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Questions tagged [martingales]

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

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27 views

Advice on correct usage of conditional expectation (proof verification)

Assume $X_1, \ldots, X_n$ are independent and identically distributed random variables where $X \in \{-1, 1\}$ with equal probabilities, and $S_k$ represents their cumulative sum of up to $k$ ...
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22 views

Showing that sequence is a martingale

Let $x_1, x_2,...,x_n$ be random variable defined on a finite probability space $(\Omega, \mathcal{P}(\Omega), P)$. For an arbitrary possible $k$ suppose that $x_k$ is measurable with respect to ...
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22 views

Showing $\text{cov} (x_d-x_c, x_b-x_a)=0$

Suppose that $(x_k, \mathcal{D}_k), k=1,...,n,$ is a martingale define on finite probability space. Show that $$\text{cov} (x_d-x_c, x_b-x_a)=0$$ for arbitrary integers $a<b<c<d$. How can I ...
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44 views

Showing $\mathbb{E}S_{\tau}^2=\mathbb{E}\tau$.

Suppose that $x_1, x_2,...x_n$ are independent copies of random variable $x$ having distribution $$P(x=1)=P(x=-1)=\frac{1}{2}.$$ In addition, suppose that $\mathcal{D}=\mathcal{D}_{x_1,...,x_k}(k=1,......
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26 views

Proving that sequence is a martingale

Suppose that $x_1, x_2,...,x_n$ are independent copies of random variable $x$ having distribution $$P(x=1)=p, \text{ } P(x=-1)=q,$$ where $p+q=1$. In addition, suppose that $\mathcal{D}_k=\mathcal{D}_{...
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1answer
46 views

Double conditional expectation of martingale

Assume $X$ a martingale. A common exercise is showing that every martingale has uncorrelated increments. That is, with $a < b < c< d$, $$ Cov( X_a - X_b, X_c - X_d) = 0 $$ While there are ...
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52 views

Prove that $(X_n)$ converges to finite limit almost surely.

Let be $(X_n)$, $(Y_n)$ and $(Z_n)$ positive sequences on filtration $(F_n)$. Assume that $E|X_n| < \infty$ and $E(X_{n+1}|F_n) \leq (1 + Z_n)X_n + Y_n$ and $\sum_n Y_n < \infty$ and $\sum_n Z_n ...
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31 views

Find the expected number of stages until one of the players is eliminated.

I have been given an example: Moreover I have been given a problem based on this example: I tried to find an appropriate martingale to solve this problem but I couldn't. I would appreciate any ...
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19 views

Heads and tails with unlimited capital

Let $ S_{n} $ be the sum of won money by player 1 until moment $ n $ He gets 1 if he wins and loses 1 if he loses Let $ X = \inf\left\{ n : S_{n} = 1\right\} $ We assume that coin is asymetric and ...
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19 views

Prove that supermartingale with specific characteristics is a martingale

Prove that if $ (X_{n}, \mathcal{F}_n)_{n=0}^\infty $ is a supermartingale such as $ EX_{n} = EX_{0} $ for all $ n $ then $ (X_{n}, \mathcal{F}_n)_{n=0}^\infty $ is a martingale Is it enough to say ...
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32 views

Bounding expectation of stopping time

Let $(X_{t})_{t\ge0}$ be adapted to $(\mathcal{F}_{t})_{t\ge0}$ with continuous trajectories. Assume that $X_{0} = 0$ and $X_{t}^{4} - 3t^{2}$ is a martingale with respect to $(\mathcal{F}_{t})$. Let ...
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3answers
23 views

Almost sure convergence of a martingale sum

Consider the senquence of iid r.v. $(Y_k)_{k\geq1}$ such that $\mathbb{P}(Y_k=1)=\mathbb{P}(Y_k=-1)=\frac{1}{2}$ and then consider the process $X=(X_k)_{n\geq1}$ such that $X_n=\sum_{k=1}^n\frac{Y_k}{...
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132 views

Proof of identity about generalized binomial sequences.

