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

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29 views

Martingale and mean squared error

In preparation for a course I am doing later in the semester I have been trying to brush up on my knowledge about martingales. But I am struggling with the following problem: Let ...
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1answer
66 views

The branching process $\mu^{-n}Z_n = \mu^{-n}\sum_{k=1}^{Z_n{-1}}X_{n,k}$ is a martingale

Let $\{X_{n,k} : n,k \geq 1\}$ be a collection of i.i.d. $\mathbb{Z}_+$-random variables with finite variance $\sigma^2 > 0$ and mean $\mu > 0$. Define $(Z_n)_{n\geq 0}$ recursively by ...
2
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1answer
71 views

Determine a sequence of random variables is a martingale

I'm trying to solve a problem from an old exam. This is an easy but a bit lengthy exercise, divided into subproblems. Since they are based on each other and probably are quite short, I was hoping that ...
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0answers
92 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 ...
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1answer
42 views

Bilinearity of quadratic variation

Fix a filtered probability space satisfying the usual conditions. Let $\mathcal{M}^2_0$ be the vector space of cadlag martingales null at $0$ bounded in $L^2$. We state without proof the following ...
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1answer
38 views

product of martingales bounded in $L^2$

Let $(M_t)_t$ and $(M_t)_t$ be two càdlàg martingales on the same filtered probability space. We know that $M_{\infty}$ and $N_{\infty}$ are orthogonal in $L^2$. Is it true that $(M_t N_t)_t$ is a ...
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1answer
40 views

Extended stochastic exponential

I have encountered a problem, which is hopefully rather easily solvable. I just can't get my head around it at the moment. First we extend a well known notion. We call a stochastic process $M$ a ...
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1answer
96 views

A weak-type submartingale inequality

Let $(X_n)_{n\in\mathbb N_0}$ be a submartingale or a supermartingale. Show that, for all $n\in\mathbb N$ and $\lambda>0$, $$\lambda P[|X|^*_n\ge \lambda ]\le 12 E[|X_0|]+9E[|X_n|].$$ ...
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1answer
152 views

What are some good books about martingales?

I'm looking for suggestions for well written books dealing with martingale theory, not necessarily exclusively. I'm also looking for a nice compilation of problems, preferably with answers, on this ...
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1answer
75 views

Textbook suggestion for studying martingales

I've been studying probability from Davar Khoshnevisan's graduate textbook since the beginning of the semester. A month ago, I came to the chapter of "Martingales". Since it's my first encounter with ...
3
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1answer
98 views

Continuous local martingale of finite variation is constant

Is a continuous local martingale $M$ of finite variation constant? We know that there exists a sequence of stopping times $T_n\nearrow \infty$ a.s. as $n\to\infty$ such that the stopped process ...
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135 views

Proving the martingale property of stochastic exponentials of pure jump processes

I am playing with different versions of compound-Poisson like processes with regime-switching features. Then I take stochastic exponentials of these to define a change of measure process. However, how ...
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155 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. ...
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1answer
61 views

Show that a process is no semimartingale

_Hello everyone! I got a little question about how to show that the process $X_t:=|B_t|^{\frac{1}{3}}$ is NOT a semimartingale. So far I tried to apply Ito. Since if $X_t$ was a semimartingale so is ...
2
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1answer
202 views

Conditional Expectation of Poisson Process

I have a Poisson Process with stationary and independent increments. Therefore I know: $$P(N_T - N_t = r) = \dfrac{\exp(-\lambda(T-t))(\lambda(T-t))^r}{r!} \mbox{ where } T>t.$$ Now suppose I am ...
3
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1answer
61 views

Azuma's inequality: Expected sum of differences

I am looking for an extension of Azuma's inequality which involves the expected sum of squared differences. In particular, recall that Azuma's inequality states \begin{align*} \Pr[X_n-X_0 \geq a] \leq ...
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1answer
66 views

Convergence of a conditional expectation

Consider a filtered probability space ($\Omega,\mathcal{F}_t,\mathcal{F},\mathbb{P}$) and a random variable $X$ defined on $\Omega$ with values in a set $E$. We consider the process $X_t = ...
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64 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 ...
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82 views

Jump diffusion process with sum of Poisson processes a martingale?

