Search Results

First for the case of time-independent coefficients (autonomous). A good setting for this question is the Feller-explosion test (see here Is this process strictly positive? for related question). The …

From the references mentioned in the comments , (section 4.4,"Renormalisation of parabolic stochastic PDEs") (RP) and (section 6.1,"Introduction to regularity structures") (IRS), we first start from t …

the links on the video and notes are not there. But indeed as they explain in "Stochastic PDEs, Regularity Structures, and Interacting Particle Systems" at page 13, the main reason is that the Schaude …

As mentioned in the comments you simply need to do a transformation. Here in "Modelling the Stochastic Correlation" pg.23 they give the concrete transformation You replace your SDE by the above. Then …

Indeed, as mentioned in the comments one can return to a geometric BM. We start with the Lamperti transformation 2.1.5/2.1.6 from "The Lamperti Transform" by de Boer. First using Ito to compute for ge …

Here we can do a time-change to get a time-changed Brownian motion solution. see here Characterization of martingale diffusions ending in $\{-1,1\}$ This is also done carefully in Shreves-Karatzas sec …

Indeed as mentioned in the comments and in the blog here https://almostsuremath.com/2010/04/20/time-changed-brownian-motion/ for $\theta(t)=\frac{t}{1-t}$, we get that $B_{\theta(t)}\stackrel{d}{=}\i …

The situation can be a bit trickier because we are dealing with distribution valued objects and thus the need to first smoothen the equations. For example, as explained in "Fluctuation and Rate of Con …

As mentioned in this answer, the conversion to Stratonovich SDE is the following: You are considering the Stratonovich SDE $$ dX = h(X) \, dt + \gamma (X) \circ dW, $$ where I suppose the following h …

Here we are assuming that by white noise $W(r)$ the OP meant $$W(r)=\frac{dB_{r}}{dr}.$$ If meant something different, please comment below. Then one interpretation can be done by letting $X=\frac{dT} …

So one good start is "Rough Stochastic PDEs". Here Hairer builds a notion of an SPDE solution using rough paths first building the invariant solution $\psi_{t}(x)$ and lifting to a rough path $\psi_{ …

here "An SDE perspective on stochastic convex optimization" they study these type of SDEs (Stochastic gradient descent) and basically ask bounded and an L2 condition for gamma_t in theorem 3.1. Here …

Based on the comment of being interested in the hitting time for OU, it turns out that finding its density is still open as mentioned here: On the First Hitting Time Density of an Ornstein-Uhlenbeck P …

One of the latest conditions on $\mu,\sigma$ are from "A Numerical Method for SDEs with Discontinuous Drift": This result states that the SDE (1) ($dX_t = \mu(X_t)dt + \sigma(X_t)dW$) admits a unique …

I am guessing in "The particular thing I'm trying to prove is that,..." you are talking about the convergence of discrete generator to continuous one. The natural topology for these questions is Skoro …