On the existence of SLE trace: finite energy drivers and non-constant $\kappa$

Abstract

Existence of Loewner trace is revisited. We identify finite energy paths (the “skeleton of Wiener measure”) as natural class of regular drivers for which we find simple and natural estimates in terms of their (Cameron–Martin) norm. Secondly, now dealing with potentially rough drivers, a representation of the derivative of the (inverse of the) Loewner flow is given in terms of a rough- and then pathwise Föllmer integral. Assuming the driver within a class of Itô-processes, an exponential martingale argument implies existence of trace. In contrast to classical (exact) SLE computations, our arguments are well adapted to perturbations, such as non-constant $\kappa$ (assuming $<2$ for technical reasons) and additional finite-energy drift terms.

Appendices

Appendix A: Decay of $|f_t'(z)|$ and existence of trace

We collect some variations on familiar results concerning existence of trace from decay of $|f_t'(iy + U_t)|$ as $y \rightarrow 0$.

Lemma 3

Suppose there exist a $\theta < 1$ and $y_0 > 0$ such that for all $y \in (0 , y_0]$

$$\begin{aligned} \sup _{ t \in [0,T]} |f_t'(iy + U_t )| \le y^{-\theta } \end{aligned}$$

(5.1)

then the trace exists.

Proof

Note that for $y_1< y_2 < y_0$,

$$\begin{aligned} | f_t(iy_2 + u_t) - f_t(iy_1 + U_t) |= & {} \left| \int _{y_1}^{y_2} f_t'( ir + U_t) dr\right| \\\le & {} \int _{y_1}^{y_2} r^{-\theta } dr = \frac{1}{1-\theta }(y_2^{1-\theta } - y_1^{1 - \theta } ) \end{aligned}$$

which implies that $f_t(iy + U_t)$ is Cauchy in y and thus

$$\begin{aligned} \gamma _t = \lim \limits _{y \rightarrow 0+ } f_t(iy + U_t) \end{aligned}$$

exists. For continuity of $\gamma$, observe that

$$\begin{aligned} |\gamma _t - f_t(iy + U_t) | \le \frac{y^{1-\theta }}{1-\theta }. \end{aligned}$$

Now,

$$\begin{aligned} |\gamma _t - \gamma _s|&\le |\gamma _t - f_t(iy + U_t) | + |\gamma _s- f_s(iy + U_s) | + | f_t(iy + U_t) - f_s(iy + U_s)| \\&\lesssim y^{1 -\theta } + | f_t(iy + U_t) - f_s(iy + U_s)|; \end{aligned}$$

it is easy to see that for $y > 0$,

$$\begin{aligned} \lim \limits _{s \rightarrow t } | f_t(iy + U_t) - f_s(iy + U_s)| = 0 \end{aligned}$$

and since y was arbitrary, this concludes the proof. $\square$

Lemma 4

If U is weakly $\frac{1}{2}$-Holder and there exist constant $b > 2$, $\theta < 1$ and $C < \infty$ such that for all $t \in [0,T]$ and $y > 0$

$$\begin{aligned} \mathbb {P}[ |f_t'(iy + U_t)| \ge y^{-\theta }] \le Cy^b \end{aligned}$$

then the trace exists.

Proof

By using of Borel–Cantelli lemma, it is easy that almost surely for n large enough,

$$\begin{aligned} |f_{k2^{-2n}}'( i 2^{-n} + U_{ k 2^{-2n}})| \le 2^{ n \theta } \end{aligned}$$

for all $k = 0, 1, .., 2^{2n} - 1$. Now applying results in section 3 of [7] (Lemma 3.7 and the Distortion Theorem in particular) completes the proof. $\square$

We also collect some familiar results on behaviour of $f_t$ which are used in the proofs above.

