We consider a natural class of $\mathbf{R}^d$-valued one-dimensional stochastic PDEs driven by space-time white noise that is formally invariant under the action diffeomorphism group on $\mathbf{R}^d$. This contains in particular KPZ equation, multiplicative heat additive and rough Burgers-type equations. exhibit one-parameter family solution theories with following properties: - For all SPDEs ...

In this paper, stability theorems for stochastic differential equations and backward stochastic differential equations driven by G-Brownian motion are obtained. We show the existence and uniqueness of solutions to forward-backward stochastic differential equations driven by G-Brownian motion. Stability theorem for forward-backward stochastic differential equations driven by G-Brownian motion is...

This paper surveys some results in stochastic differential delay equations beginning with ”On stationary solutions of a stochastic differential equations” by K. Ito and M. Nisio, 1964, and also some results in stochastic stability beginning with the ”Stability of positive supermartingales” by R. Bucy, 1964. The problems discussed in this survey are the existence and uniqueness of solutions of s...

For a backward stochastic differential equation (BSDE, for short), when the generator is not progressively measurable, it might admit adapted solutions, shown by an example. However, Volterra integral equations (BSVIEs, generators are allowed to be anticipating. This gives, among other things, essential difference between BSDEs and BSVIEs. Under some proper conditions, well-posedness of such BS...