All Questions
47 questions
4
votes
1
answer
66
views
Expectation bounds on supremum of family of martingales
Suppose I fix a filtered probability space $(\Omega, \mathcal{F}, \mathbb{F}, P)$ and on it a Brownian motion $B$. Let $\tau_\alpha$ denote a set of stopping times which satisfies $\sup_\alpha \tau_\...
- 143
2
votes
0
answers
61
views
Characterisation of Bessel process
Let $\delta \in (0, 2)$; $(X_t)_{t \ge 0}$ a nonnegative continuous Markov process. Suppose that
For each $T \ge 0$, if we write $\tau \overset{\mathrm{def}}= \inf\{t \ge T : X_t = 0\}$, then $(X_{T +...
- 177
0
votes
0
answers
90
views
Martingale defined by an integral
Consider a probability space $(\Omega,\mathcal{F},P).$ Let $f \in C^{\infty}_{c}(\mathbb{R}^d,\mathbb{R}),p \geq 2.$ $(X_r^{y})_{(r,y) \in \mathbb{R}_+ \times \mathbb{R}^d}$ is a stochastic process ...
- 573
6
votes
1
answer
396
views
Is a martingale conditioned to be large a submartingale?
Let $X$ be a continuous time martingale such that $X_\infty := \lim_{t \to \infty} X_t$ exists almost surely. Let $x \in \mathbb R$ be such that $\mathbb P(X_\infty \geq x) > 0$, and define the ...
- 6,195
7
votes
2
answers
2k
views
A curious martingale
Does there exist an almost surely continuous martingale $X$ with $X_t \to +\infty$ almost surely?
Remark: Note that such a martingale exists in discrete time, or equivalently in continuous time if the ...
- 6,195
2
votes
0
answers
121
views
Martingale regularization
Consider a submartingale $X,$ then for almost every $\omega \in \Omega,$ for every $v \in \mathbb{R},\lim_{u \in \mathbb{{Q},u \uparrow v}}X_u(\omega)$ exist in $\mathbb{R}.$
I was wondering if there ...
- 573
2
votes
1
answer
182
views
Mean of log-normal variable when exponent is replaced by runnung maximum of Ito-integral
Let $W=\{W_t\}_{t\in[0;1]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;1]}$ the filtration generated by $W$, augmented with the nullsets. Let $\{\sigma_t\}_{t\in[0;1]}$ be a continuous and ...
- 335
1
vote
1
answer
139
views
Characterization of Brownian motion: processes with right-continuous paths
I am looking for a reference with a proof for the following fact:
If a right-continuous martingale $(X_r)_{ r \geq 0}$ is such that $X_0=0,(X^2_r-r)_r,(X_r^3-3rX_r)_r,(X_r^4-6rX_r^2+3r^2)_r$ are ...
- 573
1
vote
0
answers
240
views
Where to submit a new proof of the continuous martingale convergence theorem?
There were various proofs of the discrete martingale convergence theorem, but as far as I know there is only one proof of the continuous version of this theorem using the up-crossing lemma.
I wrote a ...
- 11
1
vote
0
answers
108
views
Decomposition of reversed processes
Consider a reversed filtration $(\mathcal{F}_k)_{k \geq 0} $ $(\mathcal{F}_{k+1} \subset\mathcal{F}_k),$ $(X_k)_{k \geq0}$ is a processes in $L^1,\mathcal{F}_k$-adapted.
Is it possible to decompose $...
- 249
1
vote
0
answers
80
views
Almost supermartingale and a.s convergence
After reading a paper on the convergence of almost supermartingale, the following result appeared:
If $(X_k)_k,(Y_k)_k,(W_k)_k$ are three $(\mathcal{F}_k)$-adapted processes taking values in $\mathbb{...
- 249
1
vote
1
answer
182
views
Is a stopped Ito-integral integrable if the Ito integrand is only square-integrable on an open interval?
Assume a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\in[0;T)}, \mathbb P)$ with an $\mathbb R^n$-valued Brownian motion $\{W_t\}_{t\in[0;T)}$ and the filtration $\{\mathcal F_t\}_{t\in[0;T)...
- 335
0
votes
0
answers
71
views
Conditions for existence of a semi-martingale representing a system of probability measures
Let $(\nu_t)_{t \in [0,1]}$ be Borel probability measures on a stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_{t \in [0,1]})_t,\mathbb{P})$.
Does there exist a semi-martingale $(X_t)_{t\in[0,1]}$ ...
- 5,405
1
vote
0
answers
744
views
Local martingale but not martingale
For a 3-dimensional Brownian motion $B = (B_t, t ≥ 0)$ and $x ∈ \mathbb{R}^3 \backslash \{0\}$ define the process
$Y = (Y_t, t ≥ 0)$ via $Y_t =\frac{1}{|B_t+x|}$ how come this is a continuous local ...
