Quantum mechanics

Discrete-time stochastic process

Classification of discontinuities

Statistics

Markov process

Materials science

Differential equation

Stochastic process

Applied mathematics

Stochastic differential equation

Range (aeronautics)

Computer science

Physics

Trajectory

Mathematics

Astronomy

Flow (mathematics)

Point process

Mathematical analysis

Geometry

Composite material

Jump

Ordinary differential equation

Jump process

Piecewise

Continuous-time stochastic process

arXiv (Cornell University)

Many time series are effectively generated by a combination of deterministic continuous flows along with discrete jumps sparked by stochastic events. However, we usually do not have the equation of motion describing the flows, or how they are affected by jumps. To this end, we introduce Neural Jump Stochastic Differential Equations that provide a data-driven approach to learn continuous and discrete dynamic behavior, i.e., hybrid systems that both flow and jump. Our approach extends the framework of Neural Ordinary Differential Equations with a stochastic process term that models discrete events. We then model temporal point processes with a piecewise-continuous latent trajectory, where the discontinuities are caused by stochastic events whose conditional intensity depends on the latent state. We demonstrate the predictive capabilities of our model on a range of synthetic and real-world marked point process datasets, including classical point processes (such as Hawkes processes), awards on Stack Overflow, medical records, and earthquake monitoring.

Neural Jump Stochastic Differential Equations