Universal Thermodynamic Uncertainty Relation in Non-Equilibrium Dynamics

We derive a universal thermodynamic uncertainty relation (TUR) that applies to an arbitrary observable in a general Markovian system. The generality of our result allows us to make two findings: (1) for an arbitrary out-of-equilibrium system, both the entropy production and the $\mathit{\text{degree}}\mathit{\text{of}}\mathit{\text{non-stationarity}}$ are required to tightly bound the strength of a thermodynamic current; (2) by removing the antisymmetric constraint on observables, the TUR in physics and a fundamental inequality in theoretical finance can be unified in a single framework.