Correlated Diffusions and Multidimensional Ito Calculations

Correlated Noise and the Covariance Matrix

In a multidimensional Itô calculation, the second-order terms depend not only on each Brownian coordinate’s quadratic variation but also on the covariations between coordinates. When two Brownian motions are correlated, their product contributes a drift term. The covariance matrix records all these second-order contributions in one place.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\). Let \((B^1_t,B^2_t)\) be a two-dimensional Brownian motion with respect to this filtration, with constant correlation \(\rho\in[-1,1]\). This means that the vector has continuous paths, starts at \((0,0)\), and its increments after time \(s\) are independent of \(\mathcal{F}_s\); each increment over \([s,t]\) is a centered Gaussian vector with covariance matrix \((t-s)C\), where

C= \begin{pmatrix} 1 & \rho\\ \rho & 1 \end{pmatrix}.

In particular, each coordinate is a Brownian motion relative to the stated filtration. Merely requiring that two Brownian paths be adapted would not suffice: their future increments must be independent of the filtration’s past.

Definition: For \(B^1,B^2\) as above, their quadratic variations and covariation are \([B^1]_t=[B^2]_t=t\) and \([B^1,B^2]_t=\rho t\). In differential notation, \(dB^1_t\,dB^1_t=dt\), \(dB^2_t\,dB^2_t=dt\), \(dB^1_t\,dB^2_t=\rho\,dt\), \(dB^i_t\,dt=0\), and \(dt\,dt=0\).

The covariation follows from the covariance of the Brownian increments; equivalently, it is the limit in probability of sums of products of corresponding increments. The same identity remains valid at the degenerate endpoint correlations \(\rho=1\) and \(\rho=-1\). For example, when \(\rho=1\), the difference \(B^1-B^2\) has zero quadratic variation and is identically zero; when \(\rho=-1\), the sum is identically zero.

All stochastic integrals below are Itô integrals. On each finite interval \([0,T]\), an integrand \(H_s\) is required to be progressively measurable and to satisfy \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\), that is, \(H\in\mathcal{L}^2\). Such an integral is a square-integrable martingale: it is integrable by the Itô isometry, and its conditional expectation at a later time, given \(\mathcal{F}_s\), equals its value at time \(s\). If only local square integrability is available, the corresponding integral is in general only a local martingale; it need not be a true martingale.

Multidimensional Itô Formula and the Generator

Theorem: Let \(X=(X^1,\ldots,X^d)\) satisfy \(dX^i_t=b_i(t,X_t)\,dt+\sum_{k=1}^{m}\sigma_{ik}(t,X_t)\,dB^k_t\), where \(\mathbf B\) is an \(m\)-dimensional Brownian motion relative to the filtration, with constant covariance matrix \(C\). Suppose \(f(t,x)\) is once continuously differentiable in \(t\) and twice continuously differentiable in \(x\). Assume the drift terms are integrable on finite intervals and each stochastic integrand \(\sum_i f_{x_i}(t,X_t)\sigma_{ik}(t,X_t)\) is progressively measurable and in \(\mathcal{L}^2\) on \([0,T]\). Then the multidimensional Itô formula holds, with \(a=\sigma C\sigma^\mathsf{T}\):

df(t,X_t)= \left(f_t+\sum_i b_i f_{x_i} +\frac12\sum_{i,j}a_{ij}f_{x_ix_j}\right)(t,X_t)\,dt +\sum_k\left(\sum_i f_{x_i}(t,X_t)\sigma_{ik}(t,X_t)\right)dB^k_t.

This is the standard multidimensional Itô formula; its proof from partition sums comes later in the course. The covariance matrix enters through \(a=\sigma C\sigma^\mathsf{T}\), the instantaneous covariance matrix of the diffusion coordinates. In particular, the coefficient of \(f_{x_ix_j}\) cannot be computed by treating correlated Brownian coordinates as independent.

For a time-homogeneous diffusion, the second-order differential operator

\mathcal{L}f(x)=\sum_i b_i(x)f_{x_i}(x)+\frac12\sum_{i,j}a_{ij}(x)f_{x_ix_j}(x)

is called its generator. It is the drift coefficient in Itô’s formula applied to \(f(X_t)\). This definition concerns the drift term and does not by itself assert that \(f(X_t)\) is a martingale; that also depends on the stochastic integral and the integrability of its integrand.

