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

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Modifying a Martingale Betting System for Different Odds

I understand how Martingale works for a roulette style system, but is it possible to modify the Martingale system to work fora game with different odds? Specifically if the odds of winning are 88% ...
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30 views

Expected stopping time for random process

Consider an infinite dimension vector $X$ where each $X_i \in \{-1,1\}$, $P(X_i = 1) = 1/2$, $P(X_i = -1) = 1/2$ and all are i.i.d. Now consider an $n$-dimensional vector $Y$ with $Y_i \in ...
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1answer
16 views

Markov process which is not martingale

I have seen the examples of a discrete time martingale that is not a Markov Process. Can you construct me an example of discrete time Markov Process that is not a martingale?
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1answer
23 views

Is the sum of two compensated poisson processes always a martingale?

Let $M^{1}_t=N^{1}_t-t\lambda^{1}$ and $M^{2}_t=N^{2}_t-t\lambda^{2}$ be two compensated poisson processes, where $\lambda^{1}$ and $\lambda^{2}$ are the constant intensities of $N^{1}_t$ and ...
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2answers
16 views

Question on Doob's martingale convergence theorem

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and $(\mathcal F_k)_{k\in\mathbb N}$ a filtration of $\mathcal F$ such that $\mathcal F=\sigma(\mathcal F_k\mid k\in\mathbb N).$ Let ...
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1answer
47 views

Martingales problem how to

I am unsure how to approach the following question. Given $\{X_1,X_2,...\}$ let $\displaystyle S_n=\sum_{i}^n X_i$ and $F_n=\sigma(X_1,...X_n)$. Suppose that for all $n\geq 1$, $\mathbb ...
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0answers
42 views

Is the martingale propertey preserved by taking weak$^*$-limits?

Let $(\Omega,\mathcal F)$ be a measurable space, $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable map. Let $(\mathcal F_k)_{k\in\mathbb N}$ be a filtration of $\mathcal F$ such that ...
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15 views

Show $W_{\infty}$ is not tail measurable but {$W_{\infty}=0$} is

The random walk: ({$\omega_{n}$},$\mathbb{P}$) simple random walk on d-dimensional integer lattice $\mathbb{Z}^{d}$ and the random environment: $\eta$={$\eta(n,x):n\in\mathbb{N}, x\in \mathbb{Z}^{d}$} ...
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1answer
23 views

arbitrage free price in martingale measures

Consider a one-period market with $S^1_t,\cdots,S^n_t$, with $t=0,1$ the price process of $n$ assets, where $S_1$ is a risk-free asset: $S^1_0=1$,$S^1_1=1+R$. Assumes that this market satisfies the ...
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26 views

For $M(t)$, a mean 0 martingale with $0=t_0<t_1<\dots<t_n=t$, show that $E[\sum_{i=1}^n(M(t_i)-M(t_{i-1}))^2]=E[(M(t)-M_0)^2]$.

I'm actually not sure where to even start with this one.. I've tried expanding the LHS and rearranging, but I'm not getting anywhere with that..
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1answer
48 views

Find the distribution of the maximum of a Wiener Process with negative drift

So.. what I have now is Let $M=max\{W_t; 0\leq t <\infty\}$ since $W_0=0$, $M\geq 0$ with probability 1. So, $P(M>x)=P(T_x<\infty)$ where $T_x$ is the stopping time, so we now use the ...
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1answer
36 views

Question on Martingales and Brownian Motion

I've run into an issue and I am unsure of how to proceed. In the text I'm working through, the following is left "as an exercise to the reader." Normally proofs listed as such tend to be fairly simple ...
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1answer
67 views

Levy's Martingale Using Radon Nikodym

Let P and Q be two probability measures on the same space $(\Omega,\mathcal{F},\mathcal{P})$ and let $\mathcal{F_n}$ be filtration. Assume that $Q \ll P$. Let $X_n$ denote the Radon-Nikodym derivative ...
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21 views

probability of supremum of martingale

Let $Z_{t}$ a continuous nonnegative martingale with $\lim Z_t=0 $ a.s. for every $s\geq0$ and $ b>0$ . show 1/ $ \textbf{P}(\sup_{t>s}Z_t \geq b\mid\textbf{F}_s) =\frac{1}{b} Z_s$ on ${Z_t ...
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Compute the quadratic variation

