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Consider a parametrised martingale as follows : $$X^x_t := x+ \int_0^t\sqrt{2p_s} \, dW_s,$$ where $W$ is a standard Brownian motion and $(p_t)_{t\ge 0}$ is a locally square integrable process satisfy …

Let $W=(W_t)_{t\ge 0}$ be a standard Brownian motion starting from zero and $Z>0$ be an independent random variable. Fix $T>0$ and $C>0$. Denote by $\mathcal A$ the set of progressively measurable pro …

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_ …

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 $X=(X_t:t\ge 0)$ be a stochastic process (martingale in general) starting at $X_0=0$. For $T>0$ and $a<b$, let $U^T_{a,b}(X)$ be the number of upcrossings of $X$ across the interval $[a,b]$ over $ …

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 $(\Omega,\mathcal F, \mathbb P)$ be a probability space on which a Brownian motion $W$ is defined, and $\mathcal U$ be the set of progressively measurable (w.r.t. the Brownian filtration) processe …

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_ …

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 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 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^-< …

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 …

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 …

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_{ …