Consider the equation (d X_{t}=mathbb{1}_{left{X_{t} geq 0 ight}} d B_{t}). Prove (in a direct way) that this

Consider the equation \(d X_{t}=\mathbb{1}_{\left\{X_{t} \geq 0\right\}} d B_{t}\). Prove (in a direct way) that this equation has no solution starting from 0 . Prove that the equation \(d X_{t}=\mathbb{1}_{\left\{X_{t}>0\right\}} d B_{t}\) has a solution.