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For some filtered probability space $\big(\Omega,\mathcal F, (\mathcal F_t),\mathbb P\big)$, consider a stochastic differential equation (driven by a real-valued Brownian motion $W$) for $X=(X_t)$, wh …

Provided a martingale $X$, discrete-time $X=(X_n, n\in\mathbb N)$ or continuous-time $X=(X_t, t\ge 0)$, Doob's upcrosssing inequality states that : If $U_N(a,b)$ denotes the number of up-crossings of …

Let $X$ be a predictable process defined on some filtered probability space (as good as possible) such that $$X_t \in \{0,1\},\quad \forall t\ge 0.$$ Does this imply the left continuity of $X$? If so, …

Let $M=(M_t)_{1\le t\le 2}$ be a continuous (resp. right-continuous) martingale. Denote $x:=\mathbb E[M_1]\in\mathbb R$. Can we construct on some probability space a continuous (resp. right-continuous …

Consider a stochastic differential equation (SDE) on some filtered probability space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$ : for all $t>0$ $$dX_t = u_tf(X_{t-})dt+ u_t g(X_{t-})dW_t + u_t\int_{ …

It is known that any function that is right-continuous with left limits (càdlàg as a French abbreviation) can be approximated by continuous ones (under e.g. Skorokhod topology). Let $M=(M_t:0\le t\le …

Consider a probability space that support a standard Brownian motion $W=(W_t)$ and a random variable $Z$ that is independent of $W$. Denote by $\mathbb F^W=(\mathcal F^W_t)_t$ the natural filtration g …

As no answer or comment to Can we construct close martingales if their terminal marginal laws are close? we consider a simplified version (discrete-time) as below: Let $M=(M_k)_{0\le k\le n}$ be a rea …

Let $M=(M_t)_{0\le t\le 1}$ be a real-valued continuous martingale. Let $\mu := {\rm Law}(M_1)$ and $\varepsilon \in (0,1)$. For any $\nu$ satisfying $W_2(\mu,\nu)\le \varepsilon$, can we construct tw …

Consider for $i=1,\ldots, N\ge2$ $$X^i_t=x_i+W^i_t,\quad \forall t\ge 0,$$ where $x_1,\ldots, x_N\in (0,\infty)$ and $W^1,\ldots, W^N$ are independent Brownian motions. Denote by $\tau_i$ the first hi …

Consider the SDE (stochastic differential equation) as follows: $$dX_t=X_t\big(b(X_t)dt+a(X_t)dW_t\big)$$ where $b,a:\mathbb R\to\mathbb R$ are Lipschitz and bounded and $W$ is a real-valued Brownian …

Consider two SDEs (stochastic differential equations) as follows: $$dX_t=b^-(t,X_t) \, dt+a(t,X_t) \, dW_t;\quad dY_t = b^+(t,Y_t)\,dt+a(t,Y_t)\,dW_t,$$ where $b^-,b^+,a$ are Lipschitz such that $b^-< …

Let $X$ be the solution to some stochastic differential equation $$dX_t =b(X_t) \, dt+a(X_t) \, dW_t,\quad \forall t>0.$$ Here $b,a: \mathbb R^d \to\mathbb R^d$ are bounded and Lipschitz and $W$ denot …

Consider the delayed stochastic differential equation as below: $$dX_t^\theta=X_{(t-\theta)^+}^\theta(1-X_{(t-\theta)^+}^\theta)(dt+dW_t),\quad \forall t>0$$ $$dY_t^\theta=Y_{(t-\theta)^+}^\theta(1-Y_ …

Let $\Omega\subset \mathbb R^d$ be a bounded open (and connected) set. Consider $E\subset \Omega$ and $x\in \Omega\setminus E$. Denote by $W^x$ the standard Brownian motion starting at $x$, i.e. $W^x_ …