Stratonovich stochastic differential equation with irregular coefficients: Girsanov’s example revisited

Abstract

In this paper we study the Stratonovich stochastic differential equation dX " |X| α˝d B, α P p´1, 1q, which has been introduced by Cherstvy et al. [New Journal of Physics 15:083039 (2013)] in the context of analysis of anomalous diffusions in heterogeneous media. We determine its weak and strong solutions, which are homogeneous strong Markov processes spending zero time at 0: for α P p0, 1q, these solutions have the formwhere B θ is the θ-skew Brownian motion driven by B and starting at 1 1´α pX0q 1´α , θ P r´1, 1s, and pxq γ " |x| γ sign x; for α P p´1, 0s, only the case θ " 0 is possible. The central part of the paper consists in the proof of the existence of a quadratic covariation rf pB θ q, Bs for a locally square integrable function f and is based on the time-reversion technique for Markovian diffusions.

Well-Posedness for a Class of Degenerate Itô Stochastic Differential Equations with Fully Discontinuous Coefficients

“…Theorem 5.14 (Local Krylov type estimate) Assume (C3). Let y ∈ R d and M y be a weak solution to (25) starting from y such that (6…”

“…By this it is then also possible to obtain uniqueness in law as in Theorem 5.18 in the case m ≥ d. Theorem 5.16 (Local Itô-formula) Assume (C3). Let y ∈ R d and M y be a weak solution to (25) starting from y such that (6) holds. Let R 0 > 0 with y ∈ B R0 and…”

“…Definition 5.17 Let y ∈ R d . We say that uniqueness in law holds for (25) starting from y, if for any two weak solutions M y = ( Ω, ( F t ) t≥0 , F , P y ), ( X t ) t≥0 , ( W t ) t≥0 and M y = ( Ω, ( F t ) t≥0 , F , P y ), ( X t ) t≥0 , ( W t ) t≥0 of (25) starting from y, it holds…”

“…Here, generally speaking, any weak solution is allowed as long as it does not spend a positive amount of time at the zeros of the dispersion coefficient, we hence derive conditional uniqueness. Such kind of uniqueness appears often naturally, or as technical condition, and has been investigated in many different cases, with different constraints in the literature (see for instance [8,25,11,26] and a more detailed discussion of these papers below). Now in order to describe our main results, which are summarized in concise, quickly readable form in Section 3 below (we suggest to read these first), let us first describe the nature of the dispersion coefficient σ = 1 ψ • σ that appears in our main Theorem 3.1 about well-posedness.…”

“…Engelbert and Schmidt studied in [8] general one-dimensional Itô-SDEs including the case where the dispersion coefficient is degenerate and a similar condition to (6), called fundamental solution (see [8,Definition 4.16])), is considered to obtain well-posedness. [25] investigates the well-posedness of the Stratonovich counterpart of the Girsanov SDE and the condition of spending zero time at 0 is also imposed on the admissible solutions. In [11] equations without drift and with bounded dispersion coefficient that is locally of bounded variation are studied, and the Brownian motion is replaced by a regular Hölder path.…”