I was going through this old question about a wealthy gambler: Gambler with infinite bankroll reaching his target. The answer relies on the following identities from Concrete Mathematics by Graham, ...
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28 views

Prove that $\lim\limits_{n \to \infty} P(\Lambda_n | F_n) = 1_{\Lambda}.$

Let be $(F)_{n}$ filtration and $ A_{n} \in F_{n}$ for every $n \geq 0$. Let be $$ \Lambda_{n} = \bigcup_{m \geq n} A_m $$ and $$\Lambda = \bigcap_n A_n. $$ Prove that $\lim\limits_{n \to \infty} P(\...
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Compute $E(M_σ)$ when $σ = \inf(t ≥ 0: |B_t| = 1)$ and $M_t = 4B^2_t +e^{4B_t−8t}−4t$

Given $M_t = 4B^2_t +e^{4B_t−8t}−4t$ for $t ≥ 0$ and a Brownian motion $(B_t)_{t \geq 0}$. Compute $E(M_σ)$ when $σ = \inf(t ≥ 0: |B_t| = 1)$. I have tried to show that $E|M_σ|\leq K$ to apply Doob's ...
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64 views

A martingale that does not converge in $L^1$

I am currently studying martingales and I am working on the following problem: Let $\Omega = \mathbb N^*$ and associated probability measure $$\forall\, k \in \mathbb N^*, P({\{k\}})=\frac{1}{...
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15 views

Is Conditional Expectation of stochastic process a martingale?

I have the following: Let $Y$ be an integrable and $F_T$ -measurable random variable. Define $\{X_t\}_{t \in [0,T]}$ by $X_t =E(Y|F_t), \ \forall t \in T$. The $\{F_t\}_{t \in [0,T]}$ should be the ...
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81 views

Show a Continuous Local Martingale is a Martingale

Let $B = (B_t)_{t≥0}$ be a standard Brownian motion started at zero, let $X=(X_t)_{t≥0}$ be a nonnegative stochastic process solving $$dX_t = 3 \, dt + 2\sqrt{X_t} \, dB_t \qquad(X_0 = 0)$$ and let $$...
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1answer
34 views

The distribution of a stopping time

Let $(X_n)_{n\geq0}$ be a sequence of real $i.i.d$ random variables and $\tau = \inf\{n\geq0 : X_n\in S\}$ with $S \in \mathcal{B}(\mathbb{R}) $ I am trying to find $\tau$'s distribution. Obviously, ...
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25 views

question about martingale about maximal inequality for submartigales

Suppose ${X_n}$ is a martingale satisfying, for some $\alpha > 1$, $E\left[|X_n|^\alpha\right]<\infty$, for all n. Show $$E\left[\max_{0\leq k \leq n}|X_k| \right]\leq \frac{\alpha}{\alpha-1} E[...
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31 views

Integrability in Dynkin's formula

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $W$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $b,\sigma:\mathbb R\to\mathbb R$ be Borel measurable with $$...
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Black-scholes model exercise

Question This was the question and this is how i tried to solve it and I do not understand where I make the mistake. Can anybody help me? part1 part2
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1answer
61 views

Show that a symmetric random walk with hit any number with probability 1

I'm supposed to show this with martingale convergence thm. I have tried setting up one barrier at a fixed $n$. And the martingale convergence says it will converges to a random variable almost surely. ...
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48 views

Is the sigma algebra generated by $X$ random variable and its square equal to the sigma algebra generated by $X$ alone?

