Stratonovich Integrals

Motivation: why this matters in quant finance

Most finance texts use Itô integration because trading strategies are non-anticipative: the position over a small interval is chosen from information available at the left endpoint. Lawler's construction is explicitly this Itô construction, and his product, quadratic-variation, and generator formulas are all Itô formulas.

Stratonovich notation appears when stochastic models are imported from physics, engineering, or rough-noise limits. A quant who reads

dX_t=a(X_t)\,dt+b(X_t)\circ dW_t

must translate it before applying the Itô's lemma, infinitesimal generator, or Feynman-Kac machinery used elsewhere in the vault.

This is a careful comparison note, not a claim that Lawler develops Stratonovich integration as a separate chapter. The source grounding here is Lawler's Itô construction: left-endpoint stochastic integrals, quadratic variation, Itô's formula, and the product rule. The Stratonovich formulas are included only to explain how that alternative convention differs from the Lawler/finance convention.

The informal idea

In an ordinary Riemann integral, it does not matter whether the sample point inside each small interval is the left endpoint, midpoint, or right endpoint, provided the integrator has finite variation. Brownian motion does not have finite variation. The choice of sample point changes the limiting integral.

Itô integration uses left endpoints. That is why the integrand is adapted and why the integral has martingale properties under square-integrability conditions. It also means ordinary chain rules fail: the surviving quadratic variation produces the extra $\frac{1}{2}f''$ term in Itô's formula.

Stratonovich integration uses a symmetric, midpoint-style convention. That convention restores the ordinary-looking chain rule, but it pays for that by changing the drift when converted to Itô form. For finance, that conversion is the practical point: pricing PDEs and martingale arguments are usually written in Itô form.

Formal definitions

Lawler defines the Itô integral by approximating with simple adapted processes and, ultimately, left-endpoint information. For comparison, the Stratonovich integral is denoted with a circle:

\int_0^t H_s\circ dW_s.

For sufficiently regular semimartingale integrands, the conversion to Itô form is

\int_0^t H_s\circ dW_s =\int_0^t H_s\,dW_s+\frac{1}{2}\langle H,W\rangle_t.

In the common one-dimensional diffusion case, the Stratonovich SDE

dX_t=\alpha(X_t)\,dt+\beta(X_t)\circ dW_t

corresponds to the Itô SDE

dX_t=\left[\alpha(X_t)+\frac{1}{2}\beta(X_t)\beta'(X_t)\right]dt+\beta(X_t)\,dW_t.

Equivalently, the Itô SDE

dX_t=a(X_t)\,dt+b(X_t)\,dW_t

has Stratonovich drift

\alpha(x)=a(x)-\frac{1}{2}b(x)b'(x).

The conversion term is exactly the quadratic-variation correction. It is not a new source of economic drift; it is the accounting difference between two stochastic integration conventions.

Key properties

Stratonovich restores the ordinary-looking chain rule

For smooth $f$ , the Stratonovich notation is designed so that

df(X_t)=f'(X_t)\circ dX_t

looks like the deterministic chain rule. In Itô form, the same transformation includes the quadratic-variation term

df(X_t)=f'(X_t)\,dX_t+\frac{1}{2}f''(X_t)\,d\langle X\rangle_t.

The drift changes under conversion

The diffusion coefficient is the same, but the drift shifts by

\frac{1}{2}bb'

. Finance consequence: two SDEs that look different may describe the same stochastic model under different conventions, and two SDEs that look similar may not.

Itô is the default convention for trading gains

Lawler's betting interpretation of $\int A_s\,dW_s$ uses information available at time $s$ . That is the natural convention for self-financing strategies: the hedge held over the next small interval cannot depend on the shock realised inside that interval.

Generators are Itô objects

The generator for

dX_t=a(X_t)\,dt+b(X_t)\,dW_t

Lf(x)=a(x)f'(x)+\frac{1}{2}b^2(x)f''(x).

If the model is written in Stratonovich form, convert to Itô form before reading off the generator.

