The Cameron-Martin Space of Brownian Paths

Finite-Energy Paths on a Fixed Interval

The Cameron-Martin space is a deterministic space of paths associated with Brownian motion. Its elements start at zero and have a square-integrable rate of change. The norm measures the accumulated squared derivative, so it records the energy of a path rather than its height or its ordinary \(L^2\) size. This distinction is important: the space identifies which deterministic paths have finite energy, without asserting that a Brownian sample path itself has that regularity.

Fix \(T>0\). Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). The deterministic path space below does not depend on the choice of this probability space. For reference, \(\mathcal{L}^2(0,T)\) denotes the progressively measurable processes \(H_s\) satisfying \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\), the usual class of Ito integrands on this interval. No stochastic integral is needed to define or compute the path norms in this tutorial.

The equivalent descriptions use the fundamental theorem for absolutely continuous functions. The derivative is defined only up to equality almost everywhere, but that causes no ambiguity: changing \(h'\) on a set of Lebesgue measure zero does not change the integral defining the norm. Requiring \(h(0)=0\) also matters. Without it, the derivative would not determine the path, and the displayed formula would assign norm zero to every constant path.

Basic Structure and a Pathwise Bound

Proof. Define \(D:H_T\to L^2([0,T])\) by \(Dh=h'\), with derivatives identified when they agree almost everywhere. This map is linear. It preserves inner products because \(\langle h,k\rangle_{H_T}=\langle Dh,Dk\rangle_{L^2}\), and it is one-to-one: if \(Dh=0\) almost everywhere, then \(h(t)=\int_0^t Dh(s)\,ds=0\) for every \(t\). It is onto because for any \(g\in L^2([0,T])\), Cauchy-Schwarz gives \(\int_0^T|g(s)|\,ds\le\sqrt{T}\|g\|_{L^2}<\infty\). Thus \(h(t)=\int_0^t g(s)\,ds\) is absolutely continuous, starts at zero, has derivative \(g\) almost everywhere, and belongs to \(H_T\). Since \(L^2([0,T])\) is complete and \(D\) preserves distances and maps onto it, every Cauchy sequence in \(H_T\) converges in \(H_T\); hence \(H_T\) is complete. For the bound, use \(h(t)=\int_0^t h'(s)\,ds\) and Cauchy-Schwarz: $$ |h(t)|\le \left(\int_0^t 1^2\,ds\right)^{1/2} \left(\int_0^t |h'(s)|^2\,ds\right)^{1/2} \le \sqrt{t}\,\|h\|_{H_T}. $$ Since \(t\le T\), the second inequality follows. To see sharpness when \(t>0\), take \(h(s)=s\) for \(0\le s\le t\) and \(h(s)=t\) for \(t\le s\le T\). Then \(|h(t)|=t\) and \(\|h\|_{H_T}=\sqrt{t}\), so equality holds in the first bound. At \(t=0\), both sides of that bound are zero. \(\square\)

The theorem shows that evaluation at a fixed time is controlled by the energy norm. It also shows why the initial condition is built into the definition: the value at every later time is obtained by accumulating the derivative from time zero. The norm is not the same as the usual \(L^2\) norm of the path itself.

Computing Norms from Derivatives

For a polynomial path \(h(t)=\sum_{k=1}^m a_k t^k\), the derivative is \(\sum_{k=1}^m k a_k t^{k-1}\). Expanding its square and integrating term by term gives the useful exact formula $$ \|h\|_{H_T}^2 =\sum_{k=1}^m\sum_{\ell=1}^m \frac{k\ell\,a_k a_\ell\,T^{k+\ell-1}}{k+\ell-1}. $$ A polynomial belongs to \(H_T\) when its constant term is zero; all its derivatives are square-integrable on the finite interval.

Piecewise-linear paths are handled by their slopes. If the knots are \(0=t_0<t_1<\cdots<t_n=T\), and \(h\) is linear on each interval \([t_{i-1},t_i]\), define its slope there by \(m_i=(h(t_i)-h(t_{i-1}))/(t_i-t_{i-1})\). The derivative equals \(m_i\) in the interior of that interval, so $$ \|h\|_{H_T}^2=\sum_{i=1}^n m_i^2(t_i-t_{i-1}). $$ The finitely many knots do not affect the integral: the derivative need only be specified almost everywhere.

Membership, Interpretation, and a Common Mistake

To test membership in \(H_T\), check both conditions: the path must start at zero, and it must be absolutely continuous with square-integrable derivative. A continuous path with a jump is not absolutely continuous. A path that starts at a nonzero value fails the initial condition even if its derivative is square-integrable. A continuous piecewise-linear path with finitely many pieces and finite slopes does satisfy both conditions, and its norm is found by weighting each squared slope by the length of its interval.

The norm measures roughness through the rate of change. For example, on a fixed interval, multiplying a path by a scalar \(c\) multiplies its derivative by \(c\), so its norm becomes \(|c|\|h\|_{H_T}\). By contrast, adding a nonzero constant changes the initial value and takes the path outside \(H_T\). The space is therefore a linear space with a distinguished zero starting value, not a space of arbitrary absolutely continuous paths.

Common mistake: using \(\int_0^T h(t)^2\,dt\) as the Cameron-Martin norm. For \(h(t)=t\) on \([0,2]\), that quantity is $$ \int_0^2 t^2\,dt=\left[\frac{t^3}{3}\right]_0^2=\frac83. $$ But the Cameron-Martin derivative is \(h'(t)=1\), so the correct squared norm is $$ \|h\|_{H_2}^2=\int_0^2 1^2\,dt=2. $$ The two values differ because one measures the squared path and the other measures the squared derivative. Confusing them changes the geometry of the space and gives incorrect energies.

Another error is to assign a special derivative value at a corner of a piecewise-linear path. A finite set of corners has measure zero, so those values do not change the norm integral. What matters is the slope on each interval of positive length; both the slope and the length must be included.