Ito lemma and the stochastic chain rule

Definition Let $W$ be a standard Brownian motion with respect to a filtration $(\mathcal{F}_t)$ under $\mathbb{P}$, and let $X$ be an Itô process on $[0, T]$,

with $X_0$ an $\mathcal{F}_0$-measurable random variable, $b$ and $\sigma$ predictable, and $\int_0^T \lvert b_s \rvert \, ds < \infty$ and $\int_0^T \sigma_s^2 \, ds < \infty$ almost surely, so that both integrals are defined. In differential form $dX_t = b_t \, dt + \sigma_t \, dW_t$, and the quadratic variation of $X$ is $[X]_t = \int_0^t \sigma_s^2 \, ds$. Let $f(t, x)$ have a continuous first derivative in $t$ and continuous first and second derivatives in $x$ on $[0, T] \times \mathbb{R}$, which is written $f \in C^{1,2}$. Itô’s lemma, also called the stochastic chain rule, says that

with every derivative evaluated at $(t, X_t)$. Equivalently $df = \partial_t f \, dt + \partial_x f \, dX_t + \tfrac12 \partial_{xx} f \, d[X]_t$. The differential notation is shorthand for the identity obtained by integrating both sides from $0$ to $t$, which holds almost surely for all $t \in [0, T]$ at once. The stochastic integral in it is defined because $\partial_x f(s, X_s)$ is continuous in $s$ and so bounded on each path. It is in general a local martingale, and a martingale when $\mathbb{E}\int_0^T \sigma_s^2 (\partial_x f)^2 \, ds < \infty$. If $f$ is $C^{1,2}$ only on $[0, T] \times D$ for an open interval $D$ that contains every path of $X$, the formula still holds, and if $X$ can leave $D$ it holds for the times before $X$ first leaves $D$. The theorem is stated here with the idea of its proof, which the next card describes.

Definition The second-order correction in Itô’s lemma is the term $\tfrac12 \sigma_t^2 \, \partial_{xx} f(t, X_t) \, dt$, which equals $\tfrac12 \partial_{xx} f \, d[X]_t$ and has no counterpart in the ordinary chain rule. It comes from the Taylor expansion over a partition $0 = t_0 < \dots < t_n = t$,

with the derivatives at $(t_k, X_{t_k})$. As the mesh tends to zero, the first-order sums converge to $\int_0^t \partial_t f \, ds$ and to the left-endpoint integral $\int_0^t \partial_x f \, dX_s$. In the square $(\Delta X_k)^2$, the parts of the form $b^2 (\Delta t_k)^2$ and $2 b \sigma \, \Delta t_k \, \Delta W_k$ add up to zero in the limit, while $\sigma^2 (\Delta W_k)^2$ adds up to $\int_0^t \sigma_s^2 \, ds$ because the quadratic variation of $W$ is $t$. These limits are often recorded as the multiplication rules $(dt)^2 = 0$, $dt \, dW_t = 0$ and $(dW_t)^2 = dt$, which are bookkeeping for quadratic covariations and not identities between numbers. Controlling the Taylor remainder and the coefficients requires localization estimates that are stated, not carried out, here. Under the hypotheses of the lemma the correction is exact, not a truncation. Its sign follows the curvature of $f$ in $x$. It is positive where $f$ is convex, negative where $f$ is concave, and zero when $f$ is linear in $x$ or $\sigma = 0$, in which case Itô’s lemma reduces to the ordinary chain rule.

Definition Let $W = (W^1, \dots, W^m)^{\top}$ be a standard $m$-dimensional Brownian motion with independent coordinates, and let $X$ be an $n$-dimensional Itô process $dX_t = b_t \, dt + \Sigma_t \, dW_t$, with predictable $b_t \in \mathbb{R}^n$ and $\Sigma_t \in \mathbb{R}^{n \times m}$ whose entries satisfy the integrability conditions of the one-dimensional case. The matrix $a_t = \Sigma_t \Sigma_t^{\top}$ is symmetric and positive semidefinite, and $d[X^i, X^j]_t = a^{ij}_t \, dt$. For $f \in C^{1,2}([0, T] \times \mathbb{R}^n)$, with gradient $\nabla f$ and Hessian $\nabla^2 f$ in the state variables,

with the derivatives at $(t, X_t)$. Written out, the correction is $\tfrac12 \sum_{i,j} a^{ij}_t \, \partial_{ij} f \, dt$. With two coordinates each mixed term appears twice, for $(i, j) = (1, 2)$ and for $(2, 1)$, and since $a^{12} = a^{21}$ and $\partial_{12} f = \partial_{21} f$ the two halves combine into $a^{12}_t \, \partial_{12} f \, dt$. Independent Brownian coordinates do not make the coordinates of $X$ independent, because shared loadings in $\Sigma$ correlate them.

Definition Let $X$ and $Y$ be Itô processes driven by the same $m$-dimensional Brownian motion, $dX_t = b^X_t \, dt + \Sigma^X_t \, dW_t$ and $dY_t = b^Y_t \, dt + \Sigma^Y_t \, dW_t$, with row vectors $\Sigma^X_t, \Sigma^Y_t \in \mathbb{R}^{1 \times m}$ and the integrability conditions above. Then

where $d[X, Y]_t = \Sigma^X_t (\Sigma^Y_t)^{\top} \, dt$ is the quadratic covariation. This is the multivariate formula for $f(x, y) = xy$, whose Hessian has zeros on the diagonal and ones off it, so the correction is $\tfrac12 \cdot 2\, a^{12}_t \, dt = a^{12}_t \, dt$. When one factor has no Brownian part the covariation vanishes and the ordinary product rule holds. For a constant rate $r$, for example, $d(e^{-rt} S_t) = e^{-rt} (dS_t - r S_t \, dt)$, the rule behind discounted prices.

Definition Let $X$ solve $dX_t = b(X_t) \, dt + \Sigma(X_t) \, dW_t$ in $\mathbb{R}^n$ with continuous coefficients $b$ and $\Sigma$, let $a = \Sigma \Sigma^{\top}$, and write $\mathbb{E}^x$ for the expectation when $X_0 = x$. The generator of the diffusion acts on $f \in C^2(\mathbb{R}^n)$ by

Itô’s lemma reads $df(X_t) = \mathcal{L} f(X_t) \, dt + \nabla f(X_t)^{\top} \Sigma(X_t) \, dW_t$, so $f(X_t) - f(X_0) - \int_0^t \mathcal{L} f(X_s) \, ds$ is a local martingale. If $f$ also has compact support, the stochastic integral has a bounded integrand and is a martingale, and

so the generator is the instantaneous expected rate of change of $f$ along the diffusion. When the equation has a solution from each starting point that is unique in law, $X$ is a Markov process, a result stated here. For a function $g \in C^{1,2}$ of time and state, the drift of $g(t, X_t)$ is $(\partial_t g + \mathcal{L} g)(t, X_t)$. If $\partial_t g + \mathcal{L} g = 0$, then $g(t, X_t)$ is a local martingale, and when it is a true martingale on $[0, T]$ it satisfies $g(t, X_t) = \mathbb{E}[g(T, X_T) \mid \mathcal{F}_t]$. This equation is the starting point of the pricing equations of later lessons, which need the same martingale and integrability conditions.