The source does not develop a full Stratonovich theory

Lawler's book gives the tools needed to understand the correction term: Itô integrals, quadratic variation, Itô's formula, and the product rule. It does not present Stratonovich integration as a parallel theory, so this lesson deliberately avoids broad claims about its general construction.

Worked examples

Example 1: converting a multiplicative-noise model

Suppose a model is written in Stratonovich form as

dX_t=\alpha X_t\,dt+\sigma X_t\circ dW_t.

Here $\beta(x)=\sigma x$ and $\beta'(x)=\sigma$ , so the Itô drift is

a(x)=\alpha x+\frac{1}{2}\sigma^2x.

The Itô form is

dX_t=\left(\alpha+\frac{1}{2}\sigma^2\right)X_t\,dt+\sigma X_t\,dW_t.

If a finance calculation uses the generator, it must use the Itô drift $\left(\alpha+\frac{1}{2}\sigma^2\right)x$ , not the Stratonovich drift $\alpha x$ .

Example 2: converting Itô GBM into Stratonovich notation

Start from Itô geometric Brownian motion:

dS_t=\mu S_t\,dt+\sigma S_t\,dW_t.

With $b(s)=\sigma s$ and $b'(s)=\sigma$ , the equivalent Stratonovich drift is

\alpha(s)=\mu s-\frac{1}{2}\sigma^2s.

So the Stratonovich notation is

dS_t=\left(\mu-\frac{1}{2}\sigma^2\right)S_t\,dt+\sigma S_t\circ dW_t.

This resembles the log-price drift, which is one reason the convention can feel natural. But the Itô version is the one used for trading gains, generators, and pricing PDEs in this vault.

Example 3: why the midpoint convention changes $\int W_s\,dW_s$

Lawler computes the Itô integral

\int_0^t W_s\,dW_s=\frac{1}{2}(W_t^2-t).

The missing ordinary-calculus answer $\frac{1}{2}W_t^2$ differs by $\frac{1}{2}t$ , which is half the quadratic variation of Brownian motion. Stratonovich notation captures the ordinary-looking identity

\int_0^t W_s\circ dW_s=\frac{1}{2}W_t^2.

This example is the whole comparison in miniature: Itô preserves martingale/no-look-ahead structure; Stratonovich preserves the classical chain rule.

Common confusions and pitfalls

"Stratonovich is the correct calculus and Itô is an approximation." They are different conventions. Finance defaults to Itô because left-endpoint, adapted trading strategies match the economics of self-financing portfolios.

"The circle notation is cosmetic." The circle changes the drift after conversion. Ignoring it can move a model's expected growth rate by $\frac{1}{2}bb'$ .

"Lawler gives a full Stratonovich chapter." He does not. This note uses Lawler's Itô material to explain the comparison and flags the Stratonovich formulas as conversion rules.

"The generator can be read directly from Stratonovich drift." The generator uses the Itô drift. Convert first, then apply $Lf=af'+\frac{1}{2}b^2f''$ .

"Restoring the ordinary chain rule removes quadratic variation." Quadratic variation is still there. Stratonovich notation hides it inside the conversion between conventions.

Where this goes next

Stochastic Integrals: gives Lawler's Itô integral construction and the betting interpretation.
Quadratic Variation: explains the surviving second-order term that creates the conversion correction.
Itô Product and Quotient Rules: shows the algebra Stratonovich notation is often contrasted against.
Infinitesimal Generators and Kolmogorov Equations: uses the Itô drift after any convention conversion.
Feynman-Kac Formula: applies generator-based Itô calculus to discounted expectations.

References

Lawler, G. F. (2023). Stochastic Calculus: An Introduction with Applications. Ch. 3 §3.1-§3.4 (Itô stochastic integration and Itô's formula), Ch. 3 §3.6 (Covariation and product rule). Lawler does not develop Stratonovich integration as a standalone theory; this lesson uses these Itô sections to ground the comparison.