For question about discrete or continuous (super/sub)martingales. Often used with [probability-theory] tag.

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9
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155 views

Removing deterministic discontinuities from semi-martingales

Let $X:=(X_t)_{0 \le t \le T}$ be a solution of the SDE $$ X_t = X_0 + \int_0^t \sigma(s,X_s) dW_s + \sum_{i=1}^n f_i(X_{t_i^-}) 1_{\{t > t_i\}}$$ where $t_1,\cdots,t_n \in [0,T]$ and $(f_i)_{1 \le ...
7
votes
0answers
159 views

Sufficient condition in terms of stopping times for a stochastic process to be a local supermartingale

(Question edited in response to Nate's comment) Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin ...
6
votes
0answers
380 views

Azuma's inequality to McDiarmid's inequality?

I was going through some notes on concentration inequalities when I noticed that there are two commonly-cited forms of McDiarmid's inequality. Long story short: I know how to prove the weaker one from ...
6
votes
0answers
128 views

Proving existence of limit by Martingale.

I'm thinking about a question: Suppose $y_n > −1$ for all $n$ and $\sum |y_n| < \infty$. Show that $\prod_{m=1}^\infty (1 + y_m)$ exists. Since $\sum |y_n| < \infty$, we must be able ...
6
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0answers
282 views

An application of the Optional Sampling Theorem

let $S(k), k\geq 0$ a discrete random process. Suppose $S(N)$ is with probability one either 100 or 0 and that $S(0)=50$. Suppose further there is at least a sixty percent probability that the price ...
4
votes
0answers
58 views

Filtration and measure change

I'm reading Steven E. Shreve's "Stochastic calculus for finance II", and find myself not really understand the concept of "filtration". Yes, the definition of filtration is straight forward, it's ...
4
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0answers
92 views

Determine if this is a Martingale

I am trying to check if the process $S_t$ is a martingale, where $\mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t$, $S_0 = 1$. We know that $S_t$ is a local martingale because if we stop it ...
4
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86 views

Submartingality of generalized stochastic exponential of a BMO martingale

I attended a talk today on BMO martingales. It was my first encounter with the subject, and this may explain my inability to solve this myself. We take a continuous local martingale $L$, and say ...
4
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0answers
111 views

Time scaling of Brownian motion

Let $(B_t)_{t\geq 0}$ be a standard Brownian motion and $A_t$ be an increasing continuous process adapted to the filtration generated by the Brownian Motion and $A_0 = 0$. I am trying to prove ...
3
votes
0answers
116 views

Levy's extension of the Borel-Cantelli Lemmas

Following is the statement and proof of Levy's extension of the Borel-Cantelli Lemmas, as given in Williams' "Probability with Martingales" (1991), in section 12.15 on page 124. I understand most of ...
3
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68 views

Absolute continuity of quadratic variation of continuous local martingales

I am interested to know if there are any simple sufficient conditions on continuous local martingale to have absolutely continuous quadratic variation. In general , we know only that quadratic ...
3
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0answers
44 views

Upper bounds on the sum in a Martingale process

My question is related the hitting time of not a random walk, but a more general martingale process. Suppose we start with an arbitrary $x_0=x$ with $0\leq x\leq 1$. We compute $x_{t+1}$ from $x_t$ ...
3
votes
0answers
75 views

Discontinuous local martingales with finite variations

I would like to know if a discontinuous local martingale with paths of finite variations almost surely is a martingale. I feel that it should be the case but can't find a straightforward argument. As ...
3
votes
0answers
104 views

Maximal inequality for martingale difference sequence

I have a martingale difference sequence (MDS) $X_n$ wrt. some adapted filtration $\mathcal{F}_n$, i.e. $\mathbb{E}[|X_n|] < \infty$ and $\mathbb{E}[X_{n+1}\mid\mathcal{F}_n] = 0$ a.s. for all $n ...
3
votes
0answers
253 views

Superharmonic function and supermartingale

If $f$ is a nonnegative superharmonic function in dimension 2, how to prove that $f$ is constant? There is an exercise in R.Durrett's probability book, which gives out a method to prove it by ...
3
votes
0answers
150 views

An identity about a continuous local martingale

Let $M_t$ be a continuous local martingale with $M_0=0$ and define $I_t^0=1$ and $I_t^n= \int_0^t I_s^{n-1}\; dM_s$. Prove that we have $$n I_t^n= I_t^{n-1}- \int_0^t I_s^{n-2} \;d((M)_t) $$ where ...
3
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0answers
144 views

A Stopping Theorem for Right-Continuous Submartingales

Reading through the book "Brownian Motion & Stochastic Processes" by Karatzas and Shreve, I found the following problem (problem 3.24, page 20): Suppose that $ \{ X_t, \mathcal{F}_t \ | \ 0 \leq ...
2
votes
0answers
74 views

