English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper we study the Stratonovich stochastic differential equation$\mathrm{d} X=|X|^{\alpha}\circ\mathrm{d} B$, $\alpha\in(-1,1)$, which has beenintroduced by Cherstvy et al. [New Journal of Physics 15:083039 (2013)] in thecontext of analysis of anomalous diffusions in heterogeneous media. Wedetermine its weak and strong solutions, which are homogeneous strong Markovprocesses \chng{spending zero time at $0$: for $\alpha\in (0,1)$, thesesolutions have the form $$X_t^\theta=\bigl((1-\alpha)B_t^\theta\bigr)^{1/(1-\alpha)}, $$ where $B^\theta$is the $\theta$-skew Brownian motion driven by $B$ and starting at$\frac{1}{1-\alpha}(X_0)^{1-\alpha}$, $\theta\in [-1,1]$,} and$(x)^{\gamma}=|x|^\gamma\operatorname{sign} x$; for $\alpha\in(-1,0]$, only thecase $\theta=0$ is possible. The central part of the paper consists in theproof of the existence of a quadratic covariation $[f(B^\theta),B]$ for alocally square integrable function $f$ and is based on the time-reversiontechnique for Markovian diffusions.

Stratonovich SDE with irregular coefficients: Girsanov's example revisited