Hi Mathematics community, assume you have dynamics of a jump diffusion process consisting of a Brownian motion and a sum of compensated (not necessarily independent) Poisson processes, i.e. ...
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1answer
71 views

Expectation of (super/sub)-martingales

A stochastic process $X=(X_n)_{n\in\mathbb N}$ on the filtered probability space $(\Omega,\mathcal F,(\mathcal F_n)_{n\in\mathbb N},\mathbb P)$ that is a martingale has the property that $$\mathbb ...
2
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1answer
56 views

Hoeffding-type bound covering all partial sums? (better than naive union bound)

I think this must have been done before, possibly with martingales, but I can't find anything online! Given $X_1,\dots,X_n$ independent, each $X_i \in [a_i,b_i]$, letting $S_i = \sum_{j=1}^i X_j$, do ...
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1answer
56 views

Fake Brownian Motion

Does there exist a martingale which has Marginal distributions same as Brownian Motion marginals but the process itself not being Brownian motion? Any references are highly appreciated. Thanks.
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216 views

Can you make money on coin tosses when the odds is against you?

The strategy Given an initial investment $n$ dollars and a "bet buffer" $b$. Calculate the bet size $x=\lfloor\frac{n}{2^b-1}\rfloor$ dollars. Wager $x$ dollars on random variable $C$ that $C=1$ ...
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1answer
51 views

Show the following is Local Martingale

$X_t$ bessel square process which satisfies $$\mathop{dx_t}= 2(a+1) \mathop{dt} +2 \sqrt{x_t} \mathop{dB_t}$$ and $u$ is a function which satisfies $x^2 u'' +x u' -u(a^2 + b x^{2p+2})= 0$. How can I ...
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106 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 ...
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52 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 ...
4
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1answer
106 views

Karatzas and Shreve Problem 3.3.38

Let $X$ be a continuous process and $A$ a continuous, increasing process with $X_0 = A_0 = 0$, a.s. Suppose that for every $\theta \in \mathbb{R}$, the process $$Z_t^{\theta} = ...
2
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1answer
65 views

Prove $Y_n:=\sup|\hat{F}_n(x)-F(x)|$ is a reverse submartingale

Suppose $\{X_j,j \ge 1\}$ are iid with common distribution $F$ and let $\hat{F}_n$ be the empirical distribution based on $X_1,\dots,X_n$. Show $$Y_n:=\sup|\hat{F}_n(x)-F(x)|$$ is a reverse ...
0
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1answer
49 views

Example of a sequence of r.v.'s with constant stopping time that is not a Martingale

Could anybody give me a simple example of a sequence of random variables $(X_{n})_{n \geq 0}$ that has constant expectation, but is not a martingale?
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1answer
39 views

How $\Pi_{i=0}^n\xi_i$ converges a.s. to $0$ provided $\xi_n>0 $, iid and $E(\xi_n)=1$

suppose $\{\xi_n,n \ge 0\}$ are iid and positive random variables. $E(\xi_0)=1$. show $\Pi_{i=0}^n\xi_i$ is a positive martingale converging to $0$ provided $P[\xi_0=1]\not=1$ It's easy to prove ...
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1answer
96 views

Variance With Martingales Problem - Answered; Ignore the Bounty

Let $(X_{j})_{j \geq 1}$ be random variables such that $X_{j}$ is $\mathcal{F}$-measurable for each $j$, where $(F_{j})_{j\geq 1}$ is an increasing sequence of $\sigma$-algebras. Assume ...
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1answer
85 views

a problem about martingale, to prove $X_n \le E(X_\infty|B_n)$

Suppose $\{(X_n, B_n), n\ge0\}$ is an $L_1$-bounded martingale. If there exists an integrable random variable Y such that $X_n \le E(Y|B_n)$ then $X_n \le E(X_\infty|B_n)$ for all $n \ge0$ where ...
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87 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 ...
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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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1answer
53 views

Doob's inequality application

I'm working through an example of the application of Doob's inequality in Durrett: Let $X_m$ be a submartingale, and define $\bar{X}_n = \max\limits_{0 \leq m \leq n} X_m^+$. Let $\lambda ...
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2answers
91 views