I would like to understand which relationships hold among the sigma algebras $\sigma(X, X^2)$, $\sigma(X)$ and $\sigma(X^2)$, where X is a random variable. I would expect that $\sigma(X, X^2)=\sigma(X)...
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17 views

Expectation conditioned on a $\sigma$-algebra? (Martingale definition)

I am trying to understand the following definition: Given probability space $(\Omega, \mathcal{F}, P)$ and an increasing sequence of sub-$\sigma$-fields $\mathcal{F}_0=\{\emptyset, \Omega\} \subseteq ...
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45 views

Sum of variables of a martingale

I have the sequence $X_1, X_2,...X_n$ as a martingale, each of which is bounded. Now I want to explore some upper bound for the sum $S_n=X_1+X_2+...+X_n$, e.g., the format like Hoeffding inequality or ...
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Prove that a process (given through rsdes) is a martingale.

i have a rather complicated problem for a process which is connected to a kalman-bucy filter with a riccati equation. More precisely, let $dX_t =A(X_t)dt + R_1^{1/2} dWt, dY_=BX_tdt + R_2^{1/2}dV_t $ ...
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38 views

Show that this processes are martingales

I have to show that the following two processes are martingales $M_t=g(t)B_t-\int_0^tg'(s)B_sds$ $X_t=\exp\left(e^tB_t-\int^t_0e^sB_sds-\frac{e^{2t}}{4}+\frac{1}{4}\right)$ Where $g(t)$ is a real ...
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If $E(X_n^2)<\infty$, then for a Martingale $E(X_n^2)<M$ iff $\sum_{n=1}^\infty E[(X_n-X_{n-1})^2]<\infty$

Let $\{X_n\}_{n\geq0}$ be a martingale with $E(X_n^2)<\infty$ for all $n$. How to prove that: $E(X_n^2)<M$ for all $n$, if and only if $\sum_{n=1}^\infty E[(X_n-X_{n-1})^2]<\infty$. The ...
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32 views

Absolute value of a martingale

Given a martingale $M(t)$, can I use Doob's inequality on $|M(t)|$ to achieve the following upper bond? $$P(\sup_{0\leq t\leq T}|M(t)|>\epsilon)=P(\sup_{0\leq t\leq T}|M(t)|^2>\epsilon^2)\leq\...
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30 views

Reference Markov martingale Harmonic function

I've just finished a course of stochastic process (discret martingale and markov chain). I would like to go further, I heard it exists a link between martingale markov process and harmonic functions. ...
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1answer
88 views

Probability on first hitting time of Brownian motion with drift

I am struggling with the following problem: Let $B$ be a one dimensional Brownian motion and $a,b>0$. Show that $$P[B_t=a + bt \text{ for some } t\geq 0] = e^{-2ab}.$$ The following hint is ...
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Calculate limit: Make a stopping time $T$ bounded $T\land n$ and take the limit $n \to \infty$

Say we have a martingale $X$ and a stopping time $T$. Instead of directly studying the stopped process $X_T$, many proofs employ a trick, namely, one considers the bounded stopping time $T\land n$ ...
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The problem is about the expection of the exitpoint distance for the symmetric random walk.

Let $\nu(x)$ be a symmetric probability measure with respect to the origin on $x\in[-1,1]$ such that $\nu(\{0\})\neq 1$. Consider a random walk started at $S_0=0$, denoted $S_n=X_1+\cdots+X_n$, where ...
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1answer
41 views

Martingale of random walk and stopping time

Let $\{S_n\}$ be a symmetric random walk such that $S_0 = a$ for some $0 < a < K$. Let $T$ be the stopping time when the walk reaches $0$ or $K$. Show $$M_n = \sum_{i = 0}^n S_i - \frac{1}{3}...
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47 views

Martingales for a random walk

Firstly, I just want to check my understanding: If we have a symmetric random walk such that $P(S_{n+1} = S_n + 1|S_n)=1/2$ $P(S_{n+1} = S_n - 1|S_n)= 1/2$ then $S_n$ is a martingale, $(S_n)^2$ ...
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1answer
36 views

