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

learn more… | top users | synonyms

1
vote
2answers
24 views

Prove Kolmogorov's SLLN by martingale.

Suppose $\xi_i$ are i.i.d. and $\mathbb E(|\xi_1|)\lt\infty$ Let $X_n=\sum_{i=1}^n\xi_i$ Then we have $\frac{X_n}{n}\to \mathbb E(|\xi_1|) $a.s. In the proof of this theorem: ...
1
vote
1answer
20 views

Intuition underlying stopped martingales

Let $X$ be a martingale and $T$ a stopping time. Define the stopped martingale $X_{\min\{T,n\}}$. What is the intuition underlying this process? It is quite confusing here. $X$ is random and $T$ is ...
1
vote
1answer
43 views

An inequality in martingale

Suppose $X_n$ is a supermartingale,for $\lambda>0$ ,we have the following inequality: $$\lambda\mathbb{P}(\inf_{n\leq k}X_n\leq-\lambda)\leq\int_{[\inf_{n\leq k }X_n\leq -\lambda]}(-X_k) ...
3
votes
1answer
71 views

Expectation of Square of Stopping Time

Let $B_t$ be standard Brownian motion and $a < 0 < b$. Define stopping time $T$ as follows. $$T = \min \{t \geq 0: B_t \in \{a, b\} \}.$$ The expectation of $T$ is $\mathbb ET = |a|b$ and can be ...
0
votes
0answers
13 views

Control of the expected false discovery rate

I've been looking at why exactly the Benjamini and Hochberg procedure controls the expected false discovery proportion. More specifically, assuming $N$ hypotheses with corresponding $p$-values; for a ...
1
vote
1answer
61 views
+50

Problem about Random walk and Stopping time.

Here is an example in "Probability with Martingales" My questions are: (1)Does equation (a) hold for $T=\infty$? (2)The equation:$$\mathbb{E}M_T^\theta=1=\mathbb{E}[(sech \theta)^Te ...
1
vote
1answer
22 views

Square Integrable local martingale or locally square integrable martingale?

I have a question about martingales. What is the difference between "locally square integrable martingale" and "square integrable local martingale"? In particular, which set does $M_{loc}^2$ ...
1
vote
1answer
23 views

A problem about martingale with stopping time .

In Durrett's "Probability,theory and examples": Suppose $X_n$ is supermartingale and $H_n$ is predictable. define: $$(H\cdot X)_n\triangleq\sum^n_{m=1}H_m(X_m-X_{m-1}) $$ $N$ is stopping time and ...
0
votes
0answers
41 views

Values of $\mu$ for which $S_n=e^{\sum_{i=1}^n X_i}$, is a martingale ($X_i ~ \mathcal{N}(\mu,1)$) [closed]

Let $(X_n)_{n\geq 1}$ a sequence of $\mathcal{N}(\mu,1)$ $\mathcal{F}$-adapted and $S_n=e^{\sum_{i=1}^n X_i}$. I have to write the conditions for which S_n is a martingale, then I have to show that ...
1
vote
0answers
32 views
+50

How to get closed form solutions to stopped martingale problems?

Way back when, I took a course in stochastic processes in college. I remember being frustrated by the plethora of abstract proofs without much in the way of how to use them to get actual results. It ...
0
votes
1answer
37 views

For a Brownian motion prove that (a) $N (t) -λt $ and (b) $e^{(\log(1-u) N (t) + uλt)}$, are martingales

For a Brownian motion ${z (t)}$ and for any $β ∈ R$, be $V (t) = \exp\{ βz (t) - (t β ^ 2) / 2\}, t≥0 $ Show that ${V (t)}$ is a martingale with respect to a Brownian filtration. Also ${N (t)}$ be a ...
2
votes
2answers
38 views

Let $X$ be the value of the first die and $Y$ the sum of the two dice. Find $E(X / Y)$ [closed]

Consider an experiment of rolling two dice. Let $X$ be the value of the first die and $Y$ the sum of the two dice. Find $E(X / Y)$, ie, obtain the value of $E(x/y) (y)$ for all $y$ Good evening, I ...
1
vote
1answer
19 views

Centered independent increments process is a martingale

Let $(X_n)$ be an centered integrable process with independent increments (which as far as I understand means that $(X_{n+1}-X_n)_{n\in \mathbb N}$ is independent). While showing that $(X_n)$ is a ...
1
vote
2answers
42 views

$E[M_t|H_t]$ is a martingale with respect to $H=(H_t)_{t\geq 0}$, $H_t \subset \mathcal{F}_t \forall t$

Being $(M_t)_{t \geq 0}$ an $\mathcal{F}$-martingale, I have to show that $E[M_t|H_t]$ is a martingale with respect to $H$ ($H=(H_t)_{t\geq 0}$, $H_t \subset \mathcal{F}_t \forall t$). I proceded ...
1
vote
1answer
31 views

Do we need $\tau \leq \nu$ to show $E(X_\tau)=E(X_\nu)$?

