Questions tagged [martingales]

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

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5 views

Find the distribution for a stopped brownian motion

Let $W_t$ be a one-dimensional brownian motion. For $c>0$ define $\tau_c$ as the first time the brownian motion reaches the level c, $\tau_c(\omega)=\inf\{t \geq 0: W_t(\omega)=c \}$. Show that $\...
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a.s convergence on an event $E$

Consider a probability space $(\Omega,\mathcal{F},P),$ and sequence $(X_k)_k.$ What does it mean that $X_k$ converges a.s to a finite limit, on an event $E$ ? Is there a rigorous definition for this ...
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Is $Y_n=S_n^3-3nS_n$ a martingale?

Let $X_i$ for $i=1,\cdots,n$ random variables i.i.d., such that $P(X_1=1)=\frac{1}{2}$ and $P(X_1=-1)=\frac{1}{2}$. Let $S_n=X_1+\cdots+X_n$ Let $Y_n=S_n^3-3nS_n$, is $Y_n$ a martingale or isn't? ...
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Will the new discounted process also be a martingale?

Assume you have a filtered probability space $(\Omega, \mathcal{A},P,\mathcal{F}_t)$. Assume there is a Brownian motion $B_t$ defined on the probability space. You model a bank process $R_t$, by the ...
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1answer
33 views

A doubt involving conditional expectation.

Let $M=(M_i)_{i \in \mathbb{Z}}$ a martingale with respect to natural filtration $\mathcal{F}_{i}=\sigma(M_j; j \leq i)$. Defines $X_{i}=M_{i}-M_{i-1}$ and, for fixed $n \in \mathbb{N}$, \begin{...
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a.s convergence and well defining a processes

$(X_k)_k,(Y_k)_k,(W_k)_k$ are three $(\mathcal{F}_k)$-adapted processes taking values in $\mathbb{R}^+$ such that for all $k,$ $$E[X_{k+1}|\mathcal{F}_k] \leq X_k+Y_k-W_k \ \ \ \ \ \ \ \ (1)$$ We want ...
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local martingale?

Consider the ito-process: $X(t)= \sigma(t) dW(t)$ for $t \in [0,T]$, where $\sigma$ is a predictable process and $\int_{0}^T \sigma^2 dt <\infty$. Consider $X^2(t)$: Appyling Ito's formula to $f(x)=...
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Two exponential processes are martingales w.r.t the same measure?

Consider a filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F_t}\},P)$. Define a process $$M_t=\exp\left(\int_0^t \lambda_sW_s-\frac{1}{2}\int_0^t \lambda_s^2ds\right)$$ where $W_t$ is a $P-$...
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Quadratic variation condition and square integrability

Assume $M$ is local continuous martingale started from $0$. How would one go about showing that if it is a martingale bounded in $L^2$, then $E[\langle M\rangle_{\infty}]\lt \infty $?
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Stochastic process problem solve with R studio [closed]

I want to find a task within stochastic processes (martingales, SBM, stochastic integral, and so on), which I can solve on paper and in Rstudio. Can you suggest something with a solution? Thanks in ...
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Give a complete list of the functions of Brownian motion alone, g(Wt), which are martingales [closed]

As it says in the title. Confused by the wording really, thanks.
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28 views

construct a martingale from given conditions

If there exists a sequence of non-negative real numbers $a_n$ with $\sum_{n=1}^{\infty} a_n <\infty$ with $$\mathbb E[X_n\mid \mathcal F_{n-1}]\leq X_{n-1}+a_n,$$ prove $X_n$ converges almost ...
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Show that the following are Martingale. [closed]

Having some trobule with the following problem. I think I need to expand W(t)= W(t)-W(s)+W(s) Show that the following are Martingale. 1.E[|X(t)|]< ∞ 2.E[X(t)|Fs]=X(s) W(t) is the standard Brownian ...
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Right continuity of the potential

