General closed-form for single-increment distribution in the n-state velocity-jump model

Derive a general closed-form expression for the probability density function of a single noisy location increment Δy in the one-dimensional n-state velocity‑jump model with continuous-time Markov chain switching (rates λ_s and transition probabilities p_su), constant state-dependent velocities v_s, and Gaussian measurement noise (variance σ^2) observed at discrete times t_j=jΔt, by evaluating the infinite series in Equation P(Δy)=∑_{w=0}^{∞} P(Δy | W=w) P(W=w).

Background

The paper models single-agent motion as a velocity-jump process whose hidden state evolves according to a continuous-time Markov chain. Observations consist of noisy, discrete-time positions, and the authors seek the distribution of measured location increments Δy.

They write the exact distribution for Δy as an infinite sum over the number of state switches within the measurement interval. Due to the complexity of conditioning on all possible switch counts and timings, they introduce up-to-m-switch approximations, noting that an exact general formula is not available.

References

From Equation~Eq:P(Delta y) we can observe that there are an infinite number of terms to compute for which we cannot obtain a general formula.

Eq:P(Delta y):

P(Δy)=∑w=0∞P(Δy ∣ W=w)P(W=w),\mathbb{P}(\Delta y) =\sum_{w=0}^{\infty} \mathbb{P}(\Delta y \,|\, W=w)\mathbb{P}(W=w),

— Approximate solutions of a general stochastic velocity-jump model subject to discrete-time noisy observations  (2406.19787 - Ceccarelli et al., 2024) in Section 3, Equation \eqref{Eq:P(Delta y)}