Lemma 5

([7], Lemma 3.5) For each $z = x + iy \in \mathbb {H}$ and $t \ge s$,

$$\begin{aligned} e^{- \frac{10(t-s)}{y^2}}|f_s'(z)|\le |f_t'(z)| \le e^{\frac{10(t-s)}{y^2}}|f_s'(z)|. \end{aligned}$$

In particular, if $t-s \le y^2$, then

$$\begin{aligned} e^{- 10}|f_s'(z)|\le |f_t'(z)| \le e^{10}|f_s'(z)|. \end{aligned}$$

Also, if $t-s \le y^2$, then $|f_t(z) - f_s(z)| \le \frac{y}{5}[e^{10} - 1 ] |f_s'(z)|$.

Lemma 6

([7], Lemma 3.6) There exist constants $c , \eta < \infty$ such that for all $y > 0$ and $0 \le t-s \le y^2$,

$$\begin{aligned} |f_t'(U_t + iy)| \le c M^\eta |f_t'(U_s + iy)|, \end{aligned}$$

where $M = \max { \{1, \frac{|U_t-U_s|}{y}\}}$.

Lemma 7

([18], Lemma 2.1) For any continuous $U:[0,t]\ \rightarrow \mathbb {R}$,

$$\begin{aligned} |Re(f_t(iy + U_t)) - U_t | \le \sup _{ 0 \le r \le t } |U_t - U_r|. \end{aligned}$$

In particular, if $||U||_{\frac{1}{2},[0,t]} \le \sigma$, then $|Re(f_t(iy + U_t))| \le 2\sigma \sqrt{t}$.

Lemma 8

([18], Theorem 3.1) If $||U||_{\frac{1}{2},[0,t]} \le \sigma < 4$, there exist constant $c_\sigma > 0$ such that

$$\begin{aligned} \sqrt{\frac{4t}{1 + c_{\sigma }^2} + y^2} \le Im(f_t(iy + U_t)) \le \sqrt{4t + y^2}. \end{aligned}$$

Appendix B: Elements of Rough path theory

Here we collect some results without proofs from rough path theory useful in the present context. A detailed treatment of the subject can be found in [2, 3]. Though we have used only the elementary one dimensional case of the rough path theory, there is no extra effort in presenting the general case. Let $X:[0,T] \rightarrow \mathbb {R}^d$ be a d-dimensional path $\alpha$-Hölder continuous path for some $\alpha \in (0,1]$. The aim is to establish a rich enough integration theory of type $\int _0^T Y_rdX_r$ for appropriate paths Y against path X. The fundamental contribution of rough path theory is that such integrations can be made sense in a deterministic way, even when the path X is a sample path of a random process like Brownian motion. The theory applies in principal for any $\alpha > 0$, but we will restrict to the case of $\alpha > \frac{1}{3}$ for simplicity. In that case, the goal is acheived by “lifting” the path X to a rough path defined as follows. For a path Z and times s, t, we will follow the notation $Z_{s,t} := Z_t - Z_s$.

Definition 1

Let X be a path and $\alpha > \frac{1}{3}$. An $\alpha$-rough path is a pair of maps $\mathbf {X} = (X, \mathbb {X}) : [0,T]\times [0,T] \rightarrow \mathbb {R}^d \oplus (\mathbb {R}^d \otimes \mathbb {R}^d)$ with following properties:

• $X_{s,t} = X_t - X_s$

• For any s, u, t, 

  $$\begin{aligned} \mathbb {X}_{s,t} - \mathbb {X}_{s,u} - \mathbb {X}_{u,t} = X_{s,u}\otimes X_{u,t}. \end{aligned}$$

  (5.2)

• X and $\mathbb {X}$ have $\alpha$- and $2 \alpha$-Hölder regularity respectively, in the sense that

  $$\begin{aligned} ||X||_{\alpha } := \sup _{s,t} \frac{|X_{s,t}|}{|t-s|^{\alpha }}< \infty , \qquad ||\mathbb {X}||_{2\alpha } := \sup _{s,t} \frac{|\mathbb {X}_{s,t}|}{|t-s|^{2\alpha }} < \infty . \end{aligned}$$

  (5.3)

Further, a rough path $\mathbf {X}$ is called a geometric rough path⁴ if in addition it satisfies

$$\begin{aligned} Sym(\mathbb {X}_{s,t}) = \frac{1}{2}X_{s,t}\otimes X_{s,t}, \end{aligned}$$

(5.4)

for all $s,t \in [0,T]$, where Sym(M) denotes the symmetric part of a matrix M.