I want to know the quadratic variation of the square martingale bellow: $B_{t}^{2}-t$ And I want its proof as well. Thanks a lot!
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1answer
33 views

supremum and expectation of a martingale

Let $X_{t}$ a right continuous $\textbf{F}_{t}$ martingale and $\textbf{F}_{t}$ satisfying the usual condition Show that $ \sup_{t\geq 0}\textbf{E}(X^{2}_{t})<\infty$. I know that $X^{2}_{t}$ is ...
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1answer
60 views

Option Pricing, A Practitioners Guide, Martingale's, Drift Change and Radon-Nikodym

Im slightly confused about this section of the booklet regarding option prices byIain J. Clark. 1) Regarding the part of obtaining a martingale property we require that the last exponential term ...
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1answer
35 views

Ornstein-Uhlenbeck processs: Markov, but not martingale?

I'm puzzled about properties of the Ornstein-Uhlenbeck process, given by the Itō integral $$ X_t = x e^{-\lambda t} + \sigma \int_0^t e^{-\lambda(t-s)} d W_s \,. $$ I compute that $\{X_t\}$ is not ...
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1answer
36 views

question about martingale

In my lecture notes,I found the following problem: Let $X$ an $F_{t}$ adapted continuous process and $G_{t}\subset F_{t}$. show that $$E\left(\left. \int^{t}_{0}X_{s}ds ...
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107 views

understanding submartingale proof with discrete state space

I am reading a text about branching markov chains: My question is about the first half of page 8 where $Q(t)$ is proven to be a submartingale. Briefly the used notation: $t$ is discrete time, $n(t)$ ...
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1answer
23 views

Expectations of martingales

Consider a martingale $(M_n)_{n \geq 0}$ adapted to a filtration $(\mathcal F)_{n \geq 0}$ on a probability space $(\Omega, \mathcal F, P)$. Prove that, for each $k \leq n$; $$E(M_n M_k) = E(M_k^2)$$ ...
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51 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 ...
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38 views

Show that $E[Z_n^2]= \sum_{i=1}^n E[(Z_i-Z_{i-1})^2] $ for a martingale with $Z_0=0$

I was just wondering, if we let $(Z_n)_{n\geq 0}$be a martingale with $Z_0=0$, is it true then $$ E[Z_n^2]= \sum_{i=1}^n E[(Z_i-Z_{i-1})^2] $$ Please let me know and if it is true, can someone show ...
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33 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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1answer
19 views

A bound on the maximum of a submartingale

I'm trying to prove the following: If $\{(S_i, \mathcal{F}_i) \mid i \in [N]\}$ is a nonnegative submartingale with $\mathbb{E}S_i^p < \infty$, let $M = \max_i S_i$. Then $\|M\|_p \leq q ...
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15 views

Looking for hints on how to prove the convergence of this iterative estimator! [duplicate]

Let $X_n$ be a Poisson process with mean $\lambda^*$. The following sequence estimates the parameter of the Poisson process: $ X_{n+1} = \hat{\lambda}_{n+1} + ...
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1answer
24 views

Stochastic Integrals and Martingales

I am attempting the following proof but two aspects of the solution confuse me: Given \begin{align} I^{n}_{t} = \int^t_0 \Delta_u^ndW_u = \sum_{j=0}^{k-1}\Delta_{t_{j}}(W_{t_{j+1}}-W_{t_{j}}) + ...
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39 views

Martingales and Integrals

Could someone explain why the following is a Martingale please? \begin{align} M_s = \int_0^s(1+u^2)dW_u \end{align} (where $W_t$ is standard Brownian motion). I'm used to determining martingales ...
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38 views

Martingale and Stochastic equation

Using the Ito formula, I can show that the martingale $$ Z_{t}=\frac{1}{\sqrt{1-t}}\exp -\frac{B_t^2}{2(1-t)}\qquad 0\leq t<1 $$ admits the following differential $$ dZ_t=-\frac{B_t}{1-t}Z_tdB_t. ...
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71 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 ...
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52 views

$4^{Brownian(t)}$ martingale proof

Let $B(t)$ a Brownian motion. I like to prove that $4^{B(t)}$ = martingale I rewrote the expression into an exponential form (like $\exp(\ln(4) B)$), but then I don't know how to proceed.
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1answer
50 views

Transforming semimartingale to local martingale by change of measure

Consider a continuous $\mathbb{P}$ - semimartingale X which can be decomposed as M+A (M is local martingale and A is bounded variation process). Is it possible to change measure to $\mathbb{Q}$ s.t. ...
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1answer
28 views

Is any FV-Process a special Semimartingale?

