Analytical Survival Analysis of the Ornstein–Uhlenbeck Process

Abstract

We use asymptotic methods from the theory of differential equations to obtain an analytical expression for the survival probability of an Ornstein–Uhlenbeck process with a potential defined over a broad domain. We form a uniformly continuous analytical solution covering the entire domain by asymptotically matching approximate solutions in an interior region, centered around the origin, to those in boundary layers, near the lateral boundaries of the domain. The analytic solution agrees extremely well with the numerical solution and takes into account the non-negligible leakage of probability that occurs at short times when the stochastic process begins close to one of the boundaries. Given the range of applications of Ornstein–Uhlenbeck processes, the analytic solution is of broad relevance across many fields of natural and engineering science.

Statistics of the first passage area functional for an Ornstein–Uhlenbeck process

“…One could extend the analysis relatively easily to discuss the area statistics when the first passage time relates to a level which differs from the equilibrium level. This has been done in some detail for the statistics of the first passage time itself; for a recent overview see [39][40][41]. More generally, e.g.…”

The Thermodynamic Cost of Erasing Information in Finite-time

“…We find that these quantities are dominated by the jump statistics between the potential wells and fluctuations within the wells play only a minor role. The jump statistics are generated by representing particle paths between states as instantons and the survival probability within the individual states, for which analytical expressions are available from matched asymptotic expansion techniques [34,36]. For the case of a linearly increasing erasure force (Eq.…”

“…In Refs. [34,36] we used the method of matched asymptotic expansions to calculate such survival probabilities analytically and here we summarize the central results using the present notation;…”

“…We find that the principal sources of randomness are the transitions over the potential barrier between the two memory states, whereas the fluctuations within the potential wells have negligible effects on the work distribution. Thirdly, by using recently developed methods from stochastic resonance and early warning quantifiers [32][33][34][35][36], we provide a concrete recipe for calculating the jump-time distributions (19) using instantons and survival probabilities. By combining these results with (17), (30) and (31), the averages in (11) and (16) reduce to simple numerical quadratures.…”

Quantifying Uncertainty: Potential Medical Applications of the Heston Model of Financial Stochastic Volatility

“…While the Heston Model's square root reversion is adept at capturing the volatility dynamics observed in financial markets, the OU process's linear reversion is more suited to natural decay patterns. The OU process has been used more frequently in biological contexts because it stabilizes around some equilibrium point, making it suitable for modeling various biological phenomena [24].…”