Intereting Posts

To Find The Exponential Of a Matrix
Proving that an ideal in a PID is maximal if and only if it is generated by an irreducible
$xT=x$ iff $x=e$ implies that every $g$ may be written as $x^{-1}(xT)$ for some $x\in G$.
Is zero positive or negative?
Polynomials that pass through a lot of primes
Showing $\sup \{ \sin n \mid n\in \mathbb N \} =1$
Values of square roots
Uniqueness of memoryless property
Measure of an elementary set in terms of cardinality
I have to show $(1+\frac1n)^n$ is monotonically increasing sequence
Entailment relations that are not partial orders
What is the sum of the prime numbers up to a prime number $n$?
How to solve $x^3 + 2x + 2 \equiv 0 \pmod{25}$?
Probability distribution and their related distributions
Evaluating the integral, $\int_{0}^{\infty} \ln\left(1 – e^{-x}\right) \,\mathrm dx $

I need a little help with a problem. I am given some stochastic processes and supposed to show that they are $\mathcal{F_t}-$martingales. The first one is this, and they all look similar:

$$X_t=e^{t/2} \cdot \cos{W_t}.$$

Is this different from showing that it is a martingale? And how can and shall I use Itô’s formula here:

- Strong solutions SDE inequality with an application of Gronwall's inequality
- expectation of $\int_0^t W_s^2 dW_s $ (integral of square of brownian wrt to brownian)
- Quadratic Variation of Brownian Motion
- Proving that a process is a Brownian motion
- Law of large numbers for Brownian Motion (Direct proof using L2-convergence)
- Product rule with stochastic differentials

For some $f(t,x)$, $$f(t,W_t)=f(0,0)+\int_0^t\frac{\partial f}{\partial x}(s,W_s)dW_s+\int_0^{t}\frac{\partial f}{\partial t}(s,W_s)ds + \frac{1}{2}\int_0^t\frac{\partial^2f}{\partial x^2}(s,W_s)ds.$$

Thanks for your help!

Marie :*

- Brownian motion martingale
- Rigorous Book on Stochastic Calculus
- What is “white noise” and how is it related to the Brownian motion?
- How to show stochastic differential equation is given by an equation
- Bounded (from below) continuous local martingale is a supermartingale
- Proving Galmarino's Test
- On “for all” in if and only if statements in probability theory and stochastic calculus
- Characterization of hitting time's law. (Proof check)
- Simple formula on $X_n^{(k)} = \sum_{1 \le i_1 < … < i_k \le n} Y_{i_1} \cdot \dots \cdot Y_{i_k}$ (to show $X_n^{(k)}$ is martingale)
- What is the importance of the infinitesimal generator of Brownian motion?

Let’s assume that $\mathcal{F}_t$ is the natural filtration of the Brownian motion.

Notice that, for $s<t$

$$ \begin{eqnarray}

\mathbb{E}\left(X_t|\mathcal{F}_s\right) &=& \mathbb{E}\left(\left( \mathrm{e}^{t/2} \cos(B_s + (B_t-B_s))\right)| \mathcal{F}_s \right) \\&=& \mathrm{e}^{t/2} \mathbb{E}\left( \cos(B_s) \cos(B_{t-s}) – \sin(B_s) \sin(B_{t-s}) |\mathcal{F}_s\right) \\

&\stackrel{\color\maroon{\text{independence}}}{=}& \mathrm{e}^{t/2} \left( \cos(B_s) \mathbb{E}(\cos(B_{t-s})) – \sin(B_s) \underbrace{\mathbb{E}(\sin(B_{t-s}))}_{\text{zero}} \right) \\

&=& \mathrm{e}^{t/2} \cos(B_s) \mathbb{E}(\cos(B_{t-s})) = \mathrm{e}^{t/2} \cos(B_s) \mathrm{e}^{(s-t)/2} = X_s

\end{eqnarray}

$$

Expectation of the cosine function is easiest to extract from the characteristic function of the normal distribution:

$$

\mathbb{E}(\cos(a B_t)) = \Re\left(\mathbb{E}\left(\mathrm{e}^{i a B_t}\right)\right) = \Re\left( \mathrm{e}^{-a^2 t/2}\right) = \mathrm{e}^{-a^2 t/2}

$$

In order to establish whether $X_t$ is a square integrable martingale, we need to check that $\sup\limits_{t > 0} \left(\mathbb{E}(X_t^2)\right) < +\infty$.

$$

\mathbb{E}(X_t^2) = \mathrm{e}^t \mathbb{E}\left( \cos^2 B_t\right) =\frac{\mathrm{e}^t}{2} \mathbb{E}\left( 1+ \cos(2 B_t)\right) = \frac{\mathrm{e}^t}{2} \left( 1 + \mathrm{e}^{-2 t} \right) = \cosh(t)

$$

Since $\cosh(t)$ is unbounded, $X_t$ is not a square-integrable martingale for $t>0$, but it is a martingale, since $\mathbb{E}(|X_t|) < \exp(t) < \infty$ for all $0 \leqslant t < \infty$.

You could have used the Ito’s lemma to establish that $X_t$ is a semi-martingale, i.e. the drift coefficient is exactly zero.

- $G$ is $p$-supersoluble iff $|G : M | = p$ for each maximal subgroup $M$ in $G$ with $p | |G : M |$.
- Can a tetrahedron lying completely inside another tetrahedron have a larger sum of edge lengths?
- non homogeneous recurrence relation
- limsup and liminf of sequence of intervals in $\mathbb R$ (2)
- Why is there always a Householder transformation that maps one specific vector to another?
- Prove that $1 + \cos\alpha + \cos\beta + \cos\gamma = 0$
- Ring of $p$-adic integers $\mathbb Z_p$
- How many arrangements of a bookshelf exist where certain books must be to the left/right of other books?
- Why does Group Theory not come in here?
- On commutative unital graded rings in which no element in any homogenous part has a zero divisor
- Define a graph with segments or boundaries
- Cyclic group of prime order
- $p \leqslant q \leqslant r$. If $f \in L^p$ and $f \in L^r$ then $ f \in L^q$?
- Showing $R$ is a local ring if and only if all elements of $R$ that are not units form an ideal
- Asymptotics of sum of binomials