The Ito Product Rule for Two Processes

Why the Product Rule Has an Extra Term

For differentiable functions of time, the product rule has two terms: \(d(XY)=X\,dY+Y\,dX\). For Ito processes, fluctuations in both factors can occur on the same time interval. The product of those fluctuations contributes a further term, their quadratic covariation \(d[X,Y]\). Its value depends on the processes’ diffusion coefficients: drift terms do not contribute to this covariation.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard \(m\)-dimensional Brownian motion \(\mathbf{B}_t=(B_t^1,\ldots,B_t^m)\) with independent components. Consider continuous Ito processes of the form \(dX_t=a_t\,dt+\boldsymbol{\beta}_t\cdot d\mathbf{B}_t\) and \(dY_t=c_t\,dt+\boldsymbol{\gamma}_t\cdot d\mathbf{B}_t\). Here \(a_t,c_t\) are progressively measurable drift coefficients and \(\boldsymbol{\beta}_t,\boldsymbol{\gamma}_t\) are progressively measurable diffusion vectors. On each finite interval \([0,T]\), assume \(\mathbb{E}\int_0^T(|a_t|+|c_t|)\,dt<\infty\) and \(\mathbb{E}\int_0^T(\|\boldsymbol{\beta}_t\|^2+\|\boldsymbol{\gamma}_t\|^2)\,dt<\infty\), and take \(X_0,Y_0\) to be finite. These conditions define the drift integrals and place each Brownian integrand in \(\mathcal{L}^2\).

In particular, if \(X\) and \(Y\) share one Brownian driver, their covariation rate is the product of their diffusion coefficients. If they are driven by different independent components, that rate is zero. With correlated drivers, it is the covariance-weighted product of their coefficients; the multidimensional representation above encodes this by taking the corresponding vectors’ dot product.

The Ito Product Rule

Proof. Apply the multidimensional Ito formula to the function \(f(x,y)=xy\) and the two-dimensional Ito process \((X_t,Y_t)\). Its partial derivatives are \(f_x=y\), \(f_y=x\), \(f_{xx}=0\), \(f_{yy}=0\), and \(f_{xy}=f_{yx}=1\). The first-order terms are \(Y_t\,dX_t+X_t\,dY_t\). The second-order correction is \(\tfrac12(f_{xx}\,d[X]_t+2f_{xy}\,d[X,Y]_t+f_{yy}\,d[Y]_t)=d[X,Y]_t\). Substituting the drift and diffusion forms gives the stated drift, while the Brownian terms combine to \((X_t\boldsymbol{\gamma}_t+Y_t\boldsymbol{\beta}_t)\cdot d\mathbf{B}_t\). The formula holds locally if these product integrands are not globally square-integrable: stop the processes when their magnitudes or accumulated integrals reach finite bounds, apply Ito’s formula to the stopped processes, and then let the stopping levels increase. If the product diffusion vector is in \(\mathcal{L}^2\), its Ito integral is a true martingale: its expectation is finite by the Ito isometry, and its conditional expectation at a later time given \(\mathcal{F}_s\) equals its value at \(s\), because the Ito integral over \([s,t]\) has conditional expectation zero. Without that integrability, a local martingale need not be a true martingale. \(\square\)

The rule can also be checked using the differential multiplication table. Multiplying \(dX_t=a_tdt+\boldsymbol{\beta}_t\cdot d\mathbf{B}_t\) by \(dY_t=c_tdt+\boldsymbol{\gamma}_t\cdot d\mathbf{B}_t\), all terms containing \(dt\,dt\) or \(dt\,dB_t^i\) vanish, and the Brownian products give \(\boldsymbol{\beta}_t\cdot\boldsymbol{\gamma}_t\,dt\). Thus \(dX_t\,dY_t=d[X,Y]_t\). This multiplication is shorthand for the quadratic-covariation calculation; it does not assert that a single Brownian increment squared equals the time increment.

Worked Calculations

What the Correction Does—and Does Not—Say

The correction can be positive, negative, or zero. It is positive when the instantaneous diffusion directions reinforce each other and negative when they oppose each other. If one process has finite variation, its diffusion vector is zero, so its covariation with the other process is zero. In the special case \(X=Y\), the rule gives \(d(X_t^2)=2X_t\,dX_t+d[X]_t\), including the familiar positive quadratic-variation correction for the square of a Brownian-driven process.

The Ito product rule is not the ordinary product rule with a negligible error. The term \(d[X,Y]_t\) records the non-vanishing limit of products of simultaneous Brownian increments. Nor should it be confused with the Stratonovich product rule: for continuous semimartingales, Stratonovich integration incorporates half the relevant covariation correction, and the product rule for Stratonovich differentials has the ordinary two-term form. Converting an Ito expression to Stratonovich form requires accounting for that correction; simply deleting \(d[X,Y]\) from an Ito calculation changes the result.

Common mistake. For \(X_t=1+B_t\) and \(Y_t=2-B_t\), omitting covariation gives the incorrect calculation \(d(X_tY_t)\overset{\text{wrong}}{=}X_t\,dY_t+Y_t\,dX_t=(1-2B_t)dB_t\). The two stochastic terms do combine to \((1-2B_t)dB_t\), but this leaves out \(d[X,Y]_t=-dt\). The corrected differential is \((1-2B_t)dB_t-dt\). The direct calculation \(X_tY_t=2+B_t-B_t^2\), with \(d(B_t^2)=2B_t\,dB_t+dt\), confirms the missing drift is \(-dt\).