We extend a Discrete Time Random Walk (DTRW) numerical scheme to simulate the anomalous diffusion of financial market orders in a simulated order book. Here using random walks with Sibuya waiting times to include a time-dependent stochastic forcing function with non-uniformly sampled times between order book events in the setting of fractional diffusion. This models the fluid limit of an order book by modelling the continuous arrival, cancellation and diffusion of orders in the presence of information shocks. We study the impulse response and stylised facts of orders undergoing anomalous diffusion for different forcing functions and model parameters. Concretely, we demonstrate the price impact for flash limit-orders and market orders and show how the numerical method generate kinks in the price impact. We use cubic spline interpolation to generate smoothed price impact curves. The work promotes the use of non-uniform sampling in the presence of diffusive dynamics as the preferred simulation method.

翻译：我们扩展了一个离散时间随机游走（DTRW）数值方案，用于模拟模拟订单簿中金融市场订单的异常扩散。本研究采用具有Sibuya等待时间的随机游走，在分数阶扩散的框架下，引入依赖于时间的随机强迫函数，并在订单簿事件之间采用非均匀采样时间。该模型通过模拟信息冲击下订单的连续到达、取消和扩散过程，刻画了订单簿的流体极限。我们研究了不同强迫函数和模型参数下，经历异常扩散的订单的脉冲响应与典型化事实。具体而言，我们展示了闪限价订单和市价订单的价格冲击，并揭示了数值方法如何在价格冲击中产生拐点。我们采用三次样条插值生成平滑的价格冲击曲线。本工作推广了在扩散动力学背景下将非均匀采样作为优选模拟方法的应用。