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Consider a sequence of martingales that are right-continuous with left limits, denoted by $(X^n_t)_{0\le t\le 1}$, such that for each $n\ge 2$, \begin{eqnarray} (1) && X^n_0=0 \mbox{ and } \sup_{0\le t …

This is a continuation of Characterization of martingale diffusions ending in $\{-1,1\}$ $X=(X_t)_{0\le t\le T}$ is said to be a martingle diffusion if $X_0=0$, $X_T\in\{-1,1\}$ and $$X_t=\int_0^t a(u …

Let $T\ge 1$ be some fixed integer. Consider a discrete-time martingale $(X_t)_{t=0,1,\ldots, T}$ or a continous-time martingale $(X_t)_{0\le t\le T}$ (the latter can be continuous or cadlag if it hel …

Let $\mathcal M$ be the collection of martingle diffusions starting at zero and ending in $\{-1,1\}$. Equivalently, $X\in \mathcal M$ iff there exists a measurable function $a$ s.t. it holds almost su …

Below is far from being an answer. I solved numerically the above maximization problems via Lagrangian multipliers, which yield the Markov kernal $(p_m(x_n,\cdot): -N\le n\le N)_{0\le m\le M-1}$. Plot …

Two probability distributions $\mu, \nu$ on $\mathbb R^d$ are said to be increasing in convex order if $$\int_{\mathbb R^d} |x|\mu(dx) + \int_{\mathbb R^d} |x|\nu(dx)<\infty$$ and $$\int_{\mathbb R^d} …

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 …

This question can be seen as a continuation of Lipschitz continuity of $\mathbb P[\tau>t]$ with respect to $t$ Consider the martingale given as $$X_t=1+\int_0^t a(s,X_s)dW_s,\quad \forall t\ge 0.$$ De …