All Questions
Tagged with stochastic-calculus martingales
82 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_\...
- 5,474
3
votes
0
answers
50
views
Does double stochastic integral have exponential moments?
Consider $W=(W_1,W_2):[0,1]\to \mathbb{R}^2$ a planar Brownian motion, and $W'$ a second one, independent from the first.
Let
$I=\int_0^1\int_0^1\log (|W-W'|^{-1}) \, \mathrm{d} W_1 \, \mathrm{d} {W_1}...
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
4
votes
1
answer
3k
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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 ...
- 1,446
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 ...
CommunityBot
- 1
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 ...
- 179
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 ...
- 29.7k
5
votes
1
answer
462
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On the convergence of a martingale
Let $W$ be a standard one dimensional Brownian motion and let $A$ be the process defined by :
$$\forall \ t\geq 0: \quad A_t := \int_0^t\left(1 + e^{W_s}\right)\mathrm{d}s$$
and for $t\geq 0$, we ...
- 5,474
7
votes
2
answers
2k
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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 ...
- 29.7k
1
vote
1
answer
284
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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 ...
CommunityBot
- 1
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
4
votes
0
answers
306
views
A notion of SDE via the martingale representation theorem
$\newcommand{\d}{\mathrm{d}}$It is well-known that differentiating stochastic processes with respect to time is usually impossible in the usual sense. For instance, a Brownian motion $W$ on a ...
- 11.8k
2
votes
1
answer
219
views
Local martingale with increasing process
Here is a problem in stochastic calculus:
If $M_t$ is a continuous process and $A$ an increasing process, then $M$ is a local martingale with increasing process $A$ if and only if, for every $f\in C^2$...
- 5,474
1
vote
1
answer
369
views
Does a continuous martingale converge almost surely on the event that its quadratic variation is finite?
Let $M$ be a continuous martingale. Denote by $E$ the event that its total quadratic variation is finite, i.e.
$$E := \{\langle M, M \rangle_\infty < \infty\}.$$
Question: Is it true that as $t \to ...
- 14.2k
2
votes
1
answer
532
views
Is a martingale constant on the event that its quadratic variation is zero?
Let $M_t$ be a continuous time martingale, and assume its quadratic variation is identically zero with some positive probability less than $1$.
To be more precise, assume there exists some event $E$ ...
- 724
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 ...
- 7,514
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
2
votes
0
answers
116
views
Is a Riccati BSDE explicitly solvable?
Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $C\in (0;\infty)$ and $\{a_t\}_{t\in[0;T]}$ be a ...
- 335
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
2
answers
2k
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Expectation of Brownian motion increment and exponent of it
While reading a proof of a theorem I stumbled upon the following derivation which I failed to replicate myself. Let $\mu$ be a constant and $B(t)$ be a standard Brownian motion with $t > s$. Show ...
CommunityBot
- 1
0
votes
0
answers
121
views
Martingale representation of a stopped Brownian motion
This question follows from the previous post Question on the martingale representation theorem which has not been answered. I consider thus a particular case. Let $(B_t)_{t\ge 0}$ be a standard ...
- 1,334
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 $...
- 6,155
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)...
- 3,065
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 ...
- 63.9k
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,...
- 17.2k
2
votes
0
answers
172
views
Non-integer conditional moment of exponential functional of Brownian motion
Let $B_t$ be a standard Brownian motion.
I want to solve the following:
$$
\mathbb{E}\left[\left(\int_0^1 e^{\sigma B_t}dt \right)^{1/(1-\beta) }\mid e^{\sigma B_1}=z \right],
$$
for some fixed $0<\...
- 284
2
votes
1
answer
827
views
Calculate Radon-Nikodym derivative
For the laws of two pure-jump Markov processes $\mu_1$ and $\mu_2$ on $\mathbb R^n$, which generators are
$H_1f(x)=\int h(x,dy) (f(y)-f(x))$
and $H_2f(x)=\int e^{-g(x,y)} h(x,dy) (f(y)-f(x))$ (...
- 1,675
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 ...
- 63.9k
-2
votes
1
answer
138
views
Problem arising from martingale solutions to SPDE: $Law(u)=Law(v)$ on $C([0,T]; X)$, can $Law(u)=Law(v)$ on $C([0,t]; X)$ for $t<T$?
I ask this question because I found in some papers of martingale solutions to SPDE, to prove the approximate solutions $u_n$ is a convergent sequence, one can use "stochastic compact" method to find ...
