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

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32 views

Show that a martingale is not $L^1$ convergent

Consider the symmetric random walk $S_n$ on $\mathbb{Z}$. The process $Z_n=\exp(uS_n-n \ \log(\cosh(u)))$ for $u\in \mathbb{R}$ is a positive martingale with $E(Z_n)=1$ for all $n\geq 1$. $Z_n$ is ...
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35 views

Convergence in probability of a martingale difference sequence

I am trying to prove that $$\sum_0^T \mathbf x_t \varepsilon_t \overset {p}{\rightarrow} 0 $$ In this case $\ \mathbf x_t \varepsilon_t $ is a martingale difference sequence, $\varepsilon_t$ is ...
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1answer
21 views

$L^1$ convergence of a martingale, conditional expectation

I am trying to prove the following: Let our space be $(\Omega, \mathcal{F}, P, \{ \mathcal{F}_n \}_{n \in \mathbb{N}})$. Let $\{X_n \}_{n \in \mathbb{N}}$ be a martingale (adapted to the filtration ...
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25 views

Proving almost sure finiteness of a stopping time

Let $M_t$ be a local martingale and $S_t = \sup_{0 \leq s \leq t}M_s$ its running supremum. How can I show that the stopping time $T=\inf\{t \geq 0 : S_t - M_t = a\}$ for an arbitrary $a>0$ is ...
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26 views

Quadratic covariation of a jump process and a continuous process of finite variation always zero?

Suppose $N_t$ is a Poisson process (so $\langle N \rangle_t=N_t$, its quadratic variation is itself), and $X_t$ is a continuous process of finite variation (so $\langle X \rangle_t=0$). Then does it ...
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17 views

If $X_n$ is a martingale wrt to $F_n$, is $X_n^3$ also a martingale? proof? [duplicate]

I am having problem with this exercise question before my exam. If $X_n$ is a martingale w.r.t sigma-field $F_n$ then is $X_n^3$ also a martingale w.r.t $F_n$? This far my approach has been to try ...
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23 views

Laplace transform of the square of Brownian motion hitting time

Let $B_{\mu}(t)$ be a one-dimensional Brownian motion with drift $\mu \geq 0.$ For $a > 0,$ let $$T_a = \inf\{t > 0: B_{\mu}(t) = a\}$$ denote the first hitting time of $B.$ The Laplace ...
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49 views

Prove a thm on stopped processes given fundamental principle 'you can't beat the system'?

How does the principle below imply the thm below? From Williams' Probability w/ Martingales: Principle: Thm: What I tried: $$E[X_{T \wedge n} - X_0 | \mathscr{F_m}] =/ \le X_{T \wedge ...
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1answer
187 views

Polya's urn (martingale)

Suppose you have an urn containing one red ball and one green ball. You draw one at random; if the ball is red, put it back in the urn with an additional red ball , otherwise put it back and add a ...
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44 views

martingale problem with moment generating function

Consider iid $X_1,X_2,...N(\mu,\sigma^2)$ and let $M_n=e^{tS_n-\lambda n}$, $S_n=\sum_{i=1}^{n} X_i$. For a given $\mu,\sigma^2,t$, find the value of $\lambda$ for which $M_n$ is a martingale, ...
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85 views

Risk of ruin with tweaked Martingale

I want to calculate the expected value of using the Martingale system. But apparently I'm not clever enough to understand the complex equations, why I turn to you. Let's say that my bankroll is 8000 ...
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13 views

Martingales and Measures on forward contracts - Hull

I'm an italian student of finance, and right now I'm studying interest rate derivatives. At page 636 (8th Edition) of "Options, Futures and Other Derivatives" - Hull, the author assumes that we have ...
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1answer
49 views

Expectation of supremum of a submartingale

I have a probability space $(\mathbb{P}, \Omega, \mathcal{F})$ and in this space, I have a submartingale $(X_n)_n$ with the following two properties: $\inf_n X_n < 0$ $\mathbb{E}[X_0] \geq 0$ ...
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1answer
36 views

Find the probability $p$ such the random variables is a sub martingale and super martingale?

Consider $X_1,X_2,...,X_n$ the i.i.d. Bernoulli random variables, with $P(X_i = 1) = p$ and $P(X_i = 0) = q$. Take the process $S=(S_n)_{n\geq 0}$ where $S_n= X_1 +...+ X_n$ for $n = 1,2,\dots$ and ...
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18 views

First passage time of the Brownian motion

In an exercise (4.1 Krapinsky, "A kinetic view of statistical physics") I am asked to show that: The probability that a brownian motion on a 1D discrete lattice never reaches the site $n$ scales as ...
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19 views

semimartingale and limit

Let $X_t=X_0+M_t+A_t$ a continuous semi-martingale. Let $g: \Bbb R \to [-1,1]$, of class $C^{\infty}$, with $g(x)= \left\{ \begin{matrix} -1, & x \le 0 \\ 1, & x \ge 1 \end{matrix} \right.$. ...
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1answer
32 views

Estimate the typical numeber of times a brownian motion on ℤ starting from $0$ does a particular transition

Consider an 1D infinite lattice. The lattice is fully occupied except from a vacancy in the origin which undergoes simple diffusion (in countinuous time). At position $n>0$ in the lattice there is ...
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27 views

Calculating Expectation of Exponential with Indicator Functions (Continuous Time Martingales)

Update: So it turns out that the question I was given is actually wrong! $X_t$ should be defined with $\{T>t\}$ not $\{T<t\}$. My question is the following. It looks really easy, and so I ...
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1answer
59 views

$X_T = \lim_{n \to \infty} X_{T \wedge n}$ if X is a supermartingale and T is a finite a.s. stopping time?

