theStratonovich integral

${\displaystyle {\int }_0^T{X}_t\circ \mathrm{d}{W}_t}$

is a random variable${\displaystyle :\mathrm{\Omega }\to \mathbb{R}}$ defined as thelimit in mean square of^[1]

${\displaystyle \sum _{i=0}^{k-1}\frac{X_{t_{i+1}}+X_{t_i}}{2}\left(W_{t_{i+1}}-W_{t_i}\right)}$

as themesh of the partition of${\displaystyle [0,T]}$ tends to 0 (in the style of aRiemann–Stieltjes integral). The circle${\displaystyle \circ }$ is a notational device, used to distinguish this integral from the Itô integral.

Many integration techniques of ordinary calculus can be used for the Stratonovich integral, e.g.: if${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ is asmooth function, then

${\displaystyle {\int }_0^T{f}^{\prime }(W_t)\circ \mathrm{d}{W}_t=f(W_T)-f(W_0)}$

and more generally, if${\displaystyle f:\mathbb{R}\times \mathbb{R}\to \mathbb{R}}$ is a smooth function, then

${\displaystyle {\int }_0^T\frac{\mathrm{\partial }f}{\mathrm{\partial }W}(W_t,t)\circ \mathrm{d}{W}_t+{\int }_0^T\frac{\mathrm{\partial }f}{\mathrm{\partial }t}(W_t,t)\mathrm{d}t=f(W_T,T)-f(W_0,0).}$

This latter rule is akin to the chain rule of ordinary calculus.

Stochastic integrals can rarely be solved in analytic form, makingstochasticnumerical integration an important topic in all uses of stochastic integrals. Various numerical approximations converge to the Stratonovich integral, and variations of these are used to solve Stratonovich SDEs (Kloeden & Platen 1992).Note however that the most widely used Euler scheme (theEuler–Maruyama method) for the numeric solution ofLangevin equations requires the equation to be in Itô form.^[2]

If${\displaystyle X_t,Y_t}$, and${\displaystyle Z_t}$ are stochastic processes such that

${\displaystyle X_T-X_0={\int }_0^T{Y}_t\circ \mathrm{d}{W}_t+{\int }_0^T{Z}_t\mathrm{d}t}$

for all${\displaystyle T>0}$, we also write

${\displaystyle \mathrm{d}X=Y\circ \mathrm{d}W+Z\mathrm{d}t.}$

This notation is often used to formulatestochastic differential equations (SDEs), which are really equations about stochastic integrals. It is compatible with the notation from ordinary calculus, for instance

${\displaystyle \mathrm{d}(t^2{W}^3)=3t^2{W}^2\circ \mathrm{d}W+2t{W}^3\mathrm{d}t.}$

TheItô integral of the process${\displaystyle X}$ with respect to the Wiener process${\displaystyle W}$ is denoted by$${\displaystyle {\int }_0^T{X}_t\mathrm{d}{W}_t}$$ (without the circle). For its definition, the same procedure is used as above in the definition of the Stratonovich integral, except for choosing the value of the process${\displaystyle X}$ at the left-hand endpoint of each subinterval, i.e.,

${\displaystyle X_{t_i}}$ in place of${\displaystyle \frac{X_{t_{i+1}}+X_{t_i}}{2}}$

This integral does not obey the ordinary chain rule as the Stratonovich integral does; instead one has to use the slightly more complicatedItô's lemma.

Conversion between Itô and Stratonovich integrals may be performed using the formula

${\displaystyle {\int }_0^{T}f(W_t,t)\circ \mathrm{d}{W}_t=\frac{1}{2}{\int }_0^T\frac{\mathrm{\partial }f}{\mathrm{\partial }W}f(W_t,t)\mathrm{d}t+{\int }_0^{T}f(W_t,t)\mathrm{d}{W}_t,}$

where${\displaystyle f}$ is any continuously differentiable function of two variables${\displaystyle W}$ and${\displaystyle t}$ and the last integral is an Itô integral (Kloeden & Platen 1992, p. 101).

