Articles of brownian motion

Function of brownian motion is a martingale

Let $B_t,t\geq 0$ a brownian motion and $u(t,x)$ a function satisfying the following PDE $$\frac{\partial u}{\partial t}+\frac{1}{2}\frac{\partial^2 u}{\partial x^2}=0.$$ Then we prove that $\frac{\partial }{\partial t}\Bbb{E}(u(t,B_t))=0$, but I want to prove that $u(t,B_t)$ is a martingale (without using Itô formula). I think it suffices to show that $u(t,B_t)$ is a submartingale or supermartingale. Thanks.

Relation between $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$

Let $B_t$ be a standard Wiener motion. What can we say about $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$? Is there a relation?

Second order linear partial differential equation: $\partial_t u(t,x)+\frac12 \partial_{x,x} u(t,x)+u(t,x)v(x)=0$

Is there a way to solve $$ \partial_t u(t,x)+\frac12 \partial_{x,x}u(t,x)+u(t,x)v(x)=0? $$ This appeared as a condition for $$ X_t=u(t,B_t)e^{\int_0^tv(B_s)ds} $$ to be a martingale. With $B$ a standard Brownian motion.

quadratic variation of brownian motion doesn't converge almost surely

I just came across the following remark: If $(B_t)_{t\geq0}$ is a one dimensional Brownian motion and if we have a subdivison $0=t_0^n<…<t_{k_n}^n=t$ such that $\sup_{1\leq i\leq k_n}(t_i^n-t_{i-1}^n)$ converges to $0$ is $n$ converges to $\infty$ then $\lim_{n\to\infty}\sum_{i=1}^{k_n}(B_{t_i^n}-B_{t_{i-1}^n})^2=t$ in $L^2$. If $\sum_{n\geq 1}\sum_{i=1}^{k_n} (t_i^n-t_{i-1}^n)^2<\infty$, then we also have almost sure convergence Now I’m trying to find […]

Two Questions about Brownian Motion

How do you show $B_T\in\mathcal{F}_T$ for T is a stopping time? Note the filtration is generated by the Brownian motion (and not necessarily completed, in particular, $\mathcal{F}_T\neq\mathcal{F}_{T+}$) and a much harder question: Are all Brownian Motion stopping times previsible? (Please point me to a proof or reference)

What is the “distributional derivative” of a Brownian motion?

Let $\emptyset\ne I\subseteq\mathbb R$ be an open interval and $A:C_0^\infty(I)\to\mathbb R$ be a distribution. Then, $$\langle{\rm D}A,\varphi\rangle:=-\langle A,\varphi’\rangle\;\;\;\text{for }\varphi\in C_0^\infty(I)$$ is called distributional derivative of $A$. Let $B=(B_t)_{t\ge 0}$ be a Brownian motion. How is the distributional derivative of $B$ with respect to $t$ defined?

Expectation regarding Brownian Motion

This is a formula regarding getting expectation under the topic of Brownian Motion. \begin{align} E[W(s)W(t)] &= E[W(s)(W(t) – W(s)) + W(s)^2] \\ &= E[W (s)]E[W (t) – W (s)] + E[W(s)^2] \\ &= 0+s\\ &=\min(s,t) \end{align} How does $E[W (s)]E[W (t) – W (s)]$ turn into 0? Thanks alot!! Please let me know if you […]

Expectation of product of correlated Brownian motions at different time points

Given the information about the correlation of two Brownian motions as $E[dW_1 dW_2] = \rho dt$ and knowing that $E[W_1(t)W_1(t’)] = \min(t,t’)$, I want to compute $E[W_1(t)W_2(t’)]$ I interpret $dW_1 as \int^{t + \Delta t}_t dW_1(s)$ and I also know that $E[(dW_1)^2] = dt$ Can I follow from the correlation property that $E[W_1(t)W_2(t’)] = E[W_1(t) […]

Computing cross variation of independent brownian motions

I am familiar with computing the quadratic variation of Brownian motion, but was confused when the text I’m working through introduced cross variation of independent Brownian motions. the notation is as follows: $$\langle X,Y\rangle_t = \lim_{||\Delta||\to 0} \sum_i(X_{t_{i+1}}-X_{t_i})(Y_{t_{i+1}}-Y_{t_i}) $$ Where $X_t$ and $Y_t$ are independent Brownian motions and $\Delta$ is a partition of $[0,t]$. I […]

Show that this is a stopping time

Show that $\sigma=\inf \{ t\ge 0 : |B_t|= \log t \}$ is a stopping time with respect to $(\mathcal F_t^B)_{t\ge0}$. I’ve been trying to put the set $\{\sigma\le t\}$ equal to a countable union and then showing that this union belongs in $\mathcal F_t^B$. I am struggling to derive a countable union to this as […]