2
votes
0
answers
237
views
Semimartingale decomposition and filtrations
In short: I am trying to understand how the decomposition of a semimartingale into its local martingale and finite variation components depends on the filtration we are using.
So, taking a toy example,...
- 341
1
vote
1
answer
284
views
Martingale derivation by direct calculation
I'm reading the proof of a theorem and stumbled across the following derivation which I cannot replicate myself.
Let $W(t)$ be a $Q$-martingale and be given by $W(t) = B(t) + \mu t$ with $B(t)$ a ...
- 11
1
vote
0
answers
265
views
Wiener isometry for semimartingales
Suppose that $Y_t$ is a special square-integrable $\mathbb{R}$-valued semi-martingale and let $\mathcal{L}^2(Y)$ denote the set of $Y$-predictable processes satisfying
$$
\mathbb{E}\left[
\int_0^{\...
- 5,405
3
votes
0
answers
75
views
p-Variation distance defines semi-martingales
Question
When, does the process $\tilde{X}_t$, defined path-wise by
$$
\tilde{X}_t(\omega)\triangleq \rho_{\frac1{2}}\left((y_t,\mathbb{Y}_t),(x_t(\omega),\mathbb{X}_t(\omega))\right),
$$
define a ...
- 5,405
2
votes
0
answers
203
views
Is martingale solution equivalent to weak solution for SDE driven by stable process
Consider the following SDE
$$
d X_t=b(X_t)d t+d L_t,
$$
where $L_t$ is the symmetric $\alpha$-stable process. The corresponding generator is given by
$$
L=\Delta^{\alpha/2}+b\cdot\nabla.
$$
Is the ...
- 411
3
votes
0
answers
124
views
How can we show that the quadratic covariation of a Hilbert space valued martingale takes values in the space of nonnegative operators?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a complete filtration of $\mathcal A$
$H$ be a separable $\mathbb R$-Hilbert space
$(e_n)_{n\in\mathbb N}$ ...
- 167
0
votes
3
answers
639
views
Non-smooth Ito lemma for semi-martingales
Is there an extension of Ito's Lemma where $X_t$ is a semi-martingale and $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is a function which is not smooth?
I've been looking but have not found much, any ...
- 5,405
2
votes
2
answers
291
views
A question about Skorokhod embedding problem
The Skorokhod Embedding Problem is well known and has many solutions. Now let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion and $\tau$ be an embedding to the centered distribution $\mu$, i.e. the ...
- 1,835
2
votes
0
answers
227
views
Strong law of large number for semimartingale
I just want to know if for semimartingale $X$ we have $\lim_{t \rightarrow \infty} \frac{X_{t}}{\langle X\rangle_{t}}=0$ or when it is possible. I know it is true for Brownian motion.
Thanks
- 31
9
votes
1
answer
4k
views
Quadratic variation and predictable quadratic variation for martingales
Let $(M_{t})_{0\le t\le 1}$ be a continuous martingale with respect to the filtration $(\mathcal{F}_{t})_{0\le t\le 1}$. Assume that $E M_1^2<\infty$.
Fix $N$ and consider now a discrete version ...
- 931
3
votes
0
answers
455
views
Hitting time of two dimensional continuous martingale
Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
- 31
3
votes
1
answer
267
views
An identity for the exponential of a martingale
I am trying to understand a Lemma in Olav Kallenberg's book "Foundations of Modern Probability" (Lemma 26.19 in the second edition or 23.19 in the first edition).
The part of the lemma that I do not ...
4
votes
1
answer
441
views
Stochastic integration by parts to obtain Kailath Segall identity for iterated stochastic integrals?
If $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...
- 235
6
votes
0
answers
220
views
Reference request: Stochastic integration and martingale theory on the whole real line
I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM ...
- 617
1
vote
0
answers
218
views
question about Doob-Meyer decomposition
Given a filtered probability space and let $X$ be a cadlag local martingale defined on this space. Let $V$ be a cadlag supermartingale and assume we know the following decomposition:
$$V_t=V_0+\int_0^...
- 1,835
11
votes
2
answers
1k
views
Can every discrete martingale be embedded in a continuous martingale?
Let $(X_k)_{k=0,1,..., n}$ be a discrete martingale defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. I would like to know whether there exists a (continuous) martingale $(\tilde{X}...
- 1,835
1
vote
1
answer
411
views
a dominated convergence theorem for martingale (II)
The question is presented in
https://mathoverflow.net/questions/155392/a-dominated-convergence-theorem-for-martingale
Let $\{(X_1^n, X_2^n)\}_n$ be a sequence of martingales defined some probability ...
- 1,835
7
votes
1
answer
1k
views
a $L^1$ convergence for backward martingale
I have a question which may be naive, but I can not find the related result in the classical reference such as "Foundations of Modern Probability" and "Probability"(Billingsley). So if someone knows ...