Langevin equations exemplify the importance of specifying the interpretation (Stratonovich or Itô) in a given problem. Suppose${\displaystyle X_t}$ is a time-homogeneousItô diffusion with continuously differentiable diffusion coefficient${\displaystyle \sigma }$, i.e. it satisfies theSDE${\displaystyle \mathrm{d}{X}_t=\mu (X_t)\mathrm{d}t+\sigma (X_t)\mathrm{d}{W}_t}$. In order to get the corresponding Stratonovich version, the term${\displaystyle \sigma (X_t)\mathrm{d}{W}_t}$ (in Itô interpretation) should translate to${\displaystyle \sigma (X_t)\circ \mathrm{d}{W}_t}$ (in Stratonovich interpretation) as

${\displaystyle {\int }_0^T\sigma (X_t)\circ \mathrm{d}{W}_t=\frac{1}{2}{\int }_0^T\frac{d\sigma }{dx}(X_t)\sigma (X_t)\mathrm{d}t+{\int }_0^T\sigma (X_t)\mathrm{d}{W}_t.}$

Obviously, if${\displaystyle \sigma }$ is independent of${\displaystyle X_t}$, the two interpretations will lead to the same form for the Langevin equation. In that case, the noise term is called "additive" (since the noise term${\displaystyle d{W}_t}$ is multiplied by only a fixed coefficient). Otherwise, if${\displaystyle \sigma =\sigma (X_t)}$, the Langevin equation in Itô form may in general differ from that in Stratonovich form, in which case the noise term is called multiplicative (i.e., the noise${\displaystyle d{W}_t}$ is multiplied by a function of${\displaystyle X_t}$ that is${\displaystyle \sigma (X_t)}$).

More generally, for any twosemimartingales${\displaystyle X}$ and${\displaystyle Y}$

${\displaystyle {\int }_0^T{X}_{s-}\circ \mathrm{d}{Y}_s={\int }_0^T{X}_{s-}\mathrm{d}{Y}_s+\frac{1}{2}[X,Y{]}_T^c,}$

where${\displaystyle [X,Y{]}_T^c}$ is the continuous part of thecovariation.

The Stratonovich integral lacks the important property of the Itô integral, which does not "look into the future". In many real-world applications, such as modelling stock prices, one only has information about past events, and hence the Itô interpretation is more natural. In financial mathematics the Itô interpretation is usually used.

In physics, however, stochastic integrals occur as the solutions ofLangevin equations. A Langevin equation is a coarse-grained version of a more microscopic model (Risken 1996); depending on the problem in consideration, Stratonovich or Itô interpretation or even more exotic interpretations such as the isothermal interpretation, are appropriate. The Stratonovich interpretation is the most frequently used interpretation within the physical sciences.

TheWong–Zakai theorem states that physical systems with non-white noise spectrum characterized by a finite noise correlation time${\displaystyle \tau }$ can be approximated by a Langevin equations with white noise in Stratonovich interpretation in the limit where${\displaystyle \tau }$ tends to zero.^{[citation needed]}

Because the Stratonovich calculus satisfies the ordinary chain rule, stochastic differential equations (SDEs) in the Stratonovich sense are more straightforward to define ondifferentiable manifolds, rather than just on${\displaystyle {\mathbb{R}}^n}$. The tricky chain rule of the Itô calculus makes it a more awkward choice for manifolds.

In the supersymmetric theory of SDEs, one considers the evolution operator obtained by averaging the pullback induced on theexterior algebra of thephase space by the stochastic flow determined by an SDE. In this context, it is then natural to use the Stratonovich interpretation of SDEs.

• Øksendal, Bernt K. (2003).Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin.ISBN 3-540-04758-1.
• Gardiner, Crispin W. (2004).Handbook of Stochastic Methods (3 ed.). Springer, Berlin Heidelberg.ISBN 3-540-20882-8.
• Jarrow, Robert; Protter, Philip (2004). "A short history of stochastic integration and mathematical finance: The early years, 1880–1970".IMS Lecture Notes Monograph.45:1–17.CiteSeerX 10.1.1.114.632.
• Kloeden, Peter E.; Platen, Eckhard (1992).Numerical solution of stochastic differential equations. Applications of Mathematics. Berlin, New York:Springer-Verlag.ISBN 978-3-540-54062-5..
• Risken, Hannes (1996).The Fokker-Planck Equation. Springer Series in Synergetics. Berlin, Heidelberg:Springer-Verlag.ISBN 978-3-540-61530-9..

• Definitions of mathematical integration
• Stochastic calculus

• Articles with short description
• Short description matches Wikidata
• All articles with unsourced statements
• Articles with unsourced statements from September 2016