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Consider the following Ito diffusion $X_t$ satisfying $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x\in \mathbb{R}^n,$$ with Lipschitz coefficients $b,\sigma$. It can be shown that if $g$ is bounded a …

In my research, all the proof comes down to an estimate of the following term $$\int_t^{t+h} E|\partial_x P_{t+h-\tau}f(X_\tau)-P_{t+h-\tau}\partial_xf(X_\tau)|^2\,d\tau,\tag{1}$$ where $t>0$ is fixe …

Let $(\Omega, \mathcal{F},\mathbb{P})$ be a filtered probability space, let $b:[0,T]\times \mathbb{R}^n\to \mathbb{R}^n$ be a continuous function and Lipschitz continuous in the space variable. For e …

For each $\theta\in \mathbb{R}$, we consider a stochastic differential equation (SDE): $$ d X_t =b(t,X_t,\theta)dt+\sigma dW_t,\; t\in [0,T];\quad X_0=x_0\in \mathbb{R}, $$ where $\sigma\ge 0$ and th …

Given a deterministic function $h\in L^2([0,T]; \mathbb{R})$, we can define the associated exponential martingale \begin{align} M_t = \exp\left[\int_{0}^{t} h_s \,dB_s - \frac{1}{2}\int_{0}^{t} h_s^2 …

I am asking an extension of the question here for SDEs of the Ito form. Consider the SDE $dX_t =\sigma(X_t) dW_t$, where $W$ is a $d$-dimensional Brownian motion and $\sigma:\mathbb{R}^n\to \mathbb{R …

I am considering the transition semigroup $P_t$ associated with the Ito diffusion process $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$ where the coefficients are assumed to be Lipschitz continuous. I hope to k …