For questions about local martingales (in continuous time).
Questions tagged [local-martingales]
207 questions
18
votes
2 answers
Relative entropy for martingale measures
I need some help understanding a note given in a lot of papers I've read.
Let $(\Omega,\mathcal{F},P)$ be a complete probability Space, $\mathbb{F} = (\mathcal{F}_t)_{t\in[0,T]}$ a given filtration with usual conditions, $S$ be a locally bounded…
Gono
- 5,788
9
votes
1 answer
Exit and hitting times for the Bessel process $\textrm{d}X_t=\frac{n-1}2\frac{\textrm{d}t}{X_t}+\textrm{d}B_t$
I am trying to analyse the exit time $T_1:=\inf\{t:X_t\notin[\alpha,2]\}$ and hitting time $T_2:=\inf\{t:X_t=0\}$, where $\alpha<1$ is a constant, and $X_t$ follows the Bessel process defined by the…
user107224
- 2,268
8
votes
1 answer
Uniformly integrable local martingale
Can someone give me an example of a uniformly integrable local martingale that is not a martingale? Or are all U.I. local martingales true martingales (continuous, of course).
Protawn
- 602
6
votes
0 answers
How to show this is not a martingale.
Be advised that I cross-posted this question on MathOverflow. You can find it in this link:
https://mathoverflow.net/questions/352152/show-that-this-process-is-not-a-martingale
Assume we have the following stochastic process:
$$X_t=\int_0^t…
Chaos
- 3,417
6
votes
1 answer
Proving martingale property of $N_t = Z(M_{t\wedge s} - M_{t \wedge r})$ for martingale $M$
(Stochastic calculus and Brownian motion, LeGall, page 80).
Suppose $M = (M_t)$ is a martingale. Also, let $Z$ be a bounded random variable which is $\mathcal{F}_r$ adapted. Then we like to show that for any $0 \leq r < s$,
$$N_t = Z(M_{t\wedge…
Nick
- 81
- 5
5
votes
1 answer
Probability of stopping time being finite.
Let $X$ be a continuous non-negative local martingale with $X_0=1$ and $X_t\to0$ almost surely as $t\to\infty$. For $a>1$, let $\tau_a=\inf\{t\geq0:X_t>a\}$.
I am tasked with showing that…
verygoodbloke
- 1,033
5
votes
1 answer
Difference Between Local Martingale and Martingale
I would like to try and distinguish between the two concepts of martingale and local martingale. I have read this answer in Martingale / local martingale : some confusion which was a good start and wanted to check the following idea:
If we have a…
user258521
- 1,059
5
votes
0 answers
Prove that a martingale with a spatial parameter is differentiable
Let
$(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
$(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$
$M:\Omega\times[0,\infty)\times\mathbb R^d\to\mathbb R$ such…
0xbadf00d
- 14,208
5
votes
1 answer
Show that $(S_t - M_t) \phi(S_t) = \Phi(S_t) - \int_{0}^{t} \phi(S_s) dM_s$
Show that $(S_t - M_t) \phi(S_t) = \Phi(S_t) - \int_{0}^{t} \phi(S_s) dM_s$
(This is from Le Gall's book, Brownian Motion, Martingales, and Stochastic Calculus.)
Here, $M$ is a continuous local martingale, $S_t = \sup_{0 \leq s \leq t} M_s$,…
cgmil
- 1,509
5
votes
1 answer
Stopped local martingale as a martingale
I am reading the Dubins-Schwarz theoem on Brownian motion and stochastic calculus. It says, given a continuous local martingale $M$ such that $\lim_{n\to\infty}[M]_t = \infty$ a.s., where $[M]_t$ denotes the quadratic variation of $M$. For each $s…
Kenneth Ng
- 389
5
votes
0 answers
stochastic exponential uniformly integrable martingale
$N$ is a continuous local martingale and $T_c:=\inf\left\{t>0:\left[N\right]_t>c\right\}$, $c>0$ .
I need to show that the stochastic exponential $\mathcal{E}(-N)$ is a uniformly integrable martingale if and only if $$ \liminf_{c\rightarrow…
Mathfreak
- 257
5
votes
1 answer
Show local martingale
I have $\exp(\lambda X_t-\frac{\lambda ^2}{2}t)$ is a local martingale, now i have to know if $X_t$ is also a local martingale.
Can anybody help me how i can show this correctly?
daniäla
- 319
4
votes
1 answer
what if the square of a martingale is still a martingale?
Let $(\Omega,\mathcal{F},(\mathcal{F}_t:t\ge{0}),P)$ be a stochastic basis and $M=(M_t:t\ge{0})$ a locally square integrable martingale, which means a stochastic process such that:
$M_t\in{L^2(\Omega,\mathcal{F_t},P)}$ for all…
Roberto Palermo
- 412
4
votes
0 answers
chapter 1 ex 4.22 from Brownian motion, martingales, and stochastic calculus by Jean-Francois Le Gall
This is ex 4.22 from chapter 1 of''Brownian motion, martingales, and stochastic calculus by Jean-Francois Le Gall''.
exercise 4.22: Processes on defined on a probability space $(\Omega,\mathcal{F},P)$ equipped with a complete filtration…
neveryield
- 905
4
votes
1 answer
The expected squared increment of a continuous local martingale
Suppose $M=\{ M_t\}_{t\geq 0}$ is a continuous local martingale, and $M_0=0$. Then I often see the following equation
$$\mathbb{E}M_t^2=\mathbb{E}\sum_i(M_{t_{i+1}}^2-M_{t_i}^2)=\mathbb{E}\sum_i(M_{t_{i+1}}-M_{t_i})^2.$$
I am new to stochastic…
Yuyi Zhang
- 1,502