My lecture notes claim that if $(X_n)$ is a martingale and $\tau$ is a stopping time bounded by $N$ then $$E(X_\tau)=E(X_{\tau \wedge N})=E(X_{\tau \wedge 0})=E(X_0)$$ and then remarks that if $\tau$ ...
1
vote
0answers
13 views

Is there a Burkholder-Davis-Gundy inequality for martingale increments?

is there a Burkholder-Davis-Gundy inequality for martingale increments? More specifically, I would like to find a finite bound of order $h^{p/2}$ for the expectation $$\operatorname{E} \left[ \sup_{t ...
2
votes
1answer
42 views

Stochastic integration by parts formula to prove identity between iterated integrals

if $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...
1
vote
0answers
27 views

If a random integral has moments of all orders, is the same true for its kernel?

Suppose you have a continuous semimartingale $S_t=M_t + A_t$ where $A_t$ is the continuous finite variation part which has the form $A_t = \int_0^t b_s \, \mathrm{d} s$, where $\int_0^{\infty} |b_s| ...
4
votes
0answers
59 views

Filtration and measure change

I'm reading Steven E. Shreve's "Stochastic calculus for finance II", and find myself not really understand the concept of "filtration". Yes, the definition of filtration is straight forward, it's ...
0
votes
1answer
55 views

Suitable martingales and optional stopping theorem

Starting at value 0, the fortune of an investor increases per week by 200 with probability 3/8, remains constant with probability 3/8 and decreases by 200 with probability 2/8. The weekly increments ...
0
votes
1answer
34 views

Is $Q_n(A_n)=E(L_n A_n)$ a probability measure?

Let $(\Omega, F, F_t, P)$ be a filtered probability space and $(L_n)_{n \geq0}$ a family of positive and $F_t$ adapted random variables. I have to find the conditions for which $Q_n$, defined on ...
1
vote
2answers
70 views

On the conditional expectation.

I want to prove that: if $E[M_t\mid\mathcal{F}_s]=0$ where $\mathcal{F}_s$ is the filtration generated by a stochastic process X knowing that $E[M_t\prod_0^n h_i(X_{t_i})]=0$ for all $n\in N,\quad ...
-1
votes
0answers
33 views

Prove hitting time (with $\tau_a:=n+1$ if $S_k\leq a \forall 0\leq k\leq n$)

I have random variable $\tau$ defined as follows $\tau_a=\min\{0 \leq k \leq n : S_k>a\}$ $(a>0)$ $\tau_a:=n+1$ if $S_k\leq a$, $\forall 0\leq k\leq n$ (how should I care during the proof?) ...
1
vote
0answers
33 views

Occupancy distribution bounds for $k$ balls in $m$ bins

Suppose we throw $k$ (distinct) balls into $m$ (distinct) bins, and let $B$ count the number of non-empty bins. I am interested in lower bounds on $B$. More precisely, I wish to bound from above the ...
0
votes
1answer
21 views

Writing a martingale as the difference of two non-negative martingale

Assumption: $ (M_{t})_{t \geq 0}$ is a martingale w.r.t $(\mathcal{F}_t)_{t \geq 0}$ Question: Why can I write $M_t$ as the difference of two non-negative martingales? Attempt: $ M_t = M_t^{+} - ...
3
votes
1answer
102 views

A martingale with bounded increments either converges or diverges to both infinities a.s.