The object $\mathbb {X}_{s,t}$ in the definition above is supposed to mimic the behaviour of iterated integral $\int _{s< r_1< r_2 < t } dX_{r_1}\otimes dX_{r_2}$ via constraint equations (5.2) and (5.3). In many cases of interest, these integrals can not be analytically but via stochastic analysis. Note that smooth paths satisfies an additional algebraic constraint, namely integration by parts formula and the equation (5.4) precisely captures such a constraint. (When $d=1$ only, this determines $\mathbb {X}_{s,t} = \tfrac{1}{2} X_{s,t}^2$.) Before stating the next theorem, introduce a metric $\rho _\alpha$ and norm $|||.|||_{\alpha }$ on the space of rough path defined by

$$\begin{aligned} \rho _{\alpha }(\mathbf {X}^1, \mathbf {X}^2) := ||X^1 - X^2||_{\alpha } + ||\mathbb {X}^1- \mathbb {X}^2||_{2\alpha } \end{aligned}$$

and

$$\begin{aligned} |||\mathbf {X}|||_{\alpha } := ||X||_{\alpha } + ||\mathbb {X}||_{2\alpha }. \end{aligned}$$

Theorem 6

Let $\mathbf {X}$ be a $\alpha$-geometric rough path, then there exists a sequence of smooth paths $X^n$ such that rough path $\mathbf {X}^n$ defined by

$$\begin{aligned} \mathbf {X}^n_{0,t} := \biggl ( X^n_{0,t}, \int _0^t X_{0,r}^n \otimes dX_r^n \biggr ) \end{aligned}$$

converge uniformly to $\mathbf {X}_{0,t}$. Moreover, $\rho _{\alpha '}(\mathbf {X}^n, \mathbf {X}) \rightarrow 0 \quad \hbox {as} \quad n \rightarrow \infty$ for any $\alpha ' < \alpha$.

The rough path $\mathbf {X}$ will play the role of integrator. The natural class of the integrands in the present context are chosen to be the pairs $(f(X), f'(X))$ for $C^2$ functions f. In general, one takes the collection of pair of paths $(Y, Y')$ which mimics the behavious of $(f(X), f'(X))$, coining the definition of controlled rough paths as follows:

Definition 2

Given an $\alpha$-rough path $\mathbf {X}$, a pair of continuous paths $(Y, Y')$, with Y taking value in some $\mathbb {R}^m$ and $Y'$ in $\mathbb {R}^{m \times d }$, is called a X-controlled rough path if

• $||Y||_{\alpha } + ||Y'||_{\alpha } < \infty$

• The object $R^Y$ defined by $R^Y_{s,t} := Y_{s,t} - Y_s'X_{s,t}$ has $2 \alpha$ regularity, in the sense that

  $$\begin{aligned} ||R^Y||_{2\alpha } := \sup _{s,t }\frac{|R^Y_{s,t}|}{|t-s|^{2\alpha }} < \infty . \end{aligned}$$

The path $Y'$ is called the Gubinelli derivative of Y with respect to X.

We now present the construction of so called rough integration giving an integration theory of controlled rough paths against rough paths.

Theorem 7

[4, 13] Let $\mathbf {X}$ be a rough path and $(Y, Y')$ be a controlled rough path taking values in $(\mathbb {R}^{m \times d}, \mathbb {R}^{m \times d \times d} )$. Then,

$$\begin{aligned} \hbox {(rough integral)} \int _0^T Y_rd\mathbf {X}_r := \lim \limits _{|\mathcal {P}| \rightarrow 0 } \sum _{[s,t] \in \mathcal {P}} Y_sX_{s,t} + Y_s'\mathbb {X}_{s,t} \end{aligned}$$

exists. Furthermore there exist a constant C depending only on $\alpha$ and T such that for all s, t,

$$\begin{aligned} \biggl |\int _s ^t Y_rd\mathbf {X}_r - Y_sX_{s,t} - Y_s'\mathbb {X}_{s,t} \biggr | \le C ( ||X||_{\alpha }||R^Y||_{2\alpha } + ||\mathbb {X}||_{2\alpha }||Y'||_{\alpha }) |t-s|^{3\alpha }. \end{aligned}$$

