A multifractal boundary spectrum for $${{\,\mathrm{SLE}\,}}_\kappa (\rho )$$

We study $${{\,\mathrm{SLE}\,}}_\kappa (\rho )$$ SLE κ ( ρ ) curves, with $$\kappa $$ κ and $$\rho $$ ρ chosen so that the curves hit the boundary. More precisely, we study the sets on which the curves collide with the boundary at a prescribed “angle” and determine the almost sure Hausdorff dimensions of these sets. This is done by studying the moments of the spatial derivatives of the conformal maps $$g_t$$ g t , by employing the Girsanov theorem and using imaginary geometry techniques to derive a correlation estimate.

The Mathematics of the Martingale Approach

In this chapter we present the two main mathematical results which are needed for the application of the martingale approach to pricing and hedging. We first discuss and prove the martingale representation theorem which says that in a Wiener framework, every martingale can be represented as a stochastic integral. We then discuss and prove the Girsanov Theorem which gives us control over the class of absolutely continuous measure transformations. The abstract theory is then applied to stochastic differential equations, and to maximum likelihood estimation.

Stabilization by noise of a ℂ2-valued coupled system

D. Herzog and J. Mattingly have shown that a [Formula: see text]-valued polynomial ODE with finite-time blow-up solutions may be stabilized by the addition of [Formula: see text]-valued Brownian noise. In this paper, we extend their results to a [Formula: see text]-valued system of coupled ODEs with finite-time blow-up solutions. We show analytically and numerically that stabilization can be achieved in our setting by adding a suitable Brownian noise, and that the resulting system of SDEs is ergodic. The proof uses the Girsanov theorem to induce a time change from our [Formula: see text]-system to a quasi-[Formula: see text]-system similar to the one studied by Herzog and Mattingly.