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}, \mathbb{P})$, let $X = (X_n)_{n \geq 0}$ be a $(\{\mathscr{F_n}\}, \mathbb{P})$-supermartingale and $T$ be a finite ...
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19 views

Continuous semimartingale and limit

Let $X_t=X_0+M_t+A_t$ a continuous semi-martingale. Let $g: \Bbb R \to [-1,1]$, of class $C^{\infty}$, with $g(x)= \left\{ \begin{matrix} -1, & x \le 0 \\ 1, & x \ge 1 \end{matrix} \right.$. ...
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2answers
74 views

Expected number of tosses until 3 heads in a row - via Martingale method

(Quant job interviews - questions and answers - Question 3.8) For a fair coin, what is the expected number of tosses to get 3 heads in a row The answer is stated as : We gamble in such a way ...
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77 views

Prove $X_T$ is integrable if $X$ is a supermartingale, $T$ is stopping time and other conditions

Let $X = ({X_n})_{n \ge 1}$ be a/an $(\{\mathscr{F_n}\}_{n \ge 1}, \mathbb{P})$-supermartingale in the filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}_{n \ge 1}, \mathbb{P})$. ...
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74 views

Prove $Y_S$ is integrable if $Y$ is a bounded supermartingale and $S$ is an a.s. finite stopping time. [closed]

Let $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}_{n \in \mathbb{N}}, \mathbb{P})$ be a filtered probability space, and let $Y = ({Y_n})_{n \in \mathbb{N}}$ be a/an $(\{\mathscr{F_n}\}_{n \in \mathbb{N}}, ...
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1answer
50 views

Prove $Y_S$ is integrable if $Y$ is a supermartingale and $S$ is a bounded stopping time.

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}_{n \in \mathbb{N}}, \mathbb{P})$, let $Y = ({Y_n})_{n \in \mathbb{N}}$ be a/an $(\{\mathscr{F_n}\}_{n \in \mathbb{N}}, ...
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63 views

Is $X_t = tW\left(\frac{1}{t}\right)$ a Martingale?If not, how could it be a Brownian Motion?

As is proved, $X_t = tW\left(\frac{1}{t}\right)$ is a Brownian motion. For example see Theorem 4.2 in this paper http://math.uchicago.edu/~may/REU2012/REUPapers/Leiner.pdf I'm just confused because ...
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35 views

Does a local martingale has to be integrable?

The question seems very simple. Now, for non negative supermartingales, I can just say [we will call $X_t$ our (sub/super)martingale]: $$E[|X_t|1(\tau_N>t)]\le E[|X_{\tau_N\wedge t}|]\le ...
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34 views

Playing roulette using martingale

A player with unlimited money decides to play roulette. He bets $1$ on red, if he loses, he bets $2$, if he loses again he bets $4$ and so on till he wins. Prove that he is guaranteed to make a ...
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1answer
11 views

Confused regarding this notation for scalar Xt?

In the above, we are told that both |Xt$\preceq$| $\preceq$ |Xt| and |Xt$\succ$| $\preceq$ |Xt|. Would appreciate clarification on 1) why this is the case, and 2) what the superscripts for ...
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25 views

Number of bins in best packing of bin packing problem is Martingale

I have items $a_1,..., a_n$ with $0\leq a_i\leq 1$ and $1\leq i\leq n$, where $a_i$ is chosen independently and the capacity of each bin is 1. I want to be able to apply the Hoeffding inequality to ...
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32 views

Conditional expectation of the integral of a martingale

I'm interested in the following problem because of its relevance to the pricing of Asian-type options when the average is arithmetic and in continuous time: If $(X_t,\mathcal{F}_t)_{0\leq t\leq T}$ ...
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1answer
71 views

Questions on Doob's Optional Stopping Theorem (a) and (b)

From Williams' Probability w/ Martingales: What is $X_T$ in red box above? I am fairly certain this was not defined previously in the book. There was this though: I have a feeling $X_T = ...
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26 views

Martingale expectation value

I want to show that for $(B_t)_t$ being the Brownian motion and a stopping time $\tau:= \text{inf}_{t \ge 0} \{B_t= a+bt\}$ where $a,b>0$ we have that the expectation value $E(e^{-\lambda \tau}, ...
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1answer
20 views

Independence of $X$ and $Y_1$ and independence of $X$ and $Y_2$ implies independence of $\sigma\{X\}$ and $\sigma\{Y_1,Y_2\}$?