Products Reveal the Cross-Variation Term

Proposition: Suppose \(X\) and \(Y\) satisfy \(dX_t=b_X(t)\,dt+\alpha_1(t)\,dB^1_t+\alpha_2(t)\,dB^2_t\) and \(dY_t=b_Y(t)\,dt+\beta_1(t)\,dB^1_t+\beta_2(t)\,dB^2_t\). If the coefficients are progressively measurable, the drift terms are integrable, and the stochastic integrands in Itô’s formula for \(XY\) belong to \(\mathcal{L}^2\) on \([0,T]\), then the product differential is

d(X_tY_t)=X_t\,dY_t+Y_t\,dX_t+ \big(\alpha_1\beta_1+\alpha_2\beta_2+ \rho(\alpha_1\beta_2+\alpha_2\beta_1)\big)\,dt.

Proof. Apply the multidimensional Itô formula to \(f(x,y)=xy\). Its first derivatives are \(f_x=y\) and \(f_y=x\); its second derivatives are \(f_{xx}=f_{yy}=0\) and \(f_{xy}=f_{yx}=1\). The instantaneous covariance between \(X\) and \(Y\) is \(\alpha_1\beta_1+\alpha_2\beta_2+\rho\alpha_1\beta_2+\rho\alpha_2\beta_1\), by multiplying their Brownian coefficients using \(dB^1dB^2=\rho\,dt\). The second-order term is this covariance times \(f_{xy}=1\), yielding the displayed correction. The first-order terms are \(X\,dY+Y\,dX\). \(\square\)

Worked Example: The Product of Correlated Brownian Motions

Take \(\rho=\tfrac12\), \(X_t=B^1_t\), and \(Y_t=B^2_t\). Compute the product differential and its expectation at time \(T\).

Here \(b_X=b_Y=0\), \((\alpha_1,\alpha_2)=(1,0)\), and \((\beta_1,\beta_2)=(0,1)\). The cross-variation coefficient is \(1\cdot1\cdot\rho=\tfrac12\), so

d(B^1_tB^2_t)=B^1_t\,dB^2_t+B^2_t\,dB^1_t+\frac12\,dt.

For finite \(T\), the integrands \(B^1\) and \(B^2\) are in \(\mathcal{L}^2\), since \(\mathbb{E}\int_0^T(B^i_s)^2\,ds=\int_0^T s\,ds=T^2/2\). Each Itô integral is therefore a square-integrable martingale with conditional mean-zero increments. Taking expectations in the integrated identity gives \(\mathbb{E}[B^1_TB^2_T]=T/2\).

Check: the centered Gaussian vector \((B^1_T,B^2_T)\) has covariance \(\rho T=T/2\), so its product expectation is also \(T/2\). This independently confirms the correction term.

A Generator Calculation for a Nonlinear Function

Worked Example: A Quadratic Function of Two Correlated Diffusions

Let \(\rho=-\tfrac12\), \(U_t=1+t+B^1_t\), and \(V_t=-1+2t+B^2_t\). For \(f(u,v)=u^2+uv+2v^2\), calculate \(\mathcal{L}f\) and \(\mathbb{E}[f(U_t,V_t)]\).

The drift vector is \((1,2)\), and the diffusion coefficients of \((U,V)\) against \((B^1,B^2)\) are the identity matrix. Thus \(a=C\), with diagonal entries \(1\) and off-diagonal entries \(-\tfrac12\). The derivatives are \(f_u=2u+v\), \(f_v=u+4v\), \(f_{uu}=2\), \(f_{uv}=f_{vu}=1\), and \(f_{vv}=4\). Consequently,

\mathcal{L}f=(2u+v)+2(u+4v) +\frac12\left(2+2\left(-\frac12\right)1+4\right) =4u+9v+\frac52.

Itô’s formula therefore gives drift \(4U_t+9V_t+\tfrac52\) and stochastic terms \((2U_t+V_t)dB^1_t+(U_t+4V_t)dB^2_t\). Those integrands are in \(\mathcal{L}^2\) on every finite interval because \(U,V\) are Brownian motions plus deterministic functions and have finite second moments there.

For an independent expectation check, the means are \(m_U=1+t\) and \(m_V=-1+2t\); the variances of \(U_t,V_t\) are both \(t\), and their covariance is \(-t/2\). Hence

\begin{aligned} \mathbb{E}[f(U_t,V_t)] &=m_U^2+m_Um_V+2m_V^2+ \operatorname{Var}(U_t)+\operatorname{Cov}(U_t,V_t)+2\operatorname{Var}(V_t)\\ &=(1+2t+t^2)+(-1+t+2t^2)+2(1-4t+4t^2) +t-\frac12t+2t\\ &=2-\frac52t+11t^2. \end{aligned}

To check against the generator, its value at the means is \(4(1+t)+9(-1+2t)+\tfrac52=-\tfrac52+22t\). Its integral from \(0\) to \(t\) is \(-\tfrac52t+11t^2\); adding the initial value \(f(1,-1)=2\) reproduces the expectation above.

