3
votes
1answer
74 views

Given the SDE: $dX_t=dB_t+b(X_t) dt$ with $(x,b(x)) \leq 0, \forall x \in \mathbb{R}^n$, prove that $E[|X_t|^2] \leq nt+E[|X_0|^2]$

I'm working on this problem: Given a solution $X_t$ to the SDE $$dX_t=dB_t+b(X_t) dt$$ where $B_t$ is an $n$-dimensional Brownian motion, and $b:\mathbb{R}^n \to \mathbb{R}^n$ a Lipschitz ...
0
votes
1answer
36 views

Is it true that $X(t)^a > K \iff X(t) > K^\frac1a$

Let $a \in \mathbb{N}$, $K \in \mathbb{R^+}$ and $X(t)$ be a geometric Brownian Motion. Is the following true? $$X(t)^a > K \iff X(t) > K^\frac1a$$ The context of the above is that I want to ...