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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 $W$ be a standard Brownian motion starting at $1/2$, i.e. $W_0=1/2$. Set $$\tau := \inf\big\{t>0: W_t\notin (0,1)\big\}.$$ As $(W_t^2-t)_t$ is a martingale, one has $\mathbb P[W_\tau =0]=1/2 = \ma …

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 …

My questions are as follows : Can we construct a sequence of continuous martingales $M^n=(M^n_t:0\le t\le T)$ (that can be defined in different probability spaces) such that ${\rm Law}(M^n)\to {\rm Law …

Let $W$ be a Brownian motion and $\alpha$ be a progressively measurable process taking values in $\mathbb R_+$. Set $\beta_t:=\max(\alpha_t, 1)$ for all $t\ge 0$. Define respectively $X$, $Y$ by $$X_t …

For any $\nu$ satisfying $W_2(\mu,\nu)\le \varepsilon$, can we construct two continuous martingales $X, Y$ (on suitable probability space) such that $${\rm Law}(X)={\rm Law}(M),\quad {\rm Law}(Y_1)=\nu …

Let $W$ be a one-dimensional Brownian motion. Consider the stochastic differential equation (SDE) $$dX_t = C(t)(1-X_t)dW_t,\quad \forall t\ge 0,$$ where $C$ is a continuous and bounded function. Under …

As no answer or comment to Can we construct close martingales if their terminal marginal laws are close? … For any $\nu$ satisfying $W_2(\mu,\nu)\le \varepsilon$, can we construct two discrete-time martingales $X, Y$ (on suitable probability space) such that $${\rm Law}(X)={\rm Law}(M),\quad {\rm Law}(Y_n)= …

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 …

Let $\mu,\nu$ be probability measures on $\mathbb R$ such that $$\int_{\mathbb R}|x| \mu(d x) + \int_{\mathbb R}|x| \nu(d x)<\infty.$$ $\mu\le_{cd} \nu$ is said to hold if $$\int_{\mathbb R}(x-y)^+\mu …

Let $W$ be a Brownian motion and $\alpha$, $\beta$ be two progressively measurable processes taking values in $\mathbb R_+$ s.t. $\alpha_t\le \beta_t$ for all $t\ge 0$. Define respectively $X$, $Y$ by …

Let $\mu_0,\mu_1$ be probability measures on $\mathbb R$ that are of finite second moment and increasing in convex order, i.e. $$\int_\mathbb R f(x)\mu_0(dx) \le \int_\mathbb R f(x)\mu_1(dx)$$ holds f …

The answer is non due to the very simple example. Put $$\mu=\delta_0 \quad\mbox{and}\quad \nu=p\delta_{1}+(1-p)\delta_{-1}$$ where $p\in (0,1)$. A straightforward verification yields $\mu\le_{cd}\nu$ …

I look for a full characterisation of the continuous martingales $X=(X_t)_{0\leq t\leq T}$ (defined on some filtered probability space as nice as possible) such that $$X_0=0\quad \mbox{ and } \quad\mathbb … Does this contraction contains all desired continuous martingales? …