0
votes
0answers
25 views

a question about Dambis, Dubins-Schwarz Theorem

Let $M=(M_t)_{0\le t\le 1}$ be a continous $\mathbb{F}=\{\mathcal{F}_t\}_{0\le t\le 1}$-martingale s.t. $M_0=0$. Now my question is whether there exists a Brownin motion $B$ s.t. ...
0
votes
0answers
28 views

a question about the modification of a supermartingale

Let $\mathbf{D}\subset\mathbf{D}([0,1],\mathbb{R}_+)$ denote the space of positive cadlag functions $\mathbf{x}$ defined on $[0,1]$ with $\mathbf{x}(0)=1$. Define the canonical process ...
1
vote
1answer
54 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
votes
0answers
71 views

a question about integration by parts

Let $X$ be a cadlag martingale et $Y$ be a process of bounded variation, do we have the integration by parts formulae? $$\int_0^1Y_tdX_t=X_1Y_1-X_0Y_0-\int_0^1X_tdY_t,~ a.s.$$ Thanks for the reply!
1
vote
0answers
61 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: ...
7
votes
2answers
259 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 ...
0
votes
1answer
117 views

a dominated convergence theorem for martingale (II)

The question is presented in a dominated convergence theorem for martingale Let $\{(X_1^n, X_2^n)\}_n$ be a sequence of martingales defined some probability space. (which means ...
0
votes
0answers
56 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 ...
3
votes
1answer
106 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 ...
0
votes
1answer
62 views

construction of a approximate martingale

everyone. Given a probabilistic space $(\Omega, \mathcal{F}_t, \mathbb{P})$ and a martingale $(M_t)_{t\leq 1}$ on it. Suppose $$M_1\stackrel{\mathbb{P}}{\sim}\mu$$ where $\mu$ is a probability ...
4
votes
2answers
177 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 ...
2
votes
0answers
75 views

a generalization of Monge-Kantorovich Problem

I am thinking about the martingale version of Monge-Kantorovich Problem. Let $\mu(x)$ and $\nu(y)$ denote two density laws on $\mathbb{R}$, and define $M(\mu,\nu)$ the set of densities $f(x,y)$ on ...
1
vote
0answers
77 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 ...
0
votes
0answers
45 views

Is it possible to define a mixed normal having conditional variance almost everywhere null?

I'm trying to proving the stable limit of a martingale M_n(t). When I calculate the limit in probability of its quadratic variation, I find that it is always null except for a point. It seems to me ...
1
vote
0answers
227 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 ...
1
vote
0answers
148 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 ...
1
vote
1answer
219 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)| ...
4
votes
1answer
221 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} ...
2
votes
1answer
454 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" ...
3
votes
1answer
358 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: ...
1
vote
2answers
257 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?
9
votes
1answer
1k 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). ...
6
votes
1answer
517 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) www.math.utah.edu/~davar/ps-pdf-files/SPDEBookDK.pdf For $U,V\in \mathcal{B}(\mathbb{R}\times \mathbb{R}^+), ...
5
votes
1answer
315 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 ...
1
vote
0answers
278 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 ...
4
votes
1answer
718 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 ...
3
votes
2answers
702 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) = ...
3
votes
1answer
304 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 ...