- 29.7k
10
votes
2
answers
2k
views
Show that this process is not a martingale
I am cross-posting this question from MSE since I did not received any answer, furthermore I tried asking some professors in my university but still we could not find an answer.
The most surprising ...
- 29.7k
4
votes
0
answers
371
views
Martingale polynomial functions
If $B_t$ is a Brownian motion then using Hermite polynomials one can find that
$$1, B_t, B_t^2-t, B_t^3 - 3tB_t,...$$
are martingales.
If $X_t$ is a diffusion
$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)...
- 41
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
$$
\mathbb{E}\left[
\int_0^{\...
- 63.9k
0
votes
1
answer
1k
views
Ito integral and true martingale
Consider a twice diferentiable function $F$ on $R$ with bounded
first derivative $F'$ and a Brownian motion $W$. Show that $F(W_t)-\frac{1}{2} \int_{0}^{t} F'' (W_s)ds$ is a true martingale.
I tried ...
- 56
3
votes
0
answers
75
views
p-Variation distance defines semi-martingales
Question
When, does the process $\tilde{X}_t$, defined path-wise by
$$
\tilde{X}_t(\omega)\triangleq \rho_{\frac1{2}}\left((y_t,\mathbb{Y}_t),(x_t(\omega),\mathbb{X}_t(\omega))\right),
$$
define a ...
- 5,405
2
votes
1
answer
275
views
Martingale representation theorem for symmetric random walk
Let $X(t)$ be a martingale w.r.t. filtration generated by Brownian motion $B(t)$. There is a well-known theorem that states that there is a unique adapted process $H(t)$ such that
$$ X(t) = \int_0^t ...
2
votes
1
answer
503
views
Generalisation of Strassen's (Kellerer's) Theorem
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^d$ with finite first movements, i.e.
$$\int_{\mathbb R^d}|x|~\mu(dx),\quad \int_{\mathbb R^d}|x|~\nu(dx) \quad <\quad +\infty.$$
$\mu$...
- 345
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
1
vote
0
answers
63
views
Martingale covariation operator in infinite-dimensions
Let
$(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]},\operatorname P)$ be a filtered probability space
$U,H$ be separable $\mathbb R$-Hilbert spaces
$(e_n)_{n\in\mathbb N}$ and $(f_n)_{n\in\mathbb N}$...
- 167
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}\...
- 4,713
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|^...
- 3,946
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
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....
- 7,514
0
votes
0
answers
163
views
Using the optional stopping theorem on a stochastic process
(I'm much more used to number theory than to stochastic processes, so there are probably a lot of errors in the following:)
Consider a stochastic differential equation $dx = F(t,x) dt + \sigma dW$, ...
- 101
-3
votes
1
answer
141
views
Approximate martingales by truncation
Let $(X,Y)$ be a $\mathbb R-$valued martingale. For any $\varepsilon>0$, is it possible to find another martingale $(X',Y')$ s.t. $X'$ and $Y'$ are supported on a compact set, and
$$
\mathbb E\big[\...
- 2,720
2
votes
0
answers
227
views
Strong law of large number for semimartingale
I just want to know if for semimartingale $X$ we have $\lim_{t \rightarrow \infty} \frac{X_{t}}{\langle X\rangle_{t}}=0$ or when it is possible. I know it is true for Brownian motion.
Thanks
- 3,958
2
votes
1
answer
2k
views
Question about the stochastic integral of martingales
Let $M=(M_t)_{t\ge 0}$ be a continuous martingale defined on some filtered probability space taking values in $\mathbb{R}$. Let $H=(H_t)_{t\ge 0}$ be some bounded progressively measurable process, i.e....
- 103
4
votes
1
answer
285
views
explicit characterization of the stochastic integrand
Let $V$ be a cadlag positive supermartingale with the following decomposition:
$$V_t=V_0+\int_0^tH_sdX_s-K_t$$
where $X$ is a cadlag local martingale and $K$ is an adapted increasing process with $...
CommunityBot
- 1
1
vote
0
answers
312
views
Does the martingale property holds after changing filtration?
Let $\Omega$ be the space of continuous real-valued functions $\omega=(\omega_t)_{t\ge 0}$ starting at zero, i.e. $\omega_0=0$. Let $\Lambda=\Omega\times \mathbb R_+$ and denote by $\lambda=(\omega,\...
- 3,085