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A deterministic time-changed quadratic variation

Example

Assume the Axiom of Choice and the standing hypothesis (H) of Elementary predictable Brownian integrands, and assume that the filtration satisfies the usual conditions required by Locally square-integrable predictable Brownian integrands. Let $M_{t} = \int_{0}^{t} s d B_{s}$ , the Ito integral of the deterministic integrand $H_{s} = s$ . Then $M$ is a continuous square-integrable martingale Locally square-integrable predictable Brownian integrands, and along every deterministic partition sequence of $[0, T]$ with mesh tending to $0$ the quadratic variation of the path of $M$ is $[M]_{t} = \int_{0}^{t} s^{2} d s = \frac{t^{3}}{3}, 0 \leq t \leq T,$ the convergence being uniform in probability on $[0, T]$ as in Quadratic variation of an Ito integral. In particular the quadratic variation is a smooth deterministic function of time, of size $t^{3} / 3$ , not the elapsed time $t$ that governs Brownian motion itself.

Facts & Assumptions

Given: AC, the standing hypothesis (H), the usual conditions on the filtration, $T > 0$ , the deterministic integrand $H_{s} = s$ , its energy $A_{t} = \int_{0}^{t} s^{2} d s = t^{3} / 3$ , and a deterministic partition sequence of $[0, T]$ with mesh tending to $0$ .

[F1]

A deterministic Borel function of the time variable is a predictable process; $H_{s} = s$ is continuous, and its energy is finite at every finite time: $A_{t} = t^{3} / 3 < \infty$ . Progressively measurable and predictable processes Locally square-integrable predictable Brownian integrands

[F2]

For every locally square-integrable predictable $H$ and every deterministic vanishing-mesh partition sequence, the squared-increment partial sums of $M = H \cdot B$ converge to $\int_{0}^{t} H_{s}^{2} d s$ uniformly in probability on $[0, T]$ . Quadratic variation of an Ito integral Quadratic variation along a partition sequence

[F3]

AC is declared for the ambient interfaces. The Axiom of Choice

Verification

technique · direct

1.1

The integrand $H_{s} = s$ is deterministic and continuous, hence predictable with finite energy $A_{t} = t^{3} / 3$ at every $t$ ; under the given usual conditions its localized integral $M = H \cdot B$ is defined and is a continuous square-integrable martingale.

F1given

2.1

Applying [F2] to $H_{s} = s$ and to the given partition sequence gives $[M]_{t} = \int_{0}^{t} s^{2} d s = t^{3} / 3$ for every $t \in [0, T]$ , with the convergence uniform in probability; the value does not depend on the chosen deterministic partition sequence because the theorem holds for every such sequence.

F2step 1.1

3.1

Sanity cases: at $t = 0$ the value is $0$ ; the example's integrand grows with time, so the accumulated quadratic variation $t^{3} / 3$ is not linear, in contrast with the Brownian case $[B]_{t} = t$ ; and a constant integrand $H \equiv c$ would give $[M]_{t} = c^{2} t$ , of which this is the $c = s$ analogue. AC enters only through [F3].

F2F3step 2.1given∎

Source notes

Lawler, Theorem 3.2.6, computes the quadratic variation of an Ito integral as the integral of the squared integrand; the deterministic time-changed value $t^{3} / 3$ is the special case $H_{s} = s$ .

Depends on

Used by

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Sources

Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Theorem 3.2.6 (standard reference, not scraped)