Having a theory of rough integration, one naturally considers the differential equations driven by such rough paths. Note that given a controlled rough path $(Y, Y')$ and a $C^2$ function f, the pair $(Z, Z') := (f(Y), f'(Y)Y')$ also defines a controlled rough path. Following this remark, one can consider rough differential equations (RDE) of form

$$\begin{aligned} dZ_t = \sigma (Z_t)d\mathbf {X}_t + b(Z_t)dt, \quad Z_0 = z_0, \end{aligned}$$

(5.5)

for smooth vector fields $\sigma$ and b. With a slight abuse of notation, the solution Z is actually a controlled rough path $(Z, Z')$ with an implicitly defined Gubinelli derivative $Z' = \sigma (Z)$ and Eq. (5.5) is given the meaning

$$\begin{aligned} Z_t = z_0 + \int _0 ^t \sigma (Z_r)d\mathbf {X}_r + \int _0^t b(Z_r)dr, \end{aligned}$$

where the first integral is understood as a rough integral. Given sufficiently smooth vector fields, existence and uniqueness of a controlled rough paths $(Z, Z')$ satisfying (5.5) can be established using Picard iteration in the Banach space of controlled rough paths, see [2] for details. Furthermore, solutions to rough differential equations are also stable under approximations to rough paths, as guaranteed by the following Theorem:

Theorem 8

(Continuity of the Itô–Lyons map) Let $\mathbf {X}^n$ is a sequence of rough paths converging to a rough path $\mathbf {X}$ in rough path metric, i.e.

$$\begin{aligned} \rho _{\alpha }(\mathbf {X}^n, \mathbf {X}) \rightarrow 0 \quad \hbox {as} \quad n \rightarrow \infty \end{aligned}$$

for $\alpha \in (1/3,1/2]$. Assume $\sigma$ and b are bounded in $C^3$ and $C^1$ respectively, and write $Z^n$ and Z for the corresponding solutions to Eq. (5.5) driven by $\mathbf {X}^n$ and $\mathbf {X}$. Then

$$\begin{aligned} ||Z^n - Z||_{\alpha } \rightarrow 0 \quad \hbox {as} \quad n \rightarrow \infty . \end{aligned}$$

We now present an application of rough path theory in the context of ODEs. As above, let $\mathbf {X}$ be $\alpha$-rough path. Consider a approximation of the underlying path X by smooth paths $X^n$ converging uniformly to X as $n \rightarrow \infty$. Since each $X^n$ is a smooth path, the equation

$$\begin{aligned} dZ_t^n = \sigma (Z_t^n)dX_t^n + b(Z_t^n)dt, \quad Z_0^n = z_0, \end{aligned}$$

(5.6)

can be understood as an ODE with the unique solution $Z^n$. Also, the paths $X^n$ can be naturally lifted to a (geometric) $\alpha$-rough path $\mathbf {X}^n$ by defining

$$\begin{aligned} \mathbb {X}_{s,t}^n := \int _s^t X_{s,r}^n \otimes dX_r^n, \end{aligned}$$

where the integral in the definition is understood as a Riemann–Stieltjes integral. It can be very easily verified that rough integrals $\int _0^T (Y_r^n, Y_r^{n'})d\mathbf {X}_r^n$ matches with Riemann–Stieltjes integral $\int _0^T Y_r^ndX_r^n$ for any $X^n$-controlled rough path $(Y^n, Y^{n'})$. Thanks to this, the ODE (5.6) can be understood as RDE (5.5) driven by rough paths $\mathbf {X}^n$. There are many benefits of interpreting equation (5.6) as a RDE. As implied by Theorem 8, RDEs are stable under approximation and it readily gives the following result on the convergence of $Z^n$ as $n \rightarrow \infty$.

