Beyond Itô versus Stratonovich

Abstract

Recently, a novel framework to handle stochastic processes emerges from a series of studies in biology, showing situations beyond "Itô vs. Stratonovich". Its internal consistency can be demonstrated via the zero mass limit of a generalized Klein-Kramers equation. Moreover, the connection to other integrations becomes evident: The obtained Fokker-Planck equation defines a new type of stochastic calculus that in general differs from the α-type interpretation. A unique advantage of this new approach is a natural correspondence between stochastic and deterministic dynamics, which is useful or may even be essential in practice. The core of the framework is a transformation from a usual Langevin equation to a form that contains a potential function with two additional dynamical matrices, which reveals an underlying symplectic structure. The framework has a direct physical meaning and a straightforward experimental realization. A recent experiment offers a first empirical validation of such new stochastic integration. [This article has been published in J. Stat. Mech. (2012) P07010. Note that a typo in Sect. III B has been corrected (See also [39]). ]

Endogenous network states predict gain or loss of functions for genetic mutations in hepatocellular carcinoma

“…Other important details such as the order of mutations or cooperative effects of mutations could be incorporated in further modelling. These issues require an explicit inclusion of stochastic effects, where the potential energy landscape can be used to explore more detailed issues [35,45]. Nonetheless, without any a priori knowledge of genetic mutation propensity in HCC, our results show that such network-level analyses are indeed a powerful approach to enable the prediction and a better understanding of genetic mutations in HCC.…”

Exploring a noisy van der Pol type oscillator with a stochastic approach

“…The Langevin equation alone, however, cannot determine a random process. Specifying a stochastic interpretation is needed and leads to different consequences [1][2][3][4][5]. The most widely applied stochastic interpretations are Ito's and Stratonovich's.…”

Comment on “Construction of the landscape for multi-stable systems: Potential landscape, quasi-potential, A-type integral and beyond” [J. Chem. Phys. 144, 094109 (2016)]

“…We have also noticed and stressed this feature, 7 which is evident from the SDE decomposition but less obvious in other formulations. (v) ZL stated that for SDE decomposition, "there is no explicit stochastic integral interpretation of it in higher dimensions."…”

“…The "SDE decomposition" 6,7 decomposes an SDE into three components: a potential function φ(q) (a scalar function), a friction matrix S(q) (symmetric and semi-positive definite), and a conservative force represented by a transverse matrix A(q) (antisymmetric). Matrices S(q) and A(q) are determined by the potential condition (Eq.…”

“…Because the implicit doubling of dimension of state space, it is not the usual stochastic integration in a standard textbook, as we have already recognized as beyond Ito vs Stratonovich. 7 It is possible that a more conventional interpretation may be found. (vi) ZL asked the important question of how to generalize what has been obtained for continuous processes to discrete jump processes and speculated that the "SDE decomposition" theory would be difficult to do that.…”