Quadratic variation and stochastic limits

Definition A partition of $[0, T]$ is a finite set of times $\pi = \{t_0, t_1, \dots, t_n\}$ with $0 = t_0 < t_1 < \dots < t_n = T$ and interval lengths $\Delta t_k = t_{k+1} - t_k$. Its mesh is the length of its longest interval,

For a process $X$, the quadratic sum over $\pi$ is

The partitions in this lesson are deterministic, fixed before any path is seen. A sequence of partitions with $\lvert \pi_n \rvert \to 0$ may be nested or not, and its intervals need not be equal. The requirement is that the longest interval shrinks, so adding many points in one region while a long gap remains elsewhere does not make the mesh small.

Definition Random variables $X_n$ with $\mathbb{E}[X_n^2] < \infty$ converge in mean square, or in $L^2$, to a random variable $X$ with $\mathbb{E}[X^2] < \infty$ when

They converge in probability to $X$ when $\mathbb{P}(\lvert X_n - X \rvert > \varepsilon) \to 0$ for every fixed $\varepsilon > 0$. Chebyshev’s inequality in the form

shows that mean-square convergence implies convergence in probability. The converse fails. Neither mode implies almost-sure convergence of the whole sequence, although convergence in probability gives almost-sure convergence along some subsequence. When the limit is a constant $c$, the mean-square error splits as $\mathbb{E}[(X_n - c)^2] = \operatorname{Var}(X_n) + (\mathbb{E}[X_n] - c)^2$, variance plus squared bias.

Definition A process $X$ has quadratic variation $[X]_T$ on $[0, T]$ when $Q_{\pi_n}(X) \to [X]_T$ in probability for every sequence of partitions of $[0, T]$ with $\lvert \pi_n \rvert \to 0$. For a standard Brownian motion and every partition $\pi$ of $[0, T]$,

so $Q_{\pi_n}(W) \to T$ in mean square and in probability whenever $\lvert \pi_n \rvert \to 0$, and $[W]_T = T$. To prove it, note that each increment $\Delta W_k = W_{t_{k+1}} - W_{t_k}$ is $N(0, \Delta t_k)$, so $\mathbb{E}[(\Delta W_k)^2] = \Delta t_k$, and these add to $T$. Its fourth moment is $3(\Delta t_k)^2$, so $\operatorname{Var}((\Delta W_k)^2) = 3(\Delta t_k)^2 - (\Delta t_k)^2 = 2(\Delta t_k)^2$. Increments over disjoint intervals are independent, so the variances add, and $(\Delta t_k)^2 \le \lvert \pi \rvert \, \Delta t_k$ gives the bound. Since the mean is exactly $T$, the mean-square error equals the variance and tends to zero, and Chebyshev gives

When $\sum_n \lvert \pi_n \rvert < \infty$, for example for dyadic partitions, these bounds are summable and the Borel-Cantelli lemma upgrades the result to almost-sure convergence. The limit is taken along partitions fixed in advance and is not a supremum over partitions. The supremum of $Q_\pi(W)$ over all partitions of $[0, T]$ is infinite almost surely, a path result stated here without proof.

Definition The total variation of a path $A$ on $[0, T]$ is

the supremum over all partitions of $[0, T]$, and $A$ has finite variation when $\operatorname{TV}(A) < \infty$. Monotone paths have finite variation, continuously differentiable paths have $\operatorname{TV}(A) = \int_0^T \lvert A'(t) \rvert \, dt$, and an integral $A_t = \int_0^t \mu_s \, ds$ with $\int_0^T \lvert \mu_s \rvert \, ds < \infty$ is continuous with finite variation. If $A$ is continuous with finite variation, then

because $A$ is uniformly continuous on $[0, T]$, so its largest increment over the partition tends to zero while the total variation stays fixed. The same reasoning gives $\bigl\lvert \sum_k \Delta A_k \, \Delta X_k \bigr\rvert \le \max_k \lvert \Delta X_k \rvert \operatorname{TV}(A) \to 0$ for every continuous path $X$. Continuity of $A$ is needed. A path that adds a single jump of size $J$ to a continuous finite variation path has quadratic sums tending to $J^2$.

Definition Almost surely, a Brownian path has infinite total variation on every interval $[s, t]$ with $s < t$. For one interval $[0, T]$, choose partitions with $\sum_n \lvert \pi_n \rvert < \infty$, so that $Q_{\pi_n}(W) \to T$ almost surely. On a path with $\operatorname{TV}(W) < \infty$ the finite variation bound would force $Q_{\pi_n}(W) \to 0$, since the path is continuous, which contradicts $T > 0$. So $\mathbb{P}(\operatorname{TV}(W) < \infty) = 0$, and applying this to the countably many intervals with rational endpoints gives the statement for every interval. Consequently, almost every path is monotone on no interval, and the pathwise Stieltjes integral, which is built for integrators of finite variation, is not available for integrals against $W$. That is why the Itô integral is constructed in The Ito integral and isometry. The stronger theorem that almost every Brownian path is nowhere differentiable is stated, not proved, in the course, and it does not follow from infinite variation alone.

Definition For processes $X$ and $Y$ on $[0, T]$, the quadratic covariation $[X, Y]_T$ is the limit in probability of the cross sums

when it exists. Since $4 \Delta X_k \Delta Y_k = (\Delta X_k + \Delta Y_k)^2 - (\Delta X_k - \Delta Y_k)^2$, it equals $\tfrac14\bigl([X + Y]_T - [X - Y]_T\bigr)$ whenever both quadratic variations exist, and $[X, X]_T = [X]_T$. The Cauchy-Schwarz inequality for the sums gives $\lvert [X, Y]_T \rvert \le \sqrt{[X]_T \, [Y]_T}$. If $W$ and $Z$ are independent standard Brownian motions and $B = \rho W + \sqrt{1 - \rho^2}\, Z$ with $-1 \le \rho \le 1$, then $B$ is a standard Brownian motion and $[W, B]_T = \rho T$. Each cross term $\Delta W_k \Delta B_k$ has mean $\rho \Delta t_k$ and variance $(1 + \rho^2)(\Delta t_k)^2$, so the cross sum has mean $\rho T$ and variance at most $(1 + \rho^2) T \lvert \pi \rvert$. For a continuous Itô process

with an adapted, jointly measurable drift satisfying $\int_0^T \lvert \mu_s \rvert \, ds < \infty$ and a predictable coefficient satisfying $\int_0^T \sigma_s^2 \, ds < \infty$ almost surely, and $Y$ built in the same way from a drift $\nu$, a coefficient $\eta$ and a Brownian motion $B$ for the same filtration with $[W, B]_t = \int_0^t \rho_s \, ds$, the course states without proof

For constant coefficients these follow from the cards above, because the drift terms have finite variation and drop out of every cross sum. The shorthand $(dW_t)^2 = dt$, $dW_t \, dt = 0$, $(dt)^2 = 0$ and $dW_t \, dB_t = \rho_t \, dt$ summarizes these partition limits and is not an identity between real numbers. For processes with jumps the squared jump sizes also enter. A Poisson process $N$ has $[N]_t = N_t$, the number of jumps up to time $t$.