Correlated Geometric Diffusions

Worked Example: The Expected Product of Two Exponential Diffusions

Let \(S_0=2\), \(Y_0=3\), and consider \(dS_t=\tfrac14 S_t\,dt+S_t\,dB^1_t\), \(dY_t=-\tfrac14Y_t\,dt+2Y_t\,dB^2_t\), with \(\rho=\tfrac12\). Find \(\mathbb{E}[S_tY_t]\).

First verify the solutions. Applying one-dimensional Itô’s formula to \(2e^{B^1_t-t/4}\), whose time derivative is \(-S_t/4\), first space derivative is \(S_t\), and second space derivative is \(S_t\), gives drift \((-1/4+1/2)S_t=S_t/4\) and diffusion \(S_t\,dB^1_t\). For \(3e^{2B^2_t-9t/4}\), the time derivative is \(-9Y_t/4\), the first space derivative is \(2Y_t\), and the second is \(4Y_t\); the correction is \(2Y_t\,dt\), leaving drift \(-Y_t/4\) and diffusion \(2Y_t\,dB^2_t\). Thus

S_t=2e^{B^1_t-t/4},\qquad Y_t=3e^{2B^2_t-9t/4}.

The product formula gives drift \((\tfrac14-\tfrac14+\rho\cdot1\cdot2)S_tY_t=S_tY_t\). Its stochastic terms are \(S_tY_t\,dB^1_t+2S_tY_t\,dB^2_t\). Their integrands are in \(\mathcal{L}^2\) on finite intervals: for example, \(S_tY_t=6e^{B^1_t+2B^2_t-5t/2}\), and the Gaussian exponential moment gives \(\mathbb{E}[(S_tY_t)^2]=36e^{9t}\), which is integrable on \([0,T]\). The stochastic integrals are therefore true martingales, and taking expectations in the integrated product equation yields \(\mathbb{E}[S_tY_t]=6e^t\).

Check: \(B^1_t+2B^2_t\) is centered Gaussian with variance \(t+4t+4(\tfrac12)t=7t\). The explicit product has expectation \(6e^{-5t/2}e^{(7t)/2}=6e^t\), agreeing with the differential calculation.

Common Pitfall: Dropping the Covariation

A frequent error is to multiply differentials as if distinct Brownian motions always had zero product. In the first worked example, omitting \(dB^1_t\,dB^2_t=\tfrac12dt\) would give \(d(B^1_tB^2_t)=B^1_t\,dB^2_t+B^2_t\,dB^1_t\). Taking expectations would then incorrectly predict \(\mathbb{E}[B^1_TB^2_T]=0\). The correct calculation includes \(+\tfrac12dt\) and gives \(T/2\), in agreement with the covariance of the Gaussian pair.

The same discipline applies to the generator. Write the full covariance matrix \(a=\sigma C\sigma^\mathsf{T}\) before taking second derivatives. In two dimensions the second-order part is \(\tfrac12(a_{11}f_{xx}+2a_{12}f_{xy}+a_{22}f_{yy})\); the factor \(2\) in the mixed term combines with \(\tfrac12\), leaving \(a_{12}f_{xy}\). This is why a correlated pair contributes a mixed derivative even when both individual variances are unchanged.

Check Your Understanding

Use the covariance matrix and the hypotheses for Itô integrals in each answer.

If \(\rho=-\tfrac13\), what is \(d(B^1_tB^2_t)\), and what is \(\mathbb{E}[B^1_TB^2_T]\)?
For \(dX_t=b_1dt+\sigma_1dB^1_t\) and \(dY_t=b_2dt+\sigma_2dB^2_t\), identify the instantaneous covariance term in \(d(X_tY_t)\).
For \(f(u,v)=u^2+v^2\) and unit diffusion coefficients driven by a correlated Brownian pair, does the generator have a mixed derivative term? Explain using \(f_{uv}\).
For the quadratic function in the second worked example, evaluate \(\mathcal{L}f\) at \((u,v)=(0,1)\).
Why does local square integrability alone not justify taking expectations as though a stochastic integral were a true martingale?