We characterize the optimal signal-adaptive liquidation strategy for an agent subject to power-law resilience and zero temporary price impact with a Gaussian signal, which can include e.g an OU process or fractional Brownian motion. We show that the optimal selling speed $u_t^{\ast }$ is a Gaussian Volterra process of the form $u^{\ast }\left(t\right)=u^0\left(t\right)+\overline{u}\left(t\right)+{\int }_0^{t}k(u,t)\mathrm{d}{W}_u$ on $[0,T)$, where $k(\cdot ,\cdot )$ and $\overline{u}$ satisfy a family of (linear) Fredholm integral equations of the first kind which can be solved in terms of fractional derivatives. The term $u^0\left(t\right)$ is the (deterministic) solution for the no-signal case given in Gatheral et al. [Transient linear price impact and Fredholm integral equations. Math. Finance, 2012, 22, 445–474], and we give an explicit formula for $k(u,t)$ for the case of a Riemann-Liouville price process as a canonical example of a rough signal. With non-zero linear temporary price impact, the integral equation for $k(u,t)$ becomes a Fredholm equation of the second kind. These results build on the earlier work of Gatheral et al. [Transient linear price impact and Fredholm integral equations. Math. Finance, 2012, 22, 445–474] for the no-signal case, and complement the recent work of Neuman and Voß[Optimal signal-adaptive trading with temporary and transient price impact. Preprint, 2020]. Finally we show how to re-express the trading speed in terms of the price history using a new inversion formula for Gaussian Volterra processes of the form ${\int }_0^{t}g(t-s)\mathrm{d}{W}_s$, and we calibrate the model to high frequency limit order book data for various NASDAQ stocks.

我们用高斯信号描述了受幂律弹性和零临时价格影响的代理的最佳信号自适应清算策略，其中可以包括例如 OU 过程或分数布朗运动。我们证明了最佳销售速度${你}_{吨}^{*}$是形式为的高斯沃尔泰拉过程${你}^{*}\left(吨\right)={你}^0\left(吨\right)+\overline{你}\left(吨\right)+{\int }_0^{吨}ķ(你,吨)\mathrm{d}{W}_{你}$在$[0,吨)$， 在哪里$ķ(\cdot ,\cdot )$和$\overline{你}$满足可以用分数导数求解的第一类（线性）Fredholm 积分方程族。术语${你}^0\left(吨\right)$是 Gatheral等人给出的无信号情况的（确定性）解决方案。[瞬态线性价格影响和 Fredholm 积分方程。数学。Finance , 2012, 22 , 445–474]，我们给出了一个明确的公式$ķ(你,吨)$对于 Riemann-Liouville 价格过程作为粗略信号的典型示例。在非线性临时价格影响下，积分方程为$ķ(你,吨)$成为第二类 Fredholm 方程。这些结果建立在 Gatheral等人的早期工作之上。[瞬态线性价格影响和 Fredholm 积分方程。数学。Finance , 2012, 22 , 445–474] 用于无信号情况，并补充了 Neuman 和 Voß 最近的工作[具有临时和瞬时价格影响的最优信号自适应交易。预印本，2020]。最后，我们展示了如何使用形式为高斯沃尔泰拉过程的新反演公式根据价格历史重新表达交易速度${\int }_0^{吨}G(吨-s)\mathrm{d}{W}_s$，我们将模型校准为各种纳斯达克股票的高频限价订单数据。