All Questions
Tagged with stochastic-calculus martingales
82 questions
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
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
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 ...
- 515
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
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|^...
- 4,000
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 ...
- 6,155
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
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 ...
- 6,155
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
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}\...
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
5
votes
1
answer
462
views
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 ...
- 63
5
votes
1
answer
478
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
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
4
votes
2
answers
416
views
Probability of winding number of 2D Brownian Motion
Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau \...
- 409
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
$$X(t) = x + \int_0^t \...
- 1,399
4
votes
2
answers
2k
views
Is the truncated Brownian motion of the class DL?
Let $W$ be a standard Brownian motion under given probability space.
For a given constant $a$, $W^a$ is a truncated Brownian motion by stopping time
$T^a = \inf(t>0:W(t) = a)$. That is, $W^a(t) = ...
- 1,399
4
votes
1
answer
441
views
Stochastic integration by parts to obtain Kailath Segall identity for iterated stochastic integrals?
If $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...
- 235
4
votes
1
answer
383
views
initial condition of a diffusion approximation
I am trying to prove that a certain sequence of Markov chains $x^N_k$ converges towards a diffusion process. The invariant measure of $x^N$ is $\pi^N$ and the Markov chain $x^N$ is started in ...
- 2,133
4
votes
1
answer
66
views
Expectation bounds on supremum of family of martingales
Suppose I fix a filtered probability space $(\Omega, \mathcal{F}, \mathbb{F}, P)$ and on it a Brownian motion $B$. Let $\tau_\alpha$ denote a set of stopping times which satisfies $\sup_\alpha \tau_\...
- 143
4
votes
1
answer
285
views
explicit characterization of the stochastic integrand
Let $V$ be a cadlag positive supermartingale with the following decomposition:
$$V_t=V_0+\int_0^tH_sdX_s-K_t$$
where $X$ is a cadlag local martingale and $K$ is an adapted increasing process with $...
- 1,835
4
votes
0
answers
306
views
A notion of SDE via the martingale representation theorem
$\newcommand{\d}{\mathrm{d}}$It is well-known that differentiating stochastic processes with respect to time is usually impossible in the usual sense. For instance, a Brownian motion $W$ on a ...
- 11.8k
4
votes
0
answers
371
views
Martingale polynomial functions
If $B_t$ is a Brownian motion then using Hermite polynomials one can find that
$$1, B_t, B_t^2-t, B_t^3 - 3tB_t,...$$
are martingales.
If $X_t$ is a diffusion
$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)...
- 41
4
votes
0
answers
73
views
Existence of martingales given some constraint on laws
Let $X=(X)_{0\le t\le 1}$ be a continuous martingale starting at $0$, then denote by $\mu$ and $\nu$ the probability laws of $\int_0^1X_t \mathrm{d}t$ and $X_1$. Then it is easy to see that the couple ...
- 1,835
4
votes
0
answers
1k
views
Change of Time in Stochastic Integral
Hi everyone,
Let's be given $I(0,t)$ a Stochastic Integral with respect to a local martingale $ M_t$ of the form :
$I(0,t)=\int_0^t h(s_-)dM_s$ with $h\in L(M)$ (for example $h$ is an adapted ...
- 1,334
3
votes
1
answer
543
views
Stochastic integrals as honest martingales -- comparison criterion
We have a given positive martingale $\rho_t$, with the dynamics:
$$\textrm{d} \rho_t = \lambda_t \rho_t \textrm{d} W_t$$
where $W_t$ is a standard Brownian motion. Now we have a "dumped" process p_t:
$...
- 667
3
votes
1
answer
1k
views
On martingale representation theorem
Let $(\Omega,\mathcal{F},P)$ be a probability space and $(\mathcal{F_{t}})_{0\le t\le T}$ a filtration generated by standard Brownian motion $W_t$.
Let $f(x)$ be $C^1$ function such that $|f'(x)| >...
- 115
3
votes
1
answer
824
views
Stochastic integrals as honest martingales — exponential damping
We have a given positive martingale ρt, with the dynamics:
$$\textrm{d}\rho_t = \lambda_t \rho_t \textrm{d}W_t$$
where $W_t$ is a standard Brownian motion. Now we have an "exponentially dampened" ...
- 667
3
votes
1
answer
267
views
An identity for the exponential of a martingale
I am trying to understand a Lemma in Olav Kallenberg's book "Foundations of Modern Probability" (Lemma 26.19 in the second edition or 23.19 in the first edition).
The part of the lemma that I do not ...
3
votes
1
answer
220
views
Can $<.>$ of a martingale determine it only?
Let $\Omega$ be the space of continuous functions defined on $[0,1]$. Define the canonical process $B$ by
$$B_t(\omega)=\omega_t,~ \forall\omega\in\Omega$$
Let us equip $\Omega$ with the usual ...
- 1,835
3
votes
0
answers
50
views
Does double stochastic integral have exponential moments?
Consider $W=(W_1,W_2):[0,1]\to \mathbb{R}^2$ a planar Brownian motion, and $W'$ a second one, independent from the first.
Let
$I=\int_0^1\int_0^1\log (|W-W'|^{-1}) \, \mathrm{d} W_1 \, \mathrm{d} {W_1}...
- 131
3
votes
0
answers
75
views
p-Variation distance defines semi-martingales
Question
When, does the process $\tilde{X}_t$, defined path-wise by
$$
\tilde{X}_t(\omega)\triangleq \rho_{\frac1{2}}\left((y_t,\mathbb{Y}_t),(x_t(\omega),\mathbb{X}_t(\omega))\right),
$$
define a ...
- 5,405
3
votes
0
answers
124
views
How can we show that the quadratic covariation of a Hilbert space valued martingale takes values in the space of nonnegative operators?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a complete filtration of $\mathcal A$
$H$ be a separable $\mathbb R$-Hilbert space
$(e_n)_{n\in\mathbb N}$ ...
- 167
3
votes
0
answers
455
views
Hitting time of two dimensional continuous martingale
Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
- 31
2
votes
1
answer
503
views
Generalisation of Strassen's (Kellerer's) Theorem
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^d$ with finite first movements, i.e.
$$\int_{\mathbb R^d}|x|~\mu(dx),\quad \int_{\mathbb R^d}|x|~\nu(dx) \quad <\quad +\infty.$$
$\mu$...
- 345
2
votes
1
answer
86
views
question about the optimal decomposition of supermartingale
Given a filtered probability space $(\Omega, \mathbb{F}, \{\mathcal{F}_t\}_{0\le t\le 1}, \mathbb{P})$, let $X$ be a cadlag martingale and $V$ be cadlag supermartingale. Suppose $V$ has the following ...
- 1,835
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$...
- 21
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$ ...
- 6,155
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
2
votes
1
answer
826
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))$ (...
- 31
2
votes
1
answer
250
views
Compactness of cadlag martingales w.r.t. to the point-wise topology
Given a sequence of cadlag (right-continuous with left limits) martingales $X^n=(X^n_t)_{0\le t\le 1}$, we may use the well known criteria to determine whether it is weakly convergent, i.e. subtract a ...
- 1,835
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
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
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
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
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
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],
$$
for some fixed $0<\...
- 284
2
votes
1
answer
275
views
Martingale representation theorem for symmetric random walk
Let $X(t)$ be a martingale w.r.t. filtration generated by Brownian motion $B(t)$. There is a well-known theorem that states that there is a unique adapted process $H(t)$ such that
$$ X(t) = \int_0^t ...
2
votes
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
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