Martingale Proofs

I havent been able to find an analogous question and our textbook is lacking in good examples, so I could use a little help with this rather straight forward martingale problem: Let X=(Xn) be a ...
0
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1answer
46 views

Probablity and Expected value

Suppose you are playing a fair coin game and you win a dollar if a head shows up and lose a dollar if tail. what is the expected value of rounds you played before you lose the first dollar from your ...
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1answer
78 views

how to prove $(X_{n})_{n\in \mathbb N}$ and $(Y_{n})_{n\in \mathbb N}$ are supermartingale.and $(Y_{n})_{n\in \mathbb N}$ is convergence to -7

Let $p \in [0 , \frac{1}{2}] $ and $\eta_{i}$ be i.i.d random variables and $P(\eta_{i}=1)=p$ and $P(\eta_{i}=-1)=1-p$ and $\mathcal F_{n}=\sigma(\eta_{1},\cdots,\eta_{n})$ and ...
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1answer
63 views

weaker version of the martingale convergence theorem

Let $\mathcal{A}_n$ be a sequence of finite sigma-algebras, let $\mathcal{B}_{q,p}= \sigma(\mathcal{A}_n, q \geq n \geq p )$. Moreover, we suppose $\mathcal{A}_k \subset \mathcal{B}_{\infty,p}$ for ...
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1answer
63 views

Conditional expectation by $\sigma (G_n,Y)$ when $Y$ is $G_\infty$-measurable

Let $G_n$ be a filtration (an increasing sequence of sigma-algebras), $Y$ a random variable that is $G_\infty$-measurable, and $X$ a random variable. Is it true that in $L^2$-norm, $$ \mathbb{E}[X| ...
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1answer
154 views

Conditional Expectation of martingale at stopping time

I am trying to understand the implications of the optimal stopping theorem, which is why I tought of the following problem. Consider the continuous-time Martingale $X = (X_t)_{t \geq 0}$ and the ...
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1answer
138 views

Application of Optional Sampling Theorem

Lets assume that Brownian Motion starts from some point $x$ for which $a<x<b$ holds. Let $\tau=\inf\{t:B_t\not\in [a,b]\}$ be a stopping time. Now I want to prove that for $\theta>0$ ,an ...
2
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1answer
68 views

Finding out if a sequence is martingale

I have a sequence of sequences defined as: $$Y[k] = \alpha \prod_{n=1}^kX[k]$$ Which I want to find an $\alpha$ for which it is martingale. I have that $$E[Y[k] \mid X[0],\dots,X[k-1]] = Y[k-1]$$ ...
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1answer
89 views

Quadratic Brownian Motion

If $B(t)$ is standard Brownian Motion then can we say that $B^2(t) -t $ is a martingale?? Given the following theorem: If $$ \max_{1<k<n} (t_k-t_{k-1}) \to 0$$ as $n \to \infty$ then ...
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1answer
51 views

Verify two martingale properties

I have a process $S_n=X_1+...+X_n$ where all the $X_i$ are iid and $E[X_1]=\mu, Var(X_1)=\sigma^2$ and $\phi(\theta)=Ee^{\theta X_1}$ Now I want to prove two things. (1) $S_k^2-\sigma^2 k$ for ...
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51 views

Canonical semimartigale truncation function meaning

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon: $H = B + H^c + h(x) \ast (\mu − \nu) + (x − h(x)) \ast μ$ where $h = h(x)$ is a truncation ...
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1answer
82 views

Stochastic differential for general semimartingale

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon: "$H = B + H^c + h(x) \ast (\mu − \nu) + (x − h(x)) \ast μ$ where $h = h(x)$ is a truncation ...
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0answers
43 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 $x_t$ ...
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2answers
148 views

Is the martingale stopping theorem applicable?

Let $Z_n=\prod_{i=1}^nX_i$, where $X_i,i\ge1$ are independent random variables with $$P\{X_i=2\}=P\{X_i=0\}=1/2.$$ Let $N=\min\{n:Z_n=0\}$. Is the martingale stopping theorem applicable? Here is my ...
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0answers
50 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 ...