Let $X=(X_t)_{t \geq 0}$ be a Hunt process with transition semigroup $(P_t)_{t \geq 0}$. We define for $\alpha >0$ and a bounded Borel-function $f$ the potential $U^{\alpha}f(x)=\int^{\infty}_0 e^{-...
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$E\big[(X_{V_i}-X_{\xi_i})^2(X_{V_j}-X_{\xi_j})^2\big]=0$? [closed]

Good day! As a student who want to learn about stochastic. I encounter an existing term on manipulation involving product of sums with finite index. Assumptions: $X$ right continuous bounded $L_2$ ...
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1answer
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Symmetric random walk and martingales

Consider a symmetric random walk, where $S_n = X_1 + \cdots + X_n$ and $X_i$ are IID with probability $0.5$ of being $1$ or $-1$. I want to show that $S_{n}^2 - (n)$ is a martingale. So I started with:...
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Properties of the stochastic integral $\int_0^t\frac{|B_s|}{s}\Bbb dB_s$

For a Brownian motion $(B_t)_{t\geq0}$ we define the process $$I_t:=\int_0^t\frac{|B_s|}{s}\Bbb dB_s,\qquad (t>0).$$ But what can we say about this process? Is it even possible to define this ...
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59 views

No universal upper bound for $\mathbb{E}[\text{max}_{k \leq n}X_k]$ for nonnegatve submartingale [closed]

I want to see that there is no universal constant $C$ such that for every non-negative martingale $(X_n)_{n \geq 0}$, we have that $\mathbb{E}[\text{max}_{k \leq n}X_k] \leq C\mathbb{E}[X_n]$. I've ...
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Dynamic Programming First Order Condition with Martingale State Variable

Consider the following problem coming from an economic framework where an individual takes an action $a_t$ every period given a state variable $\mu_t$: $$ V_t(\mu_t) = \max_{a_t} \left\{ g(a_t) + \...
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Good books on Advanced Probability Theory with exercises to work with

I am interested in advanced Probability Theory books which also do have lots of exercises to work with, in the topics of convergence of random variables, conditional expectation (martingales in ...
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Why is it that we use Ito's Formula to show a process is a martingale

I am very new to stochastic processes, and one of the most common questions I see related to this topic is something along the lines of: Show that the process $M_T$ = {some function of a Brownian ...
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How to show that $B(t) - \mu t$ is a martingale

How is it possible to show that $B(t) - \mu t$ is a martingale. We of course want to show that $E[B(t) - \mu t \,|\, \mathcal{B}_s] = B(s)$, and I'm sure we'd do this using the fact that Brownian ...
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slick and quick proof martingale strategy even with added salary leads to bankruptcy

Assume James the employee initially has $X$ doolars in his bank and each day he makes $S$ doolars in his job. He then wants to make a supplemental bonus income of $B$ doolars every day, to do this he ...
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General notion and convergence of reversed submartingale

$(X_k)_k$ is a reversed submartingale (resp. supermartingale, martingale) relative to $(\mathcal{F}_k)_k$ (for all $k \in \mathbb{N},\mathcal{F}_{k+1} \subset \mathcal{F}_k$) if for all $k \in \mathbb{...
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Ito's lemma characterises martingale-ness?

Let $X_t = M_t+V_t$, where $M$ is a martingale. By Ito's lemma we have $$ df(X_t) = f'(X_t)dX_t + \frac{1}{2}f''(X_t)d\langle M \rangle _t $$ So, if on the RHS only the the term $\ .. dM_t$ remains, ...
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Why is this a martingale (SDE)?

I am reading a paper on the Fokker-Planck equation and this comes up. Let us consider the SDE in $\mathbb{R}^n$ \begin{equation} dX_t = b(X_t,t)dt + \sqrt{2}dB_t, \qquad t\in [0,T],\\ X_0 = x_0 \end{...
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Exponential of martingale is also martingale?