Qual Question concerning martingale

Suppose $X_n$ is a sequence of random variables that has the property that $\sup|X_n| \leq 1$ a.s. Then use Doob's decomposition to prove that $\sum_{n\geq 1} X_n$ converges a.s. iff the sum ...
2
votes
0answers
34 views

Rate of convergence of a martingale

I have a question related to convergence rates of martingales: Assume that there is a sequence of maximized likelihood ratios: $ \frac{f_{\hat{\theta}_{n}} \left ( Y_{1},Y_{2},\dots,Y_{n} \right ) ...
2
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0answers
27 views

Finding a pre-visible process

Question: Let $W_t$ be a standard brownian motion under P with filtration $\mathscr F_t$. Let: $$ M_t=\mathbb E[W_T^2|\mathscr F_t] $$ Show that $M$ is a P martingale. This is simple enough using ...
2
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0answers
148 views

Is the absolute value of Brownian motion a super martingale?Is it a sub martingale? Is it a Markov process?

I've just started to study random processes and I'm trying to solve the following problem: Let $W(t)$ be a Brownian motion with filtration $F(t)$ generated by $ W(t)$ (i.e., $F(t)=\sigma \left( ...
2
votes
0answers
169 views

Local martingale is locally uniformly integrable martingale?

Is a local martingale locally uniformly integrable martingale ? Here I define a local martingale to be the process with a localizing sequence $\tau_n$ such that the stopped process is martingale. ...
2
votes
0answers
51 views

Lower bound for stochastic process

Suppose the non-negative stochastic process $(X_t,Y_t)$ is such that $E\{X_t - X_a | Y_u \in A \,\,\forall u \in [a,t] \} \geq Z(A)(t-a)$. Let $T_{A}$ be the time of a visit to $A$. Assuming that the ...
2
votes
0answers
130 views

Checking proof that a given process is a martingale

I am interested in justify the well known result about the process $M^\lambda _t =\exp\left(\lambda B_t - \frac{\lambda^2}{2} t\right)$ being $\mathcal F_t$-martingale in the filtered probability ...
2
votes
0answers
57 views

Dimension free Concentration bounds for Martingales

Consider the following random process which is defined on $n$ numbers $0\leq x_1,\ldots,x_n\leq 1$: At each step, pick an arbitrary number, say $x_i$. Then randomly (and independently) change its ...
2
votes
0answers
80 views

Azuma's inequality with high probabilistic bounds

Let $(X_n)_{n \geq 0}$ be a super-martingale, that is $\mathbb{E}[X_{n+1} \mid X_1, \dots, X_n] \leq X_n$. Let's further assume that $\Pr[|X_n - X_{n-1}| < c_n] \geq 1-\delta$. Does there exist any ...
2
votes
0answers
140 views

What are the conditions for when a martingale has a continuous modification?

Update: This question has been moved to Mathoverflow. Please answer it there. There is a theorem as follows: Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. ...
2
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0answers
72 views

Doob Decomposition of American Option

I am trying to figure out the Doob decomposition of an American put option in a discrete time binomial model. I know how to price the American put, but I'm having trouble expressing it as the sum of ...
2
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0answers
133 views

Convergence of Martingale.

The question is: 5.2.11. Let $X_n$ and $Y_n$ be positive integrable and adapted to Fn. Suppose $\mathbb E(X_{n+1}|\mathcal F_n) ≤ (1 + Y_n )X_n$ with $ \sum Y_n < \infty$ a.s. Prove that ...
2
votes
0answers
148 views

how to show that the price process is a martingale

Suppose I have an $d$-dimensional semimartingale $S=\{S_t\}$ with $t\in[0,T]$ under $P$. $S $ need not to be continuous (RCLL can be assumed). Suppose $Q$ is an equivalent measure w.r.t. $P$ such that ...
2
votes
0answers
276 views

strong and weakly orthogonality for martingales

If I have two RCLL martingales $X,Y$, both bounded in $L^2$, hence uniformly integrable. Then we call $X,Y$ weakly orthogonal if $E[X_\infty Y_\infty]=0$ and we call $X,Y$ strongly orthogonal if $XY$ ...
2
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0answers
230 views

Doob inequality for continuous martingales

In our class we have proven Doob's inequality for discrete martingale, i.e. Let $(M_n)_{n \in \mathbb{N}}$ a martingale, then $$ \| \max_{0\le k\le n} M_k\|_p \le C_p \|X_n\|_p$$ for $p\in ...
2
votes
0answers
237 views

Show that the stopped sequence is a martingale

This is a series problem and I'm struggling with the last part. I assume that the last part has nothing to do with previous ones, so i won't put up the other parts. Question is : Let $\tau$ be a ...
2
votes
0answers
72 views

Optional stopping theorem for Hilbert valued martingales

Suppose $X_n$ is a Hilbert-valued martingale. Does the optional sampling theorem apply in this case? Does anyone know where I can find a proof? Thank you.
1
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0answers
12 views

Is there a Burkholder-Davis-Gundy inequality for martingale increments?

is there a Burkholder-Davis-Gundy inequality for martingale increments? More specifically, I would like to find a finite bound of order $h^{p/2}$ for the expectation $$\operatorname{E} \left[ \sup_{t ...
1
vote
0answers
27 views

If a random integral has moments of all orders, is the same true for its kernel?

