Stratonovich integral

In stochastic processes, the Stratonovich integral (developed simultaneously by Ruslan L. Stratonovich and D. L. Fisk) is a stochastic integral, the most common alternative to the Itō integral. Although the Ito 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 the Itō calculus, Stratonovich integrals are defined such that the chain rule of ordinary calculus holds.

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

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Definition

The Stratonovich integral can be defined in a manner similar to the Riemann integral, that is as a limit of Riemann sums. Suppose that W�: [0, T] \times \Omega \to \mathbb{R} is a Wiener process and X�: [0, T] \times \Omega \to \mathbb{R} is a semimartingale adapted to the natural filtration of the Wiener process. Then the Stratonovich integral

\int_0^T X_{t} \circ \mathrm{d} W_t�: \Omega \to \mathbb{R}

is defined to be the limit in probability of

\sum_{i = 0}^{k - 1} [(X_{t_{i%2B1}} %2B X_{t_i}) / 2] \left( W_{t_{i%2B1}} - W_{t_i} \right)

as the mesh of the partition 0 = t_{0} < t_{1} < \dots < t_{k} = T of [0, T] tends to 0 (in the style of a Riemann–Stieltjes integral).

Comparison with the Itō integral

Main article: Itō calculus

In the definition of the Itō integral, the same procedure is used except for choosing the value of the process X at the left-hand endpoint of each subinterval: i.e.

X_{t_{i}} in place of (X_{t_{i%2B1}} %2B X_{t_{i}}) / 2.

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

\int_{0}^{T} f(W_{t}) \circ \mathrm{d} W_{t} = \frac{1}{2} \int_{0}^{T} f'(W_{t}) \, \mathrm{d} t %2B \int_{0}^{T} f(W_{t}) \, \mathrm{d} W_{t},

where ƒ is a continuously differentiable function and the last integral is an Itō integral (Kloeden & Platen 1992, p. 101).

It follows that if Xt is a time-homogeneous Itō diffusion with continuously differentiable diffusion coefficient σ (i.e. it satisfies the SDE \mathrm{d} X_t = \mu(X_t)\,\mathrm{d} t %2B \sigma(X_t)\,\mathrm{d} W_t ), we have

\int_{0}^{T} \sigma (X_{t}) \circ \mathrm{d} W_{t} = \frac{1}{2} \int_{0}^{T} \sigma'(X_{t}) \sigma(X_{t}) \, \mathrm{d} t %2B \int_{0}^{T} \sigma (X_{t}) \, \mathrm{d} W_{t}.

More generally, for any two semimartingales X and Y

\int_{0}^{T} X_{s-} \circ \mathrm{d} Y_s = \int_0^T X_{s-}\,\mathrm{d}Y_s%2B \frac{1}{2} [X,Y]_T^c,

where [X,Y]_T^c is the continuous part of the covariation. With probability 1, a general stochastic process does not satisfy the criteria for convergence in the Riemann sense. If it did, then the Itō and Stratonovich definitions would converge to the same solution. As it is, for integrals with respect to Wiener processes, they are distinct.

Usages of the Stratonovich integral

Numerical methods

Stochastic integrals can rarely be solved in analytic form, making stochastic numerical integration an important topic in all uses of stochastic integrals. Various numerical approximations converge to the Stratonovich integral, making this important in numerical solutions of SDEs (Kloeden & Platen 1992). Note however that the most widely used Euler scheme for the numeric solution of Langevin equations requires the equation to be in Itō form.

Stratonovich integrals in real-world applications

The Stratonovich integral lacks the important property of the Itō 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 Itō interpretation is more natural. In financial mathematics the Itō interpretation is usually used.

In physics, however, stochastic integrals occur as the solutions of Langevin equations. A Langevin equation is in fact a coarse-grained version of a more microscopic model; depending on the problem in consideration, Stratonovich or Itō 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.

References