Let $X_t$ be a standard Brownian. Put $Y_t := \exp (\int _0^t udX_u)$ and determine, whether $Y$ is a martingale. We know $M_t :=\int _0^t u dX_u$ is a martingale, therefore $\langle M \rangle _t = \...
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Probability of Ruin Assumption [closed]

I was wondering if someone could explain why we take the following assumption in Insurance Mathematics (Probability of Ruin in Insurance): $E[e^{-R(c-x_1)}]=1$ and $(1-x)e^{cx}=1$ Where; c = premiums ...
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1answer
59 views

$E[|X|\ln(1+|X|)]<+\infty \implies \lim_k E[X|\mathcal{G}_k]=0$ a.s

$X$ is a random variable such that $E[X]=0,$ $E[|X|\ln(1+|X|)]<+\infty.$ If $(\mathcal{G}_k)_k$ is a sequence of independent $\sigma$-algebras of events, then prove that $E[X|\mathcal{G}_k]$ ...
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19 views

Proof finite martingale property

If I'm Trying to prove $M_n=2^{n}e^{-X_n}$ is Martingale (I've already proven Property Two ie. $E[M_{n+1}\mid F_n]=M_n$. With Point 1: $E[|2^{n}e^{-X_n}|]$ If I say $|2^{n}|e^{-X_0}E[|e^{-Y_1}|]< \...
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Make a sequence into a martingale

I want to verify my answer to the following question. Let $\{Y_n\}_{n=1}^\infty$ be i.i.d. random variables. If $Y_i \in L^3(\Omega)$, how can $\{(\sum_{k=1}^n Y_k)^3\}_{n=1}^\infty$ be made into a ...
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42 views

Exponential Martingale Problem

I'm trying to prove $M_n = 2^{n}e^{-X_n}$ is a Martingale where $X_{n+1} = X_n + Y_{n+1}$ (Random Walk) My initial thoughts were along the line of: $$E[2^{n+1}e^{-(X_n+Y_{n+1})}|F_n]$$ $$2^{n+1}E[e^{-(...
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1answer
79 views

martingale stopping time is same as total number of events?

$X:=\{X_n\}_{n\ge 1}$ is a sequence of i.i.d random variables with \begin{align*} \mathbb P\left(X_1=1\right)=p\in (0,1)\setminus \{\frac{1}{2}\} \text{ and } \mathbb P\left(X_1=-1\right)=q=(1-p) \...
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1answer
85 views

Prove that a stopped martingale is a martingale.

Let $(M_t)$ be a martingale w.r.t. $(\mathcal F_t)_{t\geq 0}$ and $\tau$ is a stopping time. I must prove that $(M_{t\wedge \tau})$ is a martingale. What should I prove ? That for $s<t$ $$\mathbb E[...
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Requirements of optional stopping theorem in random walk

Consider the iid random variables $X_1,X_2,...$ with $\mathbb{E}[X_i] = \mu$. Let $S_n = X_1+...+X_n$ be a random walk (it could be symmetric/unsymmetric in 1D or higher dimension, but I use the ...
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$L^p$-convergence of $E[X|\mathcal{F}_k],p \in ]0,1[ \cup \{\infty\}$

Let $p \in ]0,\infty],X \in L^p, (\mathcal{F}_k)_k$ a filtration, define $\mathcal{F}_{\infty}=\sigma(\bigcup_{k \in \mathbb{N}}\mathcal{F}_k).$ If $p \in [1,\infty[,$ this implies that $X \in L^1,$ ...
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Prove that $M_t=\left(\int_0^t f(s)dB_s\right)^2-\int_0^t f(s)^2ds$ is a martingale.

Let $f:\Omega \times [0,\infty )\to \mathbb R$ progressively measurable and s.t. $\mathbb E\int_0^t f(s)^2ds<\infty $ for all $t$. I would like to prove that $(M_t)_{t\geq 0}$ is a martingale where ...
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1answer
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Uncorrelated Martingales

I'm not sure how to prove that Mn+1 - Mn & Mn are uncorrelated ? If Mn = (Xn)^2 - 2nXn + n(n − 1); Where Xn is a Random walk Xn+1 = Xn +Yn+1 Where Yn - N(1,1) I already know this is a Martingale, ...
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General form of Hunt lemma

Consider on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_n)_n,P),$ a sequence of random variable $(X_n)_n$ converging a.s to $X.$ $(Y_n)_n$ is a sequence in $L^1,$ such that $(Y_n)_n$...
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1answer
65 views

