Particle filter state estimator for large urban networks

This paper applies a particle filter (PF) state estimator to urban traffic networks. The traffic network consists of signalized intersections, the roads that link these intersections, and sensors that detect the passage time of vehicles. The traffic state X(t) specifies at each time time t the state of the traffic lights, the queue sizes at the intersections, and the location and size of all the platoons of vehicles inside the system. The basic entity of our model is a platoon of vehicles that travel close together at approximately the same speed. This leads to a discrete event simulation model that is much faster than microscopic models representing individual vehicles. Hence it is possible to execute many random simulation runs in parallel. A particle filter (PF) assigns weights to each of these simulation runs, according to how well they explain the observed sensor signals. The PF thus generates estimates at each time t of the location of the platoons, and more importantly the queue size at each intersection. These estimates can be used for controlling the optimal switching times of the traffic light

High frequency trading and asymptotics for small risk aversion in a Markov renewal model

We study a an optimal high frequency trading problem within a market microstructure model designed to be a good compromise between accuracy and tractability. The stock price is driven by a Markov Renewal Process (MRP), while market orders arrive in the limit order book via a point process correlated with the stock price itself. In this framework, we can reproduce the adverse selection risk, appearing in two different forms: the usual one due to big market orders impacting the stock price and penalizing the agent, and the weak one due to small market orders and reducing the probability of a profitable execution. We solve the market making problem by stochastic control techniques in this semi-Markov model. In the no risk-aversion case, we provide explicit formula for the optimal controls and characterize the value function as a simple linear PDE. In the general case, we derive the optimal controls and the value function in terms of the previous result, and illustrate how the risk aversion influences the trader strategy and her expected gain. Finally, by using a perturbation method, approximate optimal controls for small risk aversions are explicitly computed in terms of two simple PDE's, reducing drastically the computational cost and enlightening the financial interpretation of the results.Comment: 30 pages, new asymptotic results, typos corrected, new bibliographical reference