How to Convert Between Itô and Stratonovich Integrals

The Itô integral uses left-endpoint sampling, while the Stratonovich integral for continuous semimartingales can be defined using symmetric endpoint averages. For a scalar SDE with differentiable diffusion coefficient

b

, a Stratonovich drift

a

corresponds to the Itô drift

a+\tfrac12 b\,b_x

. The same process can be written in either convention after converting its drift.

For a financial trading strategy, the natural sum multiplies the number of units held at the start of an interval by the price movement during that interval. This is the Itô convention. A symmetric approximation incorporates how the holding itself changed over the interval. If holding and price move together, their product leaves a correction. Stratonovich calculus incorporates that correction into the integral, which is why its chain rule has the familiar ordinary-calculus form.

Where the extra drift comes from

Write $\Delta W_i=W_{t_{i+1}}-W_{t_i}$ and similarly for a continuous semimartingale $H$ . The difference between the symmetric sum and the left-endpoint sum is

\sum_i \left(\frac{H_{t_i}+H_{t_{i+1}}}{2}-H_{t_i}\right)\Delta W_i =\frac12\sum_i\Delta H_i\Delta W_i.

The limit is $\tfrac12[H,W]_T$ , so

\int_0^T H_t\circ dW_t=\int_0^T H_t\,dW_t+\frac12[H,W]_T.

With $H_t=b(t,X_t)$ and sufficient smoothness, the Brownian coefficient of $H$ is $b_x(t,X_t)b(t,X_t)$ . Its covariation with $W$ integrates this coefficient over time. As a result,

dX_t=a(t,X_t)\,dt+b(t,X_t)\circ dW_t =\left(a+\tfrac12 b\,b_x\right)(t,X_t)\,dt+b(t,X_t)\,dW_t.

To get the SDE correction, isolate the noise in

dH_t

for

H_t=b(t,X_t)

. Itô's formula gives Brownian coefficient

b_x(t,X_t)b(t,X_t)

because the coefficient of

dW_t

dX_t

b

. Time and drift terms contribute zero to the covariation with

W

. This leaves

d[H,W]_t=b_xb\,dt

, so the half-covariation in the integral becomes an ordinary drift adjustment. In particular, constant additive noise has

b_x=0

and the two conventions give the same drift.

Take the Stratonovich equation $dX_t=\sigma X_t\circ dW_t$ with $X_0=x>0$ . Its chain rule gives $d\log X_t=\sigma\circ dW_t$ , and the integrand $\sigma$ is constant. Therefore $X_t=xe^{\sigma W_t}$ . In Itô form, this same process has drift $\tfrac12\sigma^2X_t$ , which follows directly by applying Itô's lemma to the exponential.

Now take the Itô equation $dY_t=\sigma Y_t\,dW_t$ with the same initial value. Its solution is $Y_t=xe^{\sigma W_t-\tfrac12\sigma^2t}$ . With $x=100$ , $\sigma=0.30$ and $t=1$ , the first process has mean $100e^{0.045}\approx104.60$ and the second has mean $100$ . Setting the displayed drift coefficient to zero in each convention describes two different processes. For a constant diffusion coefficient, $b_x=0$ and this drift adjustment vanishes.

One way to check the correction without solving an SDE is to set $H_t=W_t$ . The symmetric sum telescopes because $\tfrac12(W_{t_i}+W_{t_{i+1}})\Delta W_i=\tfrac12(W_{t_{i+1}}^2-W_{t_i}^2)$ . Thus the Stratonovich integral is $W_T^2/2$ , while the Itô integral is $(W_T^2-T)/2$ . The gap is exactly half of $[W,W]_T$ .

Where the extra drift comes from

References and next steps