Lecture 30. The Stroock-Varadhan support theorem

To conclude this course, we are going to provide an elementary proof of the StroockVaradhan support theorem which is based on rough paths theory. We first remind that the support of a random variable $X$ which defined on a metric space $X$ is the smallest closed $F$ such that $\mathbb{P}(X\in F)=1$. In particular $x \in F$ if and only if for every open ball $\mathbf{B}(x, \varepsilon)$, $\mathbb{P}( X \in \mathbf{B}(x, \varepsilon)) > 0$.

Let $(B_t)_{0 \le t \le T}$ be a $d$-dimensional Brownian motion. We can see $B$ as a random variable that takes its values in $C^{p-var}([0,T],\mathbb{R}^d)$, $p > 2$. The following theorem describes the support of this random variable.

Proposition: Let $p > 2$. The support of $B$ in $C^{p-var}([0,T],\mathbb{R}^d)$ is $C_0^{0,p-var}([0,T],\mathbb{R}^d)$, that is the closure for the $p$-variation distance of the set of smooth paths starting at 0.

Proof: The key argument is a clever application of the Cameron-Martin theorem. Let us recall that this theorem says that if
$h \in \mathbb{W}_0^{1,2} =\left\{ h : [0,T] \to \mathbb{R}^d, \exists k \in L^2([0,T], \mathbb{R}^d), h(t)=\int_0^t k (s) ds \right\},$
then the distribution of $B+h$ is equivalent to the distribution of $B$.

Let us denote by $F$ the support of $B$. It is clear that $F \subset C_0^{0,p-var}([0,T],\mathbb{R}^d)$, because the paths of $B$ have bounded $q$ variation for $2 < q < p$.
Let now $x \in F$. We have for $\varepsilon > 0$, $\mathbb{P}( d_{p-var}( B, x) < \varepsilon) > 0$. From the Cameron Martin theorem, we deduce then for $h \in \mathbb{W}_0^{1,2}$, $\mathbb{P}( d_{p-var}( B+h, x) < \varepsilon) > 0$. This shows that $x-h \in F$. We can find a sequence of smooth $x_n$ that converges to $x$ in $p$-variation. From the previous argument $x-x_n\in F$ and converges to 0. Thus $0 \in F$ and using the same argument shows then that $\mathbb{W}_0^{1,2}$ is included in $F$. This proves that $F=C_0^{0,p-var}([0,T],\mathbb{R}^d)$ $\square$

The following theorem due to Stroock and Varadhan describes the support of solutions of stochastic differential equations. As in the previous proof, we denote
$\mathbb{W}_0^{1,2} =\left\{ h : [0,T] \to \mathbb{R}^d, \exists k \in L^2([0,T], \mathbb{R}^d), h(t)=\int_0^t k (s) ds \right\}.$

Theorem: Let $\gamma > 2$ and let $V_1,\cdots,V_d$ be $\gamma$-Lipschitz vector fields on $\mathbb{R}^n$. Let $x_0 \in \mathbb{R}^n$. Let $(X_t)_{t \ge 0}$ be the solution of the Stratonovitch differential equation:
$X_t=x_0 + \sum_{i=1}^d \int_0^t V_i (X_s) \circ dB^i_s.$
Let $p > 2$. The support of $X$ in $C^{p-var}([0,T],\mathbb{R}^d)$ is the closure in the $p$-variation topology of the set:
$\left\{ x^h, h \in \mathbb{W}_0^{1,2} \right\},$
where $x^h$ is the solution of the ordinary differential equation
$x_t^h=x_0 + \sum_{i=1}^d \int_0^t V_i (x^h_s) dh^i_s$.

Proof: We denote by $B^n$ the piecewise linear process which is obtained from $B$ by interpolation along a subdivision $D_n$ which is such that $D_{n+1} \subset D_n$ and whose mesh goes to 0. We know that $B^n \in \mathbb{W}_0^{1,2}$ and that $x^{B^n}$ almost surely converges in $p$-variation to $X$. As a consequence $B$ almost surely takes its values in the closure of:
$\left\{ x^h, h \in \mathbb{W}_0^{1,2} \right\}.$
This shows that the support of $B$ is included in the closure of $\left\{ x^h, h \in \mathbb{W}_0^{1,2} \right\}$. The converse inclusion is a little more difficult and relies on the Lyons’ continuity theorem. It can be proved by using similar arguments as for $B$ (details are let to the reader) that the support of $S_2(B)$ is the is the closure in the $p$-variation topology of the set:
$\left\{ S_2(h), h \in \mathbb{W}_0^{1,2} \right\},$
where $S_2$ denotes, as usual, the lift in the Carnot group of step 2. Take $h \in \mathbb{W}_0^{1,2}$ and $\varepsilon > 0$. By the Lyons’ continuity theorem, there exists therefore $\eta > 0$ such that $d_{p-var} (S_2(h),S_2(B)) < \eta$ implies $\| X- x^h \|_{p-var} < \varepsilon$. Therefore
$0< \mathbb{P} \left( d_{p-var} (S_2(h),S_2(B)) < \eta \right) \le \mathbb{P} \left( \| X- x^h \|_{p-var} < \varepsilon\right).$
In particular, we have $\mathbb{P} \left( \| X- x^h \|_{p-var} < \varepsilon\right) > 0$. This proves that $x^h$ is in the support of $X$. So, the proof now boils down to the statement that the support of $S_2(B)$ is the closure in the $p$-variation topology of the set:
$\left\{ S_2(h), h \in \mathbb{W}_0^{1,2} \right\} \square$

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