Relation of a New Interpretation of Stochastic Differential Equations to Ito Process
By Jianghong Shi, Tianqi Chen, R. Yuan, Bo Yuan, Ping Ao. 2011
45 citations recorded by the source corpus.
Stochastic differential equations (SDE) are widely used in modeling stochastic dynamics in literature. However, SDE alone is not enough to determine a unique process. A specified interpretation for stochastic integration is needed. Different interpretations specify different dynamics. Recently, a new interpretation of SDE is put forward by one of us. This interpretation has a built-in Boltzmann-Gibbs distribution and shows the existence of potential function for general processes, which reveals both local and global dynamics. Despite its powerful property, its relation with classical ones in arbitrary dimension remains obscure. In this paper, we will clarify such connection and derive the concise relation between the new interpretation and Ito process. We point out that the derived relation is experimentally testable.
Subfields: Global and Planetary Change, Statistical and Nonlinear Physics
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In Global and Planetary Change
In Statistical and Nonlinear Physics
- Stefano Soatto
- Yian Ma
- Pratik Chaudhari
- Carlos Guestrin
- Tianqi Chen
- Yi-An Ma
- Emily B. Fox
- Alessandro Achille
Citations in the corpus
- Stochastic Gradient Hamiltonian Monte Carlo (2014, background): “(9) can be written in the following decomposed form (Yin & Ao, 2006; Shi et al., 2012)”
- Stochastic Gradient Descent Performs Variational Inference, Converges to Limit Cycles for Deep Networks (2017, background): “The Ito SDE dx =−∇ f (x) dt + √ 2β−1D(x) dW (t) is equivalent to the A-type SDE (Ao et al., 2007; Shi et al., 2012) dx =− ( D(x)+Q(x) ) ∇Φ(x) dt + √ 2β−1D(x) dW (t) (18) with the same steady-state distribution ρss ∝ e−βΦ(x) and Fokker-Planck equation (FP) if ∇ f (x) = ( D(x)+Q(x) ) ∇Φ(x)−β−1∇ · (…”
- Reversed graph embedding resolves complex single-cell developmental trajectories (2017, background): “We constructed a set of stochastic differential equations (SDEs) encoding regulatory relationships between 12 regulators important for neuronal development.”
- The free energy principle made simpler but not too simple (2022, unknown): “state density—referred to as solenoidal and gradient flows, respectively [25, 26, 27, 28, 29, 30, 9]:”
- A Complete Recipe for Stochastic Gradient MCMC (2015, background): “The equivalence of the compact form was originally proved in [16]; we include a detailed proof in the Supplement for completeness.”