Any FV-Process can be represented as the difference of two increasing (or decreasing) processes and so any FV-Process is a quasimartingale. Due to Raos Theorem any FV-Process is a special ...
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1answer
11 views

Expectation of a stochastic integral conditioned on a particular σ-algebra

Suppose that $g$ is a simple process in the class $\mathcal{V}=\mathcal{V}[U,T]$. Using the notations $g_k=g(t_k)$, $\Delta B_k = B(t_{k+1})-B(t_k)$, and $\mathcal{F}_k=\mathcal{F}_{t_k}$, with the ...
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2answers
79 views

Normal variables, uniform integrability and limit of a martingale

I'm stuck in the following problem! Would be great if you can help. Suppose you have $(Y_n)_{n\in \mathbb N}$, a sequence of independent and identically distributed random variables, $Y_1 \sim ...
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99 views

Upper bound for mean hitting time of two-dimensional random walk

Let $U_n=\sum_{i=1}^n X_i,V_n=\sum_{i=1}^n Y_i$, $n\geq 1$, be a two-dimensional random walk with iid increments $(X_n, Y_n)$, where $X_n, Y_n$ are discrete random variables with joint pmf ...
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1answer
33 views

Squared Poisson Martingale

I know that $M_t=N_t-\lambda t$ is a martingale for $N_t$ a rate $\lambda$ poisson process and that for a brownian motion, $B_t^2-t$ is a martingale. I'm wondering, is there something similar for ...
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43 views

Finding dynamics of a dividend paying stock under arbitrary numeraire

Assuming I have a dividend paying asset $S$ with dividend process $D$. Now I would like to use the bank account process $B$ as numeraire and determine the dynamics of $S$ under the the corresponding ...
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47 views

Is $8^{W(t)}$ a martingale?

I have a standard Wiener process: $W(t)$ I need to determine if the following is a Martingale: $8^{W(t)}$ I know the two conditions for a Martingale; that the expected value of the absolute value ...
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1answer
36 views

Prove that a process is a martingale

Let $W_t$ be a Wiener process, and let $N_t$ be a Poisson process with intensity $\lambda$. We define a process $Z_t = \lambda Wt^2 − N_t$ Prove that the process $Z_t$ is a martingale
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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( ...
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1answer
69 views

Is this process a martingale

I was solving some practice problems in stochastics and faced the following exercise: Given Brownian motion $W(t)$ and a stochastic process $B(t)$ defined as: $$B(t) = \begin{cases} W(t), & ...
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1answer
62 views

Probabilistic Proof That An Absolutely Continuous Function is Differentiable Almost Everywhere

Consider the probability space $([0,1), \mathcal{B}, \lambda)$ where $\mathcal{B}$ is the Borel $\sigma$-algebra and $\lambda$ is the uniform measure. Let $A_{i,n} = [(i-1)2^{-n}, i2^{-n})$ for $i \in ...
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1answer
38 views

Optional Sampling a.s. finite stopping time

Given a uniformly integrable discrete martingale $M_n$ on prob. space $(\Omega, \mathcal{F}, \mathbb{P})$, and a.s. finite stopping times $T$ and $S$ with $T\geq S$. Show that $E[M_T|\mathcal{F}_S] = ...
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75 views

Expectation of a stochastic integral

Let $M$ be a right-continuous local martingale, $s,t$ two times (stopping times, if you like). Under what conditions does the following hold: $$E\left(\int_s^t X \, dM\mid\mathcal{F}_s\right)\le 0$$ ...
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1answer
90 views

Monkey typing ABRACADABRA and gamblers

Problem: A monkey is sitting at a typewriter, typing a letter (A-Z) independently and with uniform distribution each minute. What is the expected amount of time that passes before ABRACADABRA is ...
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2answers
48 views

Conditional expectation as Borel function

Let $X,Y$ be random variables with $E|X|< \infty$. Prove that there is a Borel function $h:\mathbb{R}\rightarrow \mathbb{R}$ such that $E[X|\sigma(Y)]=h(Y)$ almost surely. (Here $\sigma(Y)$ is ...
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1answer
57 views

Find a function f(t) such that Y is a martingale

Let $(X_t)$ be a process with independent increments such that $X_0=0$ and $E(X_t)=0$ Let $F_t$ be a natural filtration of $X_t$ Let $a$ and $b$ be arbitrary real numbers and let $(Y_t)$ be a random ...
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74 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 ...
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79 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 ...