Intereting Posts

Limit of $x \log x$ as $x$ tends to $0$
Characterization of an invertible module
Prove that $\det(xI_m-AB)=x^{m-n}\det(xI_n-BA)$
Trying to prove that a group is not Cyclic
Is $W_0^{1,p}(\Omega)\cap L^\infty(\Omega)$ complete?
a follow up question about modeling with exponential distributions
Alpha and Omega limit sets (dynamical systems)
Is projection of a measurable subset in product $\sigma$-algebra onto a component space measurable?
Can you give me some concrete examples of magmas?
Confidence interval for estimating probability of a biased coin
Beginner: How to complete the induction case in a proof that all multiples are a product of the least common multiple
Why sum of two little “o” notation is equal little “o” notation from sum?
Infinite direct product of fields.
Summary: Spectrum vs. Numerical Range
From $e^n$ to $e^x$

I have a question concerning the definiton of Brownian motion. Usually (e.g. on Wikipdia) one demands a brownian motion $\lbrace B_t\rbrace_{t\in[0,\infty)}$ to satisfy the following condition:

$\forall 0\leq t_0<t_1<…<t_m: B_{t_1}-B_{t_0},…,B_{t_m}-B_{t_{m-1}}$ are independent random variables.

But today I come across with formally another definition of Brownian motion! Instead of the above condition the book demands the following property:

- Extension of Dynkin's formula, conclude that process is a martingale.
- To confirm the Novikov's condition
- Probability Brownian Motion - dependence
- Density of first hitting time of Brownian motion with drift
- process with integral is martingale
- What is the importance of the infinitesimal generator of Brownian motion?

$\forall t,h\geq 0$, $B_{t+h}-B_t$ is independent of $\lbrace B_u : 0\leq u\leq t \rbrace$.

Now I’m wondering whether both properties are indeed equivalent or not?! Hopefully someone can help me.

- Understanding the Fokker-Planck equation for non-stationary processes
- Nice references on Markov chains/processes?
- Poisson arrivals followed by locking
- Is this local martingale a true martingale?
- Expectation of Stopping Time w.r.t a Brownian Motion
- Thinning a Renewal Process - Poisson Generalization
- Why do people all the time exploiting almost sure properties of a stochastic process as if they were sure properties?
- A book/text in Stochastic Differential Equations
- Stochastic Processes Solution manuals.
- Prove $\mathbb{P}(\sup_{t \geq 0} M_t > x \mid \mathcal{F}_0)= 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$

Yes, they are indeed equivalent.

**Proof:** Let $X_1,\ldots,X_n$ arbitrary random variables and denote by

$$S_j := \sum_{k=1}^j X_k$$

the $j$-th partial sum. Since $$X_j = S_j – S_{j-1} \qquad \quad S_j = \sum_{k=1}^j X_k$$

it is not difficult to see that

$$\sigma(X_1,\ldots,X_n) = \sigma(S_1,\ldots,S_n).$$

If we set $X_j := B_{t_j}-B_{t_{j-1}}$ for a Brownian motion $(B_t)_{t \geq 0}$, then this shows

$$\sigma(B_{t_1},\ldots,B_{t_n}) = \sigma(B_{t_1}-B_{t_0},\ldots,B_{t_n}-B_{t_{n-1}}). \tag{1}$$

Consequently,

$$\begin{align*} \sigma\{B_u; u \leq t\} &\stackrel{\text{Def}}{=} \sigma \left( \bigcup_{\substack{t_1<\ldots<t_n \leq t \\n \in \mathbb{N}}} \sigma(B_{t_1},\ldots,B_{t_n}) \right) \\ &\stackrel{(1)}{=} \sigma \bigg( \underbrace{\bigcup_{\substack{t_1<\ldots<t_n \leq t \\n \in \mathbb{N}}} \sigma(B_{t_1}-B_{t_0},\ldots,B_{t_n}-B_{t_{n-1}})}_{=:\mathcal{G}} \bigg). \end{align*}$$

Since $\mathcal{G}$ is a $\cap$-stable generator of $\sigma(B_u; u \leq t)$, we conclude that $B_{t+u}-B_t$ is independent from $\sigma(B_u; u \leq t)$ if, and only if, $B_{t+h} -B_t$ is independent from $\mathcal{G}$. And this is equivalent to $B_{t+h}-B_t$ being independent from $B_{t_n}-B_{t_{n-1}},\ldots,B_{t_1}-B_{t_0}$ for all $t_1 < \ldots < t_n \leq t$.

**Literature:** See e.g. *René L. Schilling/Lothar Partzsch: Brownian Motion – An Introduction to Stochastic Processes* (Lemma 2.14).

- How to evaluate $\lim_{ n\to \infty }\frac{a_n}{2^{n-1}}$, if $a_0=0$ and $a_{n+1}=a_n+\sqrt{a_n^2+1}$?
- The cube of any number not a multiple of $7$, will equal one more or one less than a multiple of $7$
- Method to reverse a Kronecker product
- Duality. Is this the correct Dual to this Primal L.P.?
- $ \Bbb Q = \Bbb Q $
- Find the value of $\sum_0^n \binom{n}{k} (-1)^k \frac{1}{k+1}$
- Matrix graph and irreducibility
- Number of fixed points of automorphism on Riemann Surface
- Prove there is no contraction mapping from compact metric space onto itself
- Locally non-enumerable dense subsets of R
- Proximal mapping of $f(U) = -\log \det(U)$
- Is there an equation to describe regular polygons?
- A continuous function and a closed graph
- How to Prove the divisibility rule for $3$
- Why can ALL quadratic equations be solved by the quadratic formula?