Theorem 9

Suppose that $(X^n, \mathbb {X}^n)$ converges to some rough path $\mathbf {X}$ in rough path metric $\rho _{\alpha }$. Then solution $Z^n$ of ODE (5.6) converge to the solution Z of the RDE (5.5) uniformly and in $\alpha$-Hölder topology.

When X is chosen to be sample path of a d-dimensional Brownian motion B, the theory discussed above can be readily put in application with the aid of following Theorem. We also remark here that such a consideration is not limited to application for Brownian motion and can be very easily adapted to handle general semimartingales.

Theorem 10

The objects $\mathbf {B}^{Ito} := (B, \mathbb {B}^{Ito})$ and $\mathbf {B}^{Strat} := (B, \mathbb {B}^{Strat})$ defined by

$$\begin{aligned} \mathbb {B}_{s,t}^{Ito}:= & {} (\hbox {It}\hat{\mathrm{o}}\hbox { integral}) \int _s^t B_{s,r}\otimes dB_r, \qquad \\ \mathbb {B}_{s,t}^{Strat}:= & {} (\hbox {Stratonovich integral}) \int _s^t B_{s,r}\otimes \circ dB_r \end{aligned}$$

constitute rough paths with probability one.

Theorem 10 allows one to readily define rough integrals of any controlled rough paths $(Y, Y')$ (controlled by B) against Brownian rough paths defined above. This gives us way of interpreting Itô and Stratonovich integral as deterministically constructed rough integrals (with explicit knowledge of the exceptional set; that is, those $\omega$ for which $\mathbf {B}(\omega )$ fails to be a rough path or $(Y,Y')(\omega )$ fails to be controlled by $B(\omega )$. If e.g. applied to $Y=f(X),Y'=f'(X)$ for $f\in C^2$, say, the exceptional set does not depend on f, in stark contrast to classical stochastic integration theory.)

Theorem 11

Assume $Y, Y'$ are adapted processes such that almost surely $(Y, Y')$ is a $B(\omega )$-controlled rough path, then almost surely

$$\begin{aligned} (\hbox {rough integral}) \int _0 ^T Y_rd\mathbf {B}_r^{Ito} = (\hbox {It}\hat{\mathrm{o}}\hbox { integral}) \int _0 ^T Y_rdB_r. \end{aligned}$$

Furthermore, if quadratic covariation of Y and B exists, then almost surely

$$\begin{aligned} (\hbox {rough integral}) \int _0 ^T Y_rd\mathbf {B}_r^{Strat} = (\hbox {Stratonovich integral}) \int _0 ^T Y_r\circ dB_r. \end{aligned}$$

Similarly as in the previous discussion, an Itô stochastic differential equation

$$\begin{aligned} dZ_t = \sigma (Z_t)dB_t + b(Z_t)dt, \quad Z_0 = z_0 \end{aligned}$$

(5.7)

and its Stratonovich variant can be understood as RDE driven by the random rough path $\mathbf {B}^{Ito}$ and $\mathbf {B}^{Strat}$ respectively. Now, standard arguments imply that if $B^n$ is a piecewise linear approximation to Brownian motion, with mesh-size $2^{-n}$ say, the rough paths $\mathbf {B}^n$ converge almost surely to $\mathbf {B}^{Strat}$ in the rough path metric $\rho _{\alpha }$. Again with the aid of Theorem 8, one can derive the following approximation result for SDEs:

Theorem 12

(Wong–Zakai) Let $Z^n$ are solutions to ODEs

$$\begin{aligned} dZ_t^n = \sigma (Z_t^n)dB_t^n + b(Z_t^n)dt, \quad Z_0^n = z_0. \end{aligned}$$

(5.8)

Then almost surely $Z^n$ converge to solution Z of Stratonovich SDE

$$\begin{aligned} dZ_t = \sigma (Z_t)\circ dB_t + b(Z_t)dt, \quad Z_0 = z_0 \end{aligned}$$

(5.9)

in uniform and in $\alpha$-Hölder topology, any $\alpha < 1/2$.