Intereting Posts

Can you explain the “Axiom of choice” in simple terms?
Next step to take to reach the contradiction?
What's the Clifford algebra?
Infinite algebraic extension of a finite field
Projective Noether normalization?
What philosophical consequence of Goedel's incompleteness theorems?
Borel Measures: Atoms vs. Point Masses
Classsifying 1- and 2- dimensional Algebras, up to Isomorphism
How to learn from proofs?
Quadratics and divisibility
Is there limit $ \lim_{(x,y) \to (0,0)} \frac{x^3}{x^2 + y^2}$?
Line integral over ellipse in first quadrant
The probability that each delegate sits next to at least one delegate from another country
Examples of statements which are true but not provable
Solve $ x^2+4=y^d$ in integers with $d\ge 3$

On the first page of Ustunel’s lecture notes, he defines the Wiener measure in the following way:

Let $W = C_0([0,1]), \omega \in W, t\in [0,1]$, define $W_t(\omega) = \omega(t)$. If we denote by $\mathcal{B}_t = \sigma\{W_s; s\leq t\}$, then there is one and only one measure $\mu$ on $W$ such that

1) $\mu \{W_0(\omega) = 0\} = 1$

- Independence of increments of some processes
- How to show the following process is a local martingale but not a martingale?
- Extension of Dynkin's formula, conclude that process is a martingale.
- (Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$
- Strong solutions SDE inequality with an application of Gronwall's inequality
- Solving SDE: $dX(t) = udt + \sigma X(t)dB(t)$

2) $\forall f \in C_b^{\infty}$, the stochastic process

$$(t, \omega ) \mapsto f(W_t(\omega)) – \dfrac{1}{2}\int_0^tf”(W_s(\omega))ds$$

is a $(\mathcal{B}_t, \mu)$-martingale. $\mu$ is called the Wiener measure

I am more familiar with the definition which supposes that we have already a Brownian motion $B_t$ available and then define

$$\nu\left(\{\omega: \omega_{t_1} \in A_1, \cdots, \omega_{t_n} \in A_ n\}\right) = P(B_{t_1} \in A_1, \cdots, B_{t_n} \in A_ n)$$

My question is why $\mu$ and $\nu$ are the same? Of couree if we begin with $\nu$ and use Its’s formula, we can see the two conditions defning $\mu$ are verified. But if we begin with the definition of $\mu$, how can we verify the condition defining $\nu$?

In addition, in Ustunel’s notes, he first presented his definition of Wiener measure then introduced stochastic integral. So I am wondering if there is a way to begin with the definition of $\mu$, then to show $\mu$ satisfies the condition defining $\nu$ without using stochastic integral.

Of course I will still appreciate it if you help me show $\mu \implies \nu$ using stochastic integral.

Thank you!

- What is the importance of the infinitesimal generator of Brownian motion?
- Construction of Brownian Motion using Haar wavelets
- Martingale formulation of Bellman's Optimality Principle
- Why did my friend lose all his money?
- Martingale not uniformly integrable
- A Brownian motion $B$ that is discontinuous at an independent, uniformly distributed random variable $U(0,1)$
- Integral of Brownian motion is Gaussian?
- How to get closed form solutions to stopped martingale problems?
- Proving Galmarino's Test
- beginner's question about Brownian motion

For fixed $x \in \mathbb{R}$, we choose $f(x) := e^{\imath \, x \xi}$. By assumption,

$$(t,\omega) \mapsto e^{\imath \, \xi W_t(\omega)} – \frac{\xi^2}{2} \int_0^t e^{\imath \, \xi W_r(\omega)} \, dr$$

is a martingale, i.e.