Suppose you have a continuous semimartingale $S_t=M_t + A_t$ where $A_t$ is the continuous finite variation part which has the form $A_t = \int_0^t b_s \, \mathrm{d} s$, where $\int_0^{\infty} |b_s| ...
1
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0answers
33 views

Occupancy distribution bounds for $k$ balls in $m$ bins

Suppose we throw $k$ (distinct) balls into $m$ (distinct) bins, and let $B$ count the number of non-empty bins. I am interested in lower bounds on $B$. More precisely, I wish to bound from above the ...
1
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0answers
22 views

Applying Ito lemma to multi dimensional semimartingales

Let $X_t=(X^1,\dots,X^d$) be a d-dimensional semimartingale. Then the formula for Ito's lemma I have found in several places, including wikipedia, is: $f(X_T)-f(X_0)=\sum_{i=1}^d\int_0^T ...
1
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0answers
36 views

Second (centered) moment for martingales

Take the process ${x}_t$ following geometric Brownian motion (GBM) $$x_t=\mu x_t \,dt+\sigma x_t \,dW_t$$ with $x_0>0$ known. It has first moment equal to $$\text{E}[x_t]=x_0 e^{\mu t}$$ and second ...
1
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0answers
52 views

Is Zn a Martingale with mean 1?

Consider a sequence of independent tosses of a coin, and let $P_h$ be the probability of a head on any toss. Let $A$ be the hypothesis that $P_h = a$, and let $B$ be the hypothesis that $P_h = b$. Let ...
1
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0answers
43 views

Stochastic Differential Equation- When martingale?

Suppose I'd like to check the martingale property for some SDE. What do I have to require for it to be martingale? I know that no drift is one requirement, but what are the others?
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0answers
103 views

An application of the Dambis-Dubins-Schwarz theorem. Is my argument correct?

I attended a lecture today, in which the professor went through an example with a lot of tedious calculations to show something which I'd think would follow directly from the Dambis-Dubins-Schwarz ...
1
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0answers
53 views

Conditions for Martingale

Let $X{}_{1},X_{2},\ldots$be a sequence of independent RV and let $f{}_{j}$be continuous functions. Let $S{}_{0}=1$ and $S{}_{n}=\sum_{j=1}^{n}f_{j}\left(X_{j}\right)$. Find a necessary and sufficient ...
1
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0answers
94 views

Martingale property of a counting process substracting its compensator

I shall show that for a point process (counting process) $\Phi((0,t])=\sum_{n \geq 1} \mathbf{1}_{\lbrace T_n \leq t \rbrace}$, \begin{align*} M_t = \Phi((0,t]) - \int_{0}^{t} \mathbf{1}_{\lbrace s ...
1
vote
0answers
77 views

It looks like Doob's inequality, but goes the other direction.

$(X_n)$ is a nonnegative martingale with $X_0=1$ and $X_n$ converges to $0$ a.s. Suppose that $X_n$ only has finitely many possible values for each $n$. Take $S_n=\max\{x:P(X_{n+1}=x\mid F_n)>0\}$. ...
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0answers
49 views

Atypical exponential martingale

Process $\{M\}$ is a pure-jump martingale, with finite number of jumps on any finite time interval, and a compensator $a_t$ at every time $t$. It can be thus written: $$ M_t = \sum_{0<s\leq t} ...
1
vote
0answers
105 views

Forming a local martingale with continuous increasing process

If $M_t$ is continuous martingale, we know that there exists quadratic variation process which is continuous and increasing. I am interested to know if the converse is also true. To make it precise ...
1
vote
0answers
64 views

Characterization of hitting time's law. (Proof check)

Under the same assumptions of this early question, consider also a the random time $T_a := \inf\{ t > 0: B_t \geq a\}$ which is a stopping time. Since $M^\lambda$ is a continuous martingale, Doob's ...
1
vote
0answers
103 views

Stopping time and martingale for random walks

Let $X_0=0, X_1, X_2,\dots, X_N$ be i.i.d. random variables, with Gaussian distribution $\cal N (0,1)$. For $k=0,\dots, N, S_k=\sum_{i=1}^k X_i$ and $\tau=\min\{k:S_k^2\geq N-k\}$. So $\tau$ is a ...
1
vote
0answers
56 views

A generalization for martingales?

I am a computer scientist interested in analyzing stochastic processes specified as probabilistic programs. In my research, I recently encountered an idea that looks just like a martingale, but is ...