We study a financial market where the risky asset is modelled by a geometric Itô-Lévy process, with a singular drift term. This can for example model a situation where the asset price is partially controlled by a company which intervenes when the price is reaching a certain lower barrier. See e.g. Jarrow and Protter (J Bank Finan 29:2803–2820, 2005) for an explanation and discussion of this model in the Brownian motion case. As already pointed out by Karatzas and Shreve (Methods of Mathematical Finance, Springer, Berlin, 1998) (in the continuous setting), this allows for arbitrages in the market. However, the situation in the case of jumps is not clear. Moreover, it is not clear what happens if there is a delay in the system. In this paper we consider a jump diffusion market model with a singular drift term modelled as the local time of a given process, and with a delay \(\theta > 0\) in the information flow available for the trader. We allow the stock price dynamics to depend on both a continuous process (Brownian motion) and a jump process (Poisson random measure). We believe that jumps and delays are essential in order to get more realistic financial market models. Using white noise calculus we compute explicitly the optimal consumption rate and portfolio in this case and we show that the maximal value is finite as long as \(\theta > 0\). This implies that there is no arbitrage in the market in that case. However, when \(\theta \) goes to 0, the value goes to infinity. This is in agreement with the above result that is an arbitrage when there is no delay. Our model is also relevant for high frequency trading issues. This is because high frequency trading often leads to intensive trading taking place on close to infinitesimal lengths of time, which in the limit corresponds to trading on time sets of measure 0. This may in turn lead to a singular drift in the pricing dynamics. See e.g. Lachapelle et al. (Math Finan Econom 10(3):223–262, 2016) and the references therein.

我们研究了一个金融市场，其中的风险资产是通过几何Itô-Lévy流程建模的，并具有奇异的漂移项。例如，这可以为资产价格由公司部分控制的情况建模，该公司在价格达到某个较低的下限时会进行干预。参见例如Jarrow and Protter（J Bank Finan 29：2803–2820，2005），以在布朗运动案例中对该模型的解释和讨论。正如Karatzas和Shreve（《金融数学方法》，柏林，施普林格，1998年）所指出的那样（连续不断），这允许市场套利。但是，跳跃的情况尚不清楚。而且，还不清楚如果系统出现延迟会发生什么。可供交易者使用的信息流中的\（\ theta> 0 \）。我们允许股价动态取决于连续过程（布朗运动）和跳跃过程（泊松随机测度）。我们认为，跳跃和延迟对于获得更现实的金融市场模型至关重要。使用白噪声演算，在这种情况下，我们显式计算了最佳消耗率和投资组合，并且证明了只要\（\ theta> 0 \），最大值是有限的。这意味着在这种情况下市场没有套利。但是，当\（\ theta \）变为0，值变为无穷大。这与上述结果是一致的，即在没有延迟的情况下套利。我们的模型也与高频交易问题相关。这是因为高频交易通常导致密集交易发生在接近无限的时间长度上，这在一定程度上对应于按度量0的时间集进行交易。这又可能导致定价动态出现奇异漂移。参见例如Lachapelle等。（Math Finan Econom 10（3）：223-262，2016）及其中的参考文献。