Prove Martingale Property

I want to show that $M_n = (X_n)^2 -2nX_n +n(n-1)$ is a Martingale. I know that $\{X_n\}_{n\in\mathbb N}$ is a random walk process such that $X_{n+1} = X_n +Y_n$ and $\{Y_n\}$ is a sequence of i.i.d ...
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Proving the Borell-Cantelli Lemma by martingale convergence theorem

I got stuck on an exercise (Exercise 5.2.1) from the book: Ergodic Theory: with view towards Number Theory By Manfred Leopold Einsiedler and Thomas Ward. In the book they presented the following ...
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1answer
31 views

Prove that $(M_n)_n$ is a Martingale

Let $\left(S_{n}\right)_{n}$ be the simple symmetric random walk with $S_{0}=0 .$ Define $$ M_{n}=\left|S_{n}\right|-\sum_{k=0}^{n-1} 1_{S_{k}=0}, \quad \text { for } n \in \mathbb{N} $$ Let $\mathcal{...
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$\lambda P( \min_{1 \leq k \leq n}X_k\leq -\lambda) \leq \int_{\{\min_{1 \leq k \leq n}X_k> -\lambda\}}X_ndP - E[X_0]$

Let $n \in \mathbb{N}^*.$ $(X_k)_k$ is a submartingale, $\lambda>0,$ by considering, for $1 \leq k \leq n, E_k=\bigcap_{r=1}^{k-1}\{X_r >-\lambda\}\cap\{X_k \leq -\lambda\},$ prove that $$\...
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1answer
28 views

$E(X(T)\mid X(s),s\leq t)=X(t)\xi(t)$. What is $X$?

Suppose $X(t)$ is some stochastic process. We know that if $X(t)$ is a martingale, then $$E(X(T)\mid X(s),s\leq t)=X(t)$$ for all $t\geq 0.$ However, suppose $X$ is not a martingale, but it satisfies $...
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2answers
38 views

Showing a stochastic process is a (discrete) martingale

Given a sequence of iid random variables $(Y_i)_{i=1}^\infty$ on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that $\mathbb{E}|Y_i| < \infty$ and $\mathbb{E}Y_i = 0$, consider the ...
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47 views

Stopping time - theory of Martingales

Q Let $\left(X_{i}\right)_{i}$ be i.i.d. bernoulli random variables. More precisely $\mathbb{P}(X=0)=\mathbb{P}(X=1)=$ $1 / 2 .$ Let $$ T=\inf \left\{n \geq 4: X_{n-3}=0, X_{n-2}=1, X_{n-1}=0, X_{n}=1\...
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1answer
45 views

Almost Sure Convergence in Martingales

Q. Let $\left(X_{n}\right)_{n}$ be a martingale such that there exists $K$ which satisfies $$ \mathbb{P}\left(X_{n} \leq K\right)=1 $$ Define the process $M_{n}=K-X_{n}$, for $n \in \mathbb{N}$. Prove ...
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1answer
33 views

Distribution of a branching process martingale at the limit

Source: Probability with Martingale Williams, Page 11. In an attempt to get the distribution of $M_{\infty}$ for the case $\mu >1$ the author writes, you can easily check that, for $\lambda>0$ $$...
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2answers
49 views

If $\tau_n,\tau$ are stopping times with $\tau=\inf_n\tau_n$, then $\text E[X\mid\mathcal F_{\tau_n}]\to\text E[X\mid\mathcal F_\tau]$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space. Lemma 1: Let $Y\in\mathcal L^1(\operatorname P)$, $I\subseteq\mathbb R$ be countable and $(\mathcal G_t)_{t\in I}$ be a filtration on ...
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1answer
61 views

Martingale Problem - Markov process

Let $X$ be a right continuous Markov process with left limits and generator $L$. Why is $f(X_t)-f(X_0) - \int Lf(X_s) ds$ a martingale for every $f \in D(L)$? Let s<t. $E^x[M_t^f|F_s]= E^x[f(X_t)-f(...

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