Search Results

there are unique corresponding forward/backward equations (Fokker Plank) to an SDE, and unique solutions for them that correspond to transition kernels. See the nice notes here Lecture 10: Forward and …

Here is some answers for your first two questions. answer to Q1 Yes,the SDE $${\rm d}U_t= - H'(U_t) {\rm d}t+\sqrt {2} \, {\rm d} B_t,$$ has the following Fokker-Plank equation (i.e. the pde satisfied …

For d=1, this is covered in Karatzas-Shreve in the section on Feller-tests. In particular, in exercise 5.38, they have that if $$\int_{x-\epsilon}^{x+\epsilon}b^{2}(y)dy<\infty,$$ then $dX_{t}=b(X_{t} …

Following a comparison result (eg. Comparison theorem in Revuz-Yor Chap. IX §3), we will show that $X_{t}\leq b$ for all $t\geq 0$. This result requires though the Yamada-Watanabe regularity $$|\sigma …

Since we can apply Burkholder-Davis-Gundy to control the supremum of moments, we start by removing the tail simply using $1\leq \frac{|X_{t}|}{R}$. $$ \mathbb E [ |X_t|^p 1_{\{|X_t| \ge R\}} ]\leq \fr …

As mentioned in an answer here and in the blog here, A sufficient condition for the integral $\int_0^t f(\omega, s)\, dB_s$ to be a martingale on $[0,T]$ is that $f(\omega,s)$ is adapted, measurable …

The difference is $$X_t^1-X_t^2=\int_0^t\, \mu_1(X_s^1)-\mu_2(X_s^2)ds + \int_0^t\,\sigma(X_s^1)-\sigma(X_s^2)dW_s.$$ For the second term, you can use that it is a martingale and apply Burkholder-Davi …

As mentioned in the comments here Feynman-Kac formula for domains with boundary, there is a Feyman-Kac type result for heat equation with boundary data in Theorem 4.2 in Chapter 7.4 in the book "Sto …

The LHS might not make sense if g is very irregular. Consider, pth-Holder function $g(x)=x^{p}$ for $0<p<1$ and $\frac{1}{p}$ odd integer and Brownian motion $X_{t}=B_{t}$, then the process $g(X_{t})= …

since I don't see any other answers, I will turn my comment into answer. One Wiener measure and path integrals reference is "Quantum Physics: a Functional Integral Point of View" from here https://mat …

Since I don't see any other answers, I move my comments to answer. Here are some good introductory references for SPDEs, by Hairer: hairer.org/notes/SPDEs.pdf, Bréhier: hal.archives-ouvertes.fr/hal-00 …

First for the case of $\sigma\neq 0$ to get convergence as paths. As described here https://almostsuremath.com/2010/05/17/sdes-under-changes-of-time-and-measure/, the above process is a time-changed B …

I believe you meant lemma 3.8. Here at the final step of your calculation, you just apply Lebesgue-differentiation theorem since those quantities are continuous and the integral is Lebesgue. That will …

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 …

In the 2d-case a lot of results have been proved already: see textbook by Sergei Kuksin, Armen Shirikyan "Mathematics of Two-Dimensional Turbulence" (see equation 2.100 for Stochastic Stokes). In term …