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Let $b: \mathbb R_+\times\mathbb R\times [0,1]\to [\underline b,\overline b]$ and $a: \mathbb R_+\times\mathbb R\times [0,1]\to [\underline a,\overline a]$ be Lipschitz, where $\overline b>\underline …

Consider the drifted Brownian motion $X_t=1+\lambda(t)+W_t$, where $\lambda: \mathbb R\to [0,\infty)$ with $1\le \lambda'(t)\le 2$ and $(W_t)_{\ge 0}$ denotes a Brownian motion. Define the hitting tim …

I claim this is not a complete answer. I wonder whether it can be improved to obtain the Lipschitz continuity. Define $$\left(\frac{d\mathbb Q}{d\mathbb P}\right)_t := \exp\left(-\int_0^t \lambda'(s)d …

Let $$Y_t:=1+\int_0^t b_sds + W_t,\quad\forall t\ge 0,$$ where $(b_t)_{t\ge 0}$ is a bounded adapted process and $(W_t)_{t\ge 0}$ is a standard Brownian motion. Denote $\tau:=\{t\ge 0: Y_t\le 0\}$ and …

Let $$Y_t:=1+\int_0^t b(s)ds + W_t,\quad\forall t\ge 0,$$ where $b:\mathbb R_+\to[1,2]$ is continuous and $(W_t)_{t\ge 0}$ is a standard Brownian motion. Denote $\tau:=\{t\ge 0: Y_t\le 0\}$ and $X_t:= …

Let $a:\mathbb R_+\to [1,2]$ be "smooth". Given a standard Brownian motion $W$, define for $t\ge 0$ $$X_t:=\int_0^t\frac{1}{a(s)}dW_s \quad \mbox{and}\quad Y_t:=\sup_{0\le u\le t} \int_0^u a(s)dW_s.$$ …

Let $(W_t)_{t\ge 0}$ be a standard Brownian motion. For each $t\in [0,1]$, it is known that, e.g. from Burkholder-Davis-Gundy's inequality $$\mathbb E\big[\sup_{s\in [t,t+\Delta t]}|W_s-W_t|^p\big]=O( …

Let $(X_t)_{0\le t\le 1}$ be a continuous Markov martingale (with respect to its natural filtration) s.t. $X_0=0$ and $X_1\in\{-1,1\}$. Can we prove the existence of some measurable function $\sigma: …

This question follows from the previous post Question on the martingale representation theorem which has not been answered. I consider thus a particular case. Let $(B_t)_{t\ge 0}$ be a standard Browni …

Let $b: \mathbb R_+\times\mathbb R\times \mathbb R\to\mathbb R$ be a function as nice as possible, and $C^1([0,T])$ be the space of continuously differentiable functions $\alpha:[0,T]\to\mathbb R$ wit …

Let $b\in \mathbb R$ and $\sigma>0$ be given. For a fixed probability distribution $\mu_0$ on $\mathbb R$ s.t. $$\int_{(0,\infty)}\mu_0(dx)=1,$$ consider the mean field SDE : $$dX_t = \mathbf{1}_{\{X …

Assume the filtration is large enough. Then Kellerer and Strassen both proved indepently the existence of such a martingale is equivalent to $$\int fd\mu_{t_n} \le \int fd\mu_{t_{n+1}},\quad \mbox{for …

Consider the stochastic differential equation as follows: $$X_t=X_0+t+\int_0^t\frac{dW_s}{1+m(s)},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$ where $X_0>0$ is square integrable and $m(t)=\mathbb P[\i …

Given two SDEs $X^1$, $X^2$ : $$X^i_t=1+t+\int_0^t\sigma_i(s)dW_s,\quad \forall t\ge 0,$$ where $\sigma_i:\mathbb R_+\to [1/2,1]$ are non-decreasing s.t. $\sigma_1(t)\le \sigma_2(t)$ for all $t\ge 0$ …