$X,Y_1,Y_2$ are random variables. Suppose $X$ and $Y_1$ are independent, and $X$ and $Y_2$ are independent. Then by definition we have: $$Pr[X \leq x, Y_1\leq y_1] = Pr[X \leq x]Pr[Y_1\leq y_1],$$ ...
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1answer
177 views

Why is integrability needed in fundamental principle 'you can't beat the system'?

From Williams' Probability w/ Martingales: Re (iii), why do we need square integrability? I mean, why is integrability not good enough? Based on an answer in my previous question, I think ...
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54 views

Limiting sequence of exponential random variables

Let $\eta_k$ be i.i.d. random variables having an exponential distribution, $$F_\lambda(x) = P(\eta_k \leq x) = 1-e^{-\lambda x}$$ for $x \geq 0$. Consider a sequence $\xi_k = ...
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16 views

Optional Sampling Theorem - Martingales

I have problems with solving the following problem. Can anyone give me a hint how to solve it? Thanks in advance! Consider a contract that at time N will be worth either 100 or 0: Let S(n) be its ...
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$E(X_n)=E(X_{n-1})=\ldots=E(X_0)\nRightarrow (X_n)_n$ is Martingale

I am searching for an example of an adapted process $(X_n)_n$ with constant expected value which is not an martingale (I know that the reverse direction holds)
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42 views

Convergence of a stopped martingale

I have some problems with the following exercise and I need some hints (no complete solutions please) to solve it. Let $(N_t)_{t\geq 0}$ be a Poisson($\lambda$) process and $c>\lambda$. Now let ...
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51 views

Solving Stochastic Differential Equation

Let $\beta > 0$, $0 < \gamma < 1$, and let $\tau$ be the first hitting time: $$\tau = \inf\{t:t \geq 0, |W_t| = \pi /4\}$$ Solve the SDE in the random interval $0 \leq t \leq \tau$ $$dX_t = ...
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1answer
45 views

Augmented filtration martingale proof.

Part a: Consider a Wiener process, $W_t$ and denote by ${\mathscr{F}_t}_{(t \geq 0)}$ the natural filtration generated by W. Let $\mathbb{R}_{+} = \{x : x \geq 0\}$ and $\mathscr{B}$ be a sigma ...
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27 views

Stopping time distribution and transforms with 1-dimension B-motion.

Let $W_t$ be a 1-dimensional Brownian Motion. For $x>0$, we define: $$\tau_{x} = inf \{ t \geq 0; |W_t| = x\}$$ Compute $E[e^{-s\tau_x}]$ and prove that $\tau_x$ is equal in distribution to ...
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19 views

Is $\int_s^t {B_u}^2 du$ independent of $ F_s$

In order to prove that $(B_t^4)_{t \in \mathbb{R+}}$ is a continuous semi-martingale I need to compute. $E(\int_s^t B_u^2 du |F_s)$ I was first thinking that $\int_s^t B_u^2 du$ independent of $ ...
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35 views

Nonnegative martingale stays zero once it hits zero

I am not sure if my proof of the following statement is correct. Let $(M_t)_{t\geq 0}$ be a nonnegative càdlàg martingale (on $(\Omega, \mathcal{F}, P, (\mathcal{F_t})_{t\geq 0}$ ) and ...
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29 views

How to apply Ito's Lemma to this problem?

We have the martingale representation theorem: $G = E[G]+\int_0^t\theta_sdW_s$ Now, given $G=1_A$, where $A=\{exp(W_t)>K\}$, how to find the corresponding $\theta_t$? The hint I received was to ...
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1answer
116 views

Problem 3.24 of “Brownian Motion & Stochastic Processes” by Karatzas and Shreve - Submartingales and stopping times

I'm doing the problem 3.24 of Brownian Motion and Stochastic Processes by Karatzas and Shreve. There is two specific parts troubling me, I need some help to see what to do. Here is the problem: ...
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31 views

How does the sample space remain constant in filtered

I am trying to understand the concept of filtered event space from the axiomatic probability. From my reference (lecture script by Ramon Handel, Princeton) the filtered probability space looks like ...
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1answer
177 views

Extension of Dynkin's formula, conclude that process is a martingale.

Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial derivatives are uniformly bounded (or, more generally, have at most polynomial growth as $|x| \to ...
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1answer
200 views

Prove Z is a martingale by defining it is a product of random variables

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}, \mathbb{P})$ where $\mathscr{F_n} = \mathscr{F_n}^{Z} \doteq \sigma(Z_0, Z_1, \ldots, Z_n)$, show that $Z = (Z_n)_{n \geq ...
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1answer
34 views

Obtaining martingales from Poisson process

All processes here are continuous. Suppose we have a Poisson process $(N_t)_{t\geq 0}$ with parameter $\lambda > 0$ and adapted to the filtration $(\mathcal{F}_t)_{t\geq 0}$. Fix $u\in\mathbb{C}$, ...
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1answer
46 views

Karatzas and Shreve - Problem 3.3.19

I'm struggling here to solve this problem, but with no success. I was able to prove a $\implies$ b, but the next implication is troubling me. In the book, they give a solution, but I think there ...