论文标题
$ h = 0 $分数布朗田地的混乱
The Multiplicative Chaos of $H=0$ Fractional Brownian Fields
论文作者
论文摘要
我们考虑一个$ \ mathbb {r}^{d} $上的分数布朗尼田地$ \ {b^{h} \} _ {h \ in（0,1）} $，其中$ h $表示他们的hurst参数。我们首先定义了一类规范化核$ψ$的等级类别，以使$$ x^{h}（x）=γ（h）=γ（h）^{\ frac {1} {2}} {2}} \ left（b^{h}（h}（h}（x）） - x）du \ right），$$在$ h \ downrow 0 $时收敛到对数相关的高斯字段的协方差。 然后，我们使用Berestycki的“好处”方法，以得出限制量的限制度量，即分数Brownian Field $$ M^{h}_γ（dx）= E^e^{γX^{γx^{h} { e [x^{h}（x）^{2}]} dx，$$ as $ h \ downarrow 0 $ 0 $ for in（0，γ^{*}（d）] $，其中$γ^{*}（d）] $ l^{2} $收敛的$ m^{h}_γ$在``好处''的集合上，其中$ x^h $具有典型的行为，作为收敛结果的副产品，我们证明，该概率是$ hurst compotity of polatitions of nog of pote n oc $ $ $ $ $ $。此外，在这些集合中，波动率以$ l^2 $收敛到多重随机步行的波动性。
We consider a family of fractional Brownian fields $\{B^{H}\}_{H\in (0,1)}$ on $\mathbb{R}^{d}$, where $H$ denotes their Hurst parameter. We first define a rich class of normalizing kernels $ψ$ such that the covariance of $$ X^{H}(x) = Γ(H)^{\frac{1}{2}} \left( B^{H}(x) - \int_{\mathbb{R}^{d}} B^{H}(u) ψ(u, x)du\right), $$ converges to the covariance of a log-correlated Gaussian field when $H \downarrow 0$. We then use Berestycki's ``good points'' approach in order to derive the limiting measure of the so-called multiplicative chaos of the fractional Brownian field $$ M^{H}_γ(dx) = e^{γX^{H}(x) - \frac{γ^{2}}{2} E[X^{H}(x)^{2}] }dx, $$ as $H\downarrow 0$ for all $γ\in (0,γ^{*}(d)]$, where $γ^{*}(d)>\sqrt{\frac{7}{4}d}$. As a corollary we establish the $L^{2}$ convergence of $M^{H}_γ$ over the sets of ``good points'', where the field $X^H$ has a typical behaviour. As a by-product of the convergence result, we prove that for log-normal rough volatility models with small Hurst parameter, the volatility process is supported on the sets of ``good points'' with probability close to $1$. Moreover, on these sets the volatility converges in $L^2$ to the volatility of multifractal random walks.