Stopping time of Feller process

Let $X$ be a Feller process on $\mathbb{R}$ with generator $Gf=\frac{1}{2}f''-f'$ on $C_c^2$. Let $\tau_b$ be the first time that $X$ hits $b\in\mathbb{R}$. Show that for $x > 0$, $bP_x(\tau_b < ...
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17 views

Conditional expectation of martingale and two bounded stopping times

I am trying to prove the following: Let $(X_n)$ be a martingale with respect to $(\mathcal{F}_n)$ and suppose $\tau_1$ and $\tau_2$ are bounded stopping times such that $\tau_1\le \tau_2 < B$, ...
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Why is $(S_n)^2 - n(q-p)$ a martingale for a random walk

If we have a random walk such that $$P(S_{n+1} = S_n + 1|S_n)=p$$ and $$P(S_{n+1} = S_n - 1|S_n)= 1-p=q$$ then why is $$(S_n)^2 - n(q-p)$$ a martingale. I understand that $(S_n)^2$ is a sub-...
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Don't fully understand Theorem $5.13$ from Durrett

Can anyone shed some light on this theorem? Not sure of the significance of the result or how one would apply it. If $M_n$ is a supermartingale with respect to $X_n$ and $T$ is a stopping time then ...
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11 views

Alternative formulation of a markov process

I'm wondering how the markov property can be specified as follows, if anyone can provide more details (this looks awfully like the definition for a martingale): $$E[f(X_t)|\mathcal{F}_s]=E[f(X_t)|\...
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10 views

Discounted stock is a martingale?

I'm currently confused with the discount martingale. From my class, I have been told the discounted stock is a martingale. I know the fact under risk neutral, we have $$E(e^{-r(T-t)}S_T)=S_t$$ So we ...
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1answer
42 views

Prove that $\mathbf{E}(X_{\tau_2}|\mathcal{F}_{\tau_1})=X_{\tau_1}$

Let $(X_n)$ be a martingale with respect to $(\mathcal{F}_n)$ and suppose $\tau_1$ and $\tau_2$ are bounded stopping times such that $\tau_1\leq \tau_2<B<\infty.$ Then $$\mathbf{E}(X_{\tau_2}|\...
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1answer
33 views

Conditional expectation of exponent to the power of BM, representation theorem

Let $T > 0$ and $M(t) = E[e^{W(T)}|F(t)]$ where $\{F(t) : t \geq 0\}$ is a natural filtration generated by W and $t \leq T$ I need to show that M(t) is a martingale and I also need to find a unique ...
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2answers
30 views

Existence of nonnegative and nonconstant martingales

Are there martingales that are nonnegative and nonconstant? If so, are their any intuitive examples for such?
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37 views

Multiple Martingale convergences: almost surely, but not in $L^{1}$ & almost surely and in $L^{2}$ [closed]

Let $X_{i},i\in \mathbb{N}$ be a sequence of independent standard normal random variables, $\mathscr{F}_{n}$ the associated natural filtration, and $S_{n}=\sum_{i=1}^{n}X_{i}$. You are allowed to use ...
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1answer
41 views

Show that this process is not a martingale

I have to show that the following process $(X_t)_{t\in [0,\infty)}$ is no martingale. Let $Y_n$ be a sequence of independent random variables with $$P(Y_n=n)=\frac{1}{2n^2},\quad P(Y_n=-n)=\frac{1}{...
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1answer
89 views

Sufficient condition for $L^\infty$-convergence in martingale convergence theorem.

Let $I$ denote the unit interval and $f:I\to I$ be a Borel measurable function. For each $n$ let $P_n$ denote the partition of $I$ defined as $$P_n= \left\{\left[0, \frac{1}{2^n}\right), \left[\frac{1}...
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29 views

Proof of a martingale

Consider a martingale $X_{n}, n \in \mathbb{N} $ with independent increments. Assume that the variance, $\sigma^{2},$ of the increments is constant. Define $Y_{n}=X_{n}^{2}-n\sigma^{2}.$ Prove ...