7
votes
Accepted
Uniformly integrable local martingale
It is not the case that all uniformly integrable local martingales are true martingales. In fact, it is not even true that $L^2$-bounded local martingales must be true martingales. Since a family of $...
- 17.5k
6
votes
Relative entropy for martingale measures
$M_e$ contains $P$, because $P\in M_a$ and $\frac{dP}{dP}=1$ and therefore $H[P|P]=0.$
I will show that $M_e$ contains more than just $P$ in the case where $S$ is a Brownian motion.
In that case, ...
- 685
6
votes
Accepted
Show local martingale
Solution 1: Recall the following two statements.
Lemma 1: Let $(Y_t)_{t \geq 0}$ be a supermartingale and $f$ an increasing concave function, then $(f(Y_t))_{t \geq 0}$ is a supermartingale.
Lemma 2: ...
- 113k
5
votes
Continuous Local Martingale and Quadratic Variation
For the first inclusion note that if $\lim_{t \to \infty} X_t(\omega)$ exists, then it follows from the continuity of the sample paths that $t \mapsto X_t(\omega)$ is bounded. This, in turn, implies $\...
- 113k
5
votes
Accepted
Potential Local Martingale property derived from its quadratic variation
"$ M $ may not be $ o(t) $", the following is an example. Let $ B=\{B_t,t\ge 0\} $ be the Brownian motion with $\langle B\rangle_t =t $ and
\begin{equation*}
M_t=B_{t^2}, \qquad t\ge 0.
\...
- 3,745
5
votes
Probability of stopping time being finite.
Using the hint: $$\mathbb{E}(X_{t\wedge\tau_a})=a\mathbb{P}(\tau_a\leq t)+\mathbb{E}(X_t1_{\tau_a>t}).$$
Now $0\leq X_t1_{\tau_a>t}\leq X_t\to0$ almost surely as $t\to\infty$. Additionally $...
- 1,001
4
votes
Accepted
About local martingales..
To answer your first question, the result was published in 1972 by P.A. Meyer in the "Lecture Notes in Mathematics Series". The title is Martingales and Stochastic Integrals I, and the result is ...
- 1,432
4
votes
Accepted
Stopped local martingale as a martingale
Let $(\sigma_n)_{n \in \mathbb{N}}$ be a localizing sequence of the local martingale $M$, i.e. an increasing sequence of stopping times such that $\sigma_n \to \infty$ and $(M_{t \wedge \sigma_n})_{t \...
- 113k
4
votes
Accepted
Bound on supremum of local martingale
Define a stopping time $\tau$ by $$\tau := \inf\{t>0; \langle M \rangle_t>b\}.$$ Since
\begin{align*} \mathbb{P} \left( \sup_{s \leq t} M_s>a, \langle M \rangle_t \leq b \right) &= \...
- 113k
4
votes
Accepted
An example to be a local martingale but not a martingale
Take some random variable $Y$ which is not integrable (e.g. Cauchy distributed) and which is independent of $(W_t)_{t \geq 0}$. Define $\varphi_s(\omega):=Y(\omega)$, then
$$\int_0^t \varphi_s \, dW_s ...
- 113k
4
votes
Accepted
Inequality on expectation of exponential martingale
Be careful. Uniform integrability is not sufficient to apply optional stopping theorem to local martingales. Counter-example: if $R$ is a Bessel process of dimension 3 starting at 1, the local ...
- 3,375
3
votes
Exponential martingales and changes of measure
You are right to be suspicious. The recipe you describe specifies a finitely additive set function $Q$ on $\cup_t\mathcal F_t$ such that $Q|_{\mathcal F_t} = M_t\cdot P|_{\mathcal F_t}$ for each $t\ge ...
- 22.9k
3
votes
Uniformly $L^p$ continuous local martingale
As the following example shows, $(X_t)_{t \geq 0}$ is, in general, not uniformly integrable:
Let $(X_t)_{t \geq 0}$ be a one-dimensional Brownian motion. Then
$$\mathbb{E}(|X_t| 1_{\{|X_t| \geq R\}}...
- 113k
3
votes
If $M$ is a local martingale and $τ:=\inf\left\{t\ge0:\left|M_t\right|\ge\varepsilon\right\}$, then $M^τ$ is a martingale
Let $s < t$. We have to show that $E[M_t^\tau \mid \mathcal{F}_s] = M_s^\tau$ almost surely.
For each $n$ we know, as you mentioned, that $(M^\tau)^{\sigma_n} = M^{\tau \wedge \sigma_n}$ is a ...
- 90.7k
3
votes
Accepted
Is there a process that is not square-integrable but still gives rise to a martingale when integrated w.r.t. Brownian motion?
Yes, that's possible.
Example Let $Y$ be a random variable which is independent from $(B_t)_{t \geq 0}$ and satisfies $$\mathbb{E}(|Y|) < \infty \quad \text{and} \quad \mathbb{E}(Y^2)=\infty$$ (e....