- 1,835
2
votes
0
answers
134
views
Supermartingale inequality on a particular event
Say, I have a supermartingale $Y_t$ with respect to the filtration $F_t$. Let $T$ and $S$ two stopping times greater than $t>0$ such that on the event $A$, $T>S$, then since $Y_t$ is a ...
- 23
6
votes
3
answers
2k
views
Iterated Ito Integral, Gaussian Volterra Process
Let me define
$$
J^f_{n}(t) = \, \int_0^t \int_0^{t_1} \ldots \int_0^{t_{n-1}} f(t, t_1, \ldots, t_n) \; dB_{t_n} ...dB_{t_1}
$$
where $f:[0,1]^{n+1} \to \mathbb{R}$ is a nice deterministic function....
- 271
1
vote
0
answers
1k
views
What conditions on a filtration guarantee that a (sub)martingale has a continuous modification?
There is a theorem as follows:
Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. Assume $M_t$ is a submartingale adapted to $\mathcal{F}_t$ such that $t \mapsto \...
- 6,287
8
votes
3
answers
2k
views
What is the optimal growth of the constant in BDG?
Let $X$ be a continuous local martingale, and $\langle X \rangle$ be its quadratic variation process. The "standard" proof of Burkholder-Davis-Gundy inequalities found in books yields $(\mathsf{E} |X|^...
- 4,010
3
votes
1
answer
824
views
Stochastic integrals as honest martingales — exponential damping
We have a given positive martingale ρt, with the dynamics:
$$\textrm{d}\rho_t = \lambda_t \rho_t \textrm{d}W_t$$
where $W_t$ is a standard Brownian motion. Now we have an "exponentially dampened" ...
- 667
3
votes
1
answer
543
views
Stochastic integrals as honest martingales -- comparison criterion
We have a given positive martingale $\rho_t$, with the dynamics:
$$\textrm{d} \rho_t = \lambda_t \rho_t \textrm{d} W_t$$
where $W_t$ is a standard Brownian motion. Now we have a "dumped" process p_t:
$...
- 667
1
vote
2
answers
316
views
Martingale part of the discontinuous put payoff
I need the martingale part of the put payoff (not $C^2$..). Where $S_t=exp(\sigma W_t -\frac{\sigma^2t}{2})$
$d[(S_t -K)^+ ]$ ??
I guess I need to use local times but how?
- 111
4
votes
1
answer
3k
views
The only continuous martingales with stationary increments are Brownian motions
I know that the above statement is true, but I can't demonstrate it.
It's a pretty powerful theorem, here is its mathematical formulation:
Theorem: The only continuous martingales with stationary ...
- 41
12
votes
1
answer
5k
views
Martingales in both discrete and continuous setting
I am wondering, polynomials like
$S_n^4-6n S_n^2+3n^2+2n$ for $$S_n=\sum_{i=1}^n{X_i}$$ where $$\mathbb{P}(X_i=1)=\mathbb{P}(X_i=-1)=\frac{1}{2}$$ is a martingale (under the conventional filtration). ...
- 255
6
votes
1
answer
653
views
Change of space-time in Walsh's stochastic integral
One can read about Walsh's construction of martingale integral in the paper (pp.16-23)
http://www.math.utah.edu/~davar/ps-pdf-files/SPDEBookDK.pdf (Wayback Machine)
For $U,V\in \mathcal{B}(\mathbb{R}\...
5
votes
1
answer
479
views
Stieltjes integrals of predictable processes
I am looking for a direct proof of the fact that, roughly speaking, if $S=S_0+A+M$ is an $L^2$ semimartingale, and $M$ (the martingale part) has the martingale representation property, then for any ...
- 71
4
votes
0
answers
1k
views
Change of Time in Stochastic Integral
Hi everyone,
Let's be given $I(0,t)$ a Stochastic Integral with respect to a local martingale $ M_t$ of the form :
$I(0,t)=\int_0^t h(s_-)dM_s$ with $h\in L(M)$ (for example $h$ is an adapted ...
- 1,334
4
votes
1
answer
2k
views
Distribution of running maximum of a local martingale
Let $(\Omega, \mathcal{F}, \mathbb{P}, \mathcal{F}_t)$ be a given
probability space with usual conditions, on which $W$ is a standard
Brownian motion. For $x \ge 0$, consider
$$X(t) = x + \int_0^t \...
- 1,399
4
votes
2
answers
2k
views
Is the truncated Brownian motion of the class DL?
Let $W$ be a standard Brownian motion under given probability space.
For a given constant $a$, $W^a$ is a truncated Brownian motion by stopping time
$T^a = \inf(t>0:W(t) = a)$. That is, $W^a(t) = ...
- 1,399
4
votes
1
answer
383
views
initial condition of a diffusion approximation
I am trying to prove that a certain sequence of Markov chains $x^N_k$ converges towards a diffusion process. The invariant measure of $x^N$ is $\pi^N$ and the Markov chain $x^N$ is started in ...
- 2,133