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I look for references on the existence/uniqueness of the solution to SDE $$dX_t = b(t,X_t)dt + a(t,X_t)dW_t,\quad \forall t\ge 0,$$ where $b :\mathbb R_+\times\mathbb R\to\mathbb R$, $a :\mathbb R_+\t …

Let $b:\mathbb R\to [-1,1]$ and $a_1, a_2:\mathbb R\to [1,2]$ be Lipschitz functions. Consider the stochastic differential equation (SDE) as follows : $$dX_t = b(X_t)dt + a(X_t)dW_t,$$ where $(W_t)_t$ …

Consider a diffusion process $$X_t=X_0+\int_0^t {\bf 1}_{\{X_s>0\}}b(s,X_s)ds+ \int_0^t {\bf 1}_{\{X_s>0\}} a(s,X_s)dW_s,\quad \forall t\ge 0,$$ where $a: \mathbb R_+\times \mathbb R\to [1,2]$ and $b: …

Consider the following SDE driven by real-valued Brownian motion $W=(W_t)_{t\ge 0}$: $$dX_t = \left(\sigma {\bf 1}_{\{X_t>1\}} + \sigma' {\bf 1}_{\{0<X_t\le 1\}}\right)dW_t,\quad \forall t>0,$$ where …

Let $X^n$ and $X$ be stochastic processes defined by $$X^n_t=1+\int_0^tb_n(s)ds+\int_0^t\sigma_n(s)dW_s \quad\mbox{and}\quad X_t=1+\int_0^tb(s)ds+\int_0^t\sigma(s)dW_s,$$ where $b_n, \sigma_n, b, \sig …

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 …

I look for examples of SDEs (stochastic differential equations) s.t. the uniqueness of the solution fails, i.e. $$dX_t = B(t,X_t)dt + \Sigma(t,X_t)dW_t,\quad \forall t\ge 0.$$ More precisely, the coef …

Consider a sequence of parametrized SDEs : $$X^{a}_t = z + \int_0^t b(a,s,X^a_s)ds+\int_0^t\frac{\sigma(a,s,X^a_s)}{1+{\bf 1}_{\{b(a,s,X^a_s)>0\}}}dW_s,\quad \forall t\ge 0,~~~~~~~~~~~~~~~~~~~~~(\ast) …

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 …

Let $X_t:=W_{t\wedge \tau}$ for $t\ge 0$, where $(W_t)_{t\ge 0}$ is a standard Brownian motion and $\tau:=\inf\{t\ge 0: |W_t|=1\}$. It holds $$X_t=\int_0^t {\bf 1}_{\{|X_s|<1\}}dW_s,\quad \forall t\ge …

Consider $$X_t=X_0 + \int_0^t b(s)ds+ \int_0^t \sigma(s)dW_s,\quad \forall t\ge 0,$$ where $X_0\ge 0$ is a random variable of density $\rho$, $(W_t)_{t\ge 0}$ is an independent Brownian motion and $b, …

Let $W$ be a standard Brownian motion, and $a,b:\mathbb R_+\times \mathbb R\to\mathbb R$ be Lipschitz. Consider the stochastic differential equations: $$X_t=1+\int_0^ta(s,X_s)ds + W_t,\quad\quad Y_t=1 …

Consider the PDE of $p(t,x)\ge 0$ given as $$\partial_t p = \frac{\partial^2_{xx}p}{(1+m(t))^2} - \partial_x p,\quad \forall t,~x \in (0,\infty)$$ with initial and boundary conditions $p(0,\cdot)=\rho …

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 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 …