- 113k
3
votes
Accepted
If $\mathbb{E}[\underset{0 \leq s \leq t}{\sup} \rvert X_s \rvert] < \infty$ for $X_t$ a local martingale, is $X_t$ a proper martingale?
$X_t$ is indeed a martingale. We don't need anything as complicated as Burkholder-Davis-Gundy to see this. Instead, let's go back to the definitions. Let $\tau_n$ be a sequence of stopping times such ...
- 17.5k
3
votes
Accepted
Local martingale property for stochastic integrals.
This is not exactly an elementary proof. Assuming that you are well versed in the properties of the Ito integral for the class of integrands $\{f: E[\int_a^b f^2(s)\, ds] < \infty \}, $ and know ...
- 1,134
3
votes
Accepted
Quotient of continuous local martingale with quadratic variation
I believe that you need the further condition that $\langle M,M \rangle_\infty = \infty$ a.s., otherwise you can take $M$ to be something like Brownian motion stopped at $t=1$. Then you can use the ...
- 9,057
3
votes
Accepted
Interpretation of Gisarnovs theorem
Since we also have that $[M,X]=[\hat{M},X]$ are the same for all $X$ both under $\mathbb{P}$ and $\mathbb{Q}$ we know that $M$ and $\hat{M}$ are the same up to a constant. Therefore $X=\hat{M}+B$ is ...
- 1,715
3
votes
Accepted
local martingale remains local martingale when lowering the localizing stopping times
This statement is correct and your proof works. We don't need $S_k \le T_k$ to say that $(X^{T_k})^S_k$ is a martingale. A slightly more general statement is that if $(S_k) \rightarrow \infty$ is a ...
- 9,057
3
votes
Accepted
The expected squared increment of a continuous local martingale
While it certain isn't true that $M_{t_{i+1}}^2 - M_{t_i}^2 = (M_{t_{i+1}} - M_{t_i})^2$, their expectations are the same. To see this, first assume that $M$ is a martingale, and use the tower ...
- 10.3k
3
votes
Accepted
Show that a local martingale is a true martingale if and only if it is a process of class DL
Suppose that the $M$ is a local martingale with respect to the filtration $\{ {\mathcal F}_s\}$ and the stopping times $S_n \uparrow \infty$. Then for $s<t$, we have
$$\forall n, \quad E[M_{t \...
- 16k
2
votes
Accepted
Question regarding local martingales.
In answer to the question in your comment on my comment: Let $X$ be a martingale, $T$ a stopping time, and $Y_t=X_{t\wedge T}$ the stopped process. By Doob, for $0\le s<t$, $E[Y_t\,|\,\mathcal F_{s\...
- 22.9k
2
votes
Accepted
continuous local martingale brownian motion
Ito's lemma is one way to prove it. Using the definition of $f_{1-t}$, we have
$$X_t = f_{1-t}(B_t) = \frac{1}{\sqrt{2\pi (1-t)}} \exp\left\{\frac{-B_t^2}{2(1-t)}\right \}:=F(t,B_t),$$
and according ...
- 1,350
2
votes
Is an $L^2$-bounded continuous local martingale already a strict martingale?
It needn't be the case that an $L^2$-bounded continuous local martingale is a true martingale. Below I outline an example that can be found in "Diffusions, Markov Processes and Martingales: Volume 2" ...
- 17.5k
2
votes
Is a local continuous martingale square integrable.
If you are asking whether a continuous local martingale is square integrable process the answer is NO, as the counterexample of @Jochen demonstrated.
But if you are asking whether a continuous local ...
- 2,268
2
votes
Accepted
Show that $X \in \mathcal{M}^{loc}_c$ implies: $X$ is locally square summable and locally bounded.
Hints: Since $X$ is a local martingale, there exists a sequence of stopping times $(\tau_k)_{k \in \mathbb{N}}$ such that $\tau_k \uparrow \infty$ and $(X_{t \wedge \tau_k})_{t \geq 0}$ is a ...
- 113k
2
votes
Accepted
Show that $M-M_0$ is an $L^2$ bounded martingale if $E([M]_\infty)<\infty$.
WLOG $M_0=0$ (otherwise replace $M_t$ by $M_t-M_0$). By the very definition of the quadratic variation,
$$N_t := M_{t \wedge T_n}^2-[M]_{t \wedge T_n}$$
is a martingale. In particular,
$$\mathbb{E}(...
- 113k
2
votes
Accepted
Showing $E(X_T ^2)\le E([X]_T)$ for bounded stopping times $T$
Hints: Let $T$ be a bounded stopping time.
Fix a localizing sequence $(\sigma_n)_{n \in \mathbb{N}}$ of stopping times. Sinc $(X_t)_{t \geq 0}$ has continuous sample paths, we can choose the sequence ...
- 113k
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