$$\mathbb{E}\left( e^{\imath \, \xi W_t} + \frac{\xi^2}{2} \int_0^t e^{\imath \, \xi W_r} \, dr \mid \mathcal{B}_s \right) = e^{\imath \, \xi W_s}+ \frac{\xi^2}{2} \int_0^s e^{\imath \, \xi W_r} \, dr.$$

for any $s \leq t$. Multiplying both sides with $e^{-\imath \, \xi W_s}$ yields

$$\mathbb{E}(e^{\imath \, \xi (W_t-W_s)} \mid \mathcal{B}_s) = 1 – \frac{\xi^2}{2} \mathbb{E} \left( \int_s^t e^{\imath \, \xi (W_r-W_s)} \, dr \mid \mathcal{B}_s \right).$$

By Fubini’s theorem, we can interchange the conditional expectation and integration:

$$\mathbb{E}(e^{\imath \, \xi (W_t-W_s)} \mid \mathcal{B}_s) = 1 – \frac{\xi^2}{2}\int_s^t \mathbb{E}(e^{\imath \, \xi (W_r-W_s)} \mid \mathcal{B}_s) \, dr.$$

This shows that $\varphi(t) := \mathbb{E}(e^{\imath \, \xi (W_t-W_s)} \mid \mathcal{B}_s)$ is a solution to the ordinary differential equation (ODE) $$\varphi'(t) = – \frac{\xi^2}{2} \varphi(t) \qquad \varphi(s)=1.$$

Obviously, the (unique) solution to this ODE is $$\varphi(t) = \exp \left( – (t-s) \frac{\xi^2}{2} \right).$$ In particular, we find that

$$\mathbb{E}(e^{\imath \, \xi (W_t-W_s)} \mid \mathcal{B}_s) = \mathbb{E}e^{\imath \, \xi (W_t-W_s)} = \exp \left(- (t-s) \frac{\xi^2}{2} \right).$$

We conclude that:

- $W_t-W_s$ is Gaussian with mean $0$ and variance $t-s$.
- $\mathcal{B}_s$ is independent from $W_t-W_s$. This implies that $(W_t)_{t \geq 0}$ has independent increments. In particular, $(W_t)_{t \geq 0}$ is a Markov process. Using the Markov property, one can easily obtain the finite-dimensional distributions from the distributions of $W_t$ for each $t \geq 0$.

**Remarks:**

- The proof shows that it suffices to have the martingale property for functions $f$ of the form $f(x) = e^{\imath \, x \xi}$, $\xi \in \mathbb{R}$.
- The definition of $\mu$ is chosen such that $(W_t)_{t \geq 0}$ is the unique solution to the martingale problem $$f(X_t)- \int_0^t Af(X_s) \, ds$$ for the (Laplace) Operator $Af := \frac{1}{2} f”$. Invoking certain theorems from this area, also yields the claim (but is overshoot in this particular case).
- As @LiuGang already mentioned in a comment, Lévy’s characterization of Brownian motion can also be used to prove the claim. To this end, we have to overcome some technical issues since $f(x) := x$ and $f(x) := x^2$ are not bounded.

- What restricts the number of cohomologies?
- Does a closed and bounded set in $\mathbb{R}$ necessarily contain its supremum and infimum?
- Asymptotic expansion for Fresnel Integrals
- Every subspace of the dual of a finite-dimensional vector space is an annihilator
- A sequence of real numbers such that $\lim_{n\to+\infty}|x_n-x_{n+1}|=0$ but it is not Cauchy
- When can we can conclude that if a function vanishes on a non-degenerate interval, then that function must vanish everywhere?
- If the quotient group $G/N$ be isomorphic to $G_2$, is $G$ isomorphic to $G_1\times G_2$ where $N$ is isomorphic to $G_1$?
- An issue with the substitution $u=\sin x$
- Two questions regarding Ordinal Numbers.
- Textbook suggestion for studying martingales
- What is the simplest lower bound on prime counting functions proof wise?
- Intuition, proof, one-sided group definition – Any set with Associativity, Left Identity, Left Inverse is a Group – Fraleigh p.49 4.38
- If $f$ continuous and $\lim_{x\to-\infty }f(x)=\lim_{x\to\infty }f(x)=+\infty $ then $f$ takes its minimum.
- Bijection between ideals of $R/I$ and ideals containing $I$
- On the properties of the Sobolev Spaces $H^s$