Stratonovich integral

Itô integral is a construction that allows integration w.r.t. Brownian motion (or semimartingales). But it has a key feature:

\int H_{t} d W_{t}

is defined via non-anticipating integrands and uses left endpoints of partitions. This choice makes it very useful for probability/martingale theory, but it breaks the classical chain rule (you get the Itô lemma instead).

In physics and engineering, people often want a stochastic integral that behaves more like classical calculus. For instance, when doing a change of variables, they expect the ordinary chain rule to hold without the “extra Itô term.”
The Stratonovich integral is defined by using the midpoints of the partition instead of left endpoints. Symbolically,

\int H_{t} \circ d W_{t}

(circle notation for Stratonovich) represents the limit of

\sum H (s_{i}) (W_{t_{i + 1}} - W_{t_{i}}),

with $s_{i} = \frac{t_{i} + t_{i + 1}}{2}$ instead of $s_{i} = t_{i}$ .
With this choice, we recover the classical chain rule, so it fits naturally into differential geometry, stochastic mechanics, etc.

Conversion rule

Every Stratonovich integral can be rewritten as an Itô integral with a correction term:

\int_{0}^{t} H (s, X_{s}) \circ d W_{s} = \int_{0}^{t} H (s, X_{s}) d W_{s} + \frac{1}{2} \int_{0}^{t} \partial_{x} H (s, X_{s}) H (s, X_{s}) d s

or, more in general,

\int_{0}^{t} Y_{s} \circ d W_{s} = \int_{0}^{t} Y_{s} d W_{s} + \frac{1}{2} [Y, W]_{t},

where $[Y, W]_{t}$ is the quadratic covariation between $Y$ and $W$ .

If $Y_{s} = H (s)$ is deterministic, then $[Y, W]_{t} = 0$ .
If $Y_{s} = W_{s}$ , then $[W, W]_{t} = t$ .

Key differences: Stratonovich has the chain rule, Itô has martingale properties.

Boat example

Think of a boat being hit by random waves $d W_{t}$ , and a steering strategy $H_{t}$ deciding how to react (see Itô integral#Boat on a choppy sea).

With Itô integration the helmsman can only use past information (left endpoints). This makes the boat’s motion a martingale: no matter the strategy, its expected position stays at zero. You cannot create a systematic forward drift.
With Stratonovich integration the helmsman reacts in a smoother way (midpoints):

\int_{0}^{T} H_{s} \circ d W_{s} \approx \sum \frac{1}{2} (H_{t_{k}} + H_{t_{k + 1}}) (W_{t_{k + 1}} - W_{t_{k}}) .

which restores the classical chain rule. Here, certain steering strategies do produce forward motion. For example, with $H_{t} = W_{t}$ :

\int_{0}^{t} W_{s} \circ d W_{s} = \frac{1}{2} W_{t}^{2} \geq 0,

whose expectation is $\frac{t}{2}$ . Thus, the boat drifts forward on average, something impossible in the Itô setting.

Example:

Step	ΔW	W	Stratonovich ∫ W∘dW	Itô ∫ W dW
0	–	0	0.0	0
1	+1	1	0.5	0
2	+1	2	2.0	1
3	−1	1	0.5	−1
4	+1	2	2.0	0
5	+1	3	4.5	2
6	−1	2	2.0	−1
7	−1	1	0.5	−3
8	−1	0	0.0	−4

Computations:

Stratonovich: $S_{n} = \sum_{i = 1}^{n} \frac{W_{i - 1} + W_{i}}{2} Δ W_{i}$
Itô: $I_{n} = \sum_{i = 1}^{n} W_{i - 1} Δ W_{i}$ Observe: $S_{n} = I_{n} + \frac{1}{2} \sum_{i = 1}^{n} (Δ W_{i})^{2}$ .