I am reading page 236 "Probability : theory and examples" by R. Durrett. Theorem 31. Let $X_1, X_2,\ldots$ be a martingale with $|X_{n+1}-X_n|\leq M<\infty$. Let $C=\{\lim X_n \;\;\; \text{exists ...
2
votes
1answer
32 views

Prove that the discrete time martingale can be represented by $E (Y_{n +1} \mid F_n) = 0$ if $Y_{n +1} = X_{n +1}-X_n$, for $n = 0,1, \ldots $

Prove that the discrete time martingale can be represented by $E (Y_{n +1} \mid F_n) = 0$ if $Y_{n +1} = X_{n +1}-X_n$, for $n = 0,1, \ldots $ I want to use the sequence $(y_n)$ called "martingale ...
3
votes
1answer
82 views

Every Lipschitz function is the primitive of a measurable function

I was doing exercise 5 of this exercise sheet and I don't know how to conclude. I need to prove that if $f \colon [0,1]\to \mathbb{R}$ is Lipshitz, $X$ is a uniform$(0,1)$ random variable and ...
0
votes
1answer
56 views

Martingale based on normal PDF evaluated at normalized i.i.d. sums

I have the following problem. $(X_n)_{n\geq0}, n\in\mathrm{R}$, is a family of iid r.v., normally distributed $\mathcal{N}(0,1)$ $\mathcal{F_n} := \sigma((X_i)_{1\leq i\leq n})$ $x\in\mathrm{R}, ...
2
votes
2answers
50 views

Limit value of a product martingale

This question came from a problem i was solving for self-study. I'll state the problem first: Let $Y_n \sim \mathcal N(0,\sigma^2)$ be independent normally distributed variables, $X_n = ...
1
vote
0answers
22 views

Applying Ito lemma to multi dimensional semimartingales

Let $X_t=(X^1,\dots,X^d$) be a d-dimensional semimartingale. Then the formula for Ito's lemma I have found in several places, including wikipedia, is: $f(X_T)-f(X_0)=\sum_{i=1}^d\int_0^T ...
0
votes
0answers
19 views

Exponential Inequalities for Martingales

So I was having a read of the paper here: Exponential Inequalities for Self-Normalized Martingales. I am particularly interested in Remark 4.2, which states that if $M_n$ is a Gaussian martingale (and ...
2
votes
1answer
116 views

Deducing an optimal gambling strategy (using martingales).

Apologies in advance for the length, I tried being precise. Suppose a game where in each turn you can gamble a certain amount of money on the result of a fair coin toss. If the coin comes out tails ...
0
votes
1answer
27 views

Product of independent continuous local martingales is local martingale

Revuz-Yor's book mentioned if $M$ and $N$ are independent continuous local martingales, then $MN$ is still local martingale. But I don't know how to prove it. Any help, thanks!
0
votes
1answer
20 views

Finding the unique Martingale Convergence Representation of a given r.v.

According to the martingale representation there exists a unique $g(t,\omega) \in \mathcal{V}(0,T)$ such that $M_t = E[M_0]+\int^{t}_{0} g(s,\omega) dB(s); \ \ \ t \in [0,T]$ Find g in the case ...
0
votes
1answer
37 views

martingale difference

I am trying to solve the following question. {$ξ_k$} is $F_n$-martingale difference (i.e. for every $n$, $E[ξ_n|F_{n-1}]=0 $ a.s. ) Also, for every $n$ , $E[ξ_n^2]<\infty$ Show that ...
3
votes
0answers
75 views

Qual Question concerning martingale

Suppose $X_n$ is a sequence of random variables that has the property that $\sup|X_n| \leq 1$ a.s. Then use Doob's decomposition to prove that $\sum_{n\geq 1} X_n$ converges a.s. iff the sum ...
2
votes
1answer
35 views

More preliminaries of the Martingale Convergence Theorem

Really struggling with this lemma. Not sure about the general structure of the proof. Why have we chosen g to be orthogonal to all functions of the form 4.3.1? Why should $G(\lambda)=0$, does it ...
2
votes
1answer
32 views

Preliminaries of the Martingale Representation Theorem

I cannot understand why we are taking a dense subset of $[0,T]$. Furthermore, I cannot see a result that would allow each such $g_n(B_{t_1},\ldots,B_{t_n})$ to be approximated in ...
0
votes
0answers
9 views

About the space $((R^n)^{[0,\infty)},\mathcal{B},\tilde{Q}^x)$ in Oksendal SDE book

I am reading the book Stochastic differential equations (6th ed.) by Oksendal. I am not sure about the meaning on P.146. (Below Theorem 8.3.1) It says that ``if we identify each $\omega \in \Omega$ ...
4
votes
2answers
59 views

Why are stochastic processes with decreasing expected value called supermartingales?

I am curious to know why a process which has decreasing expected value is called a supermartingale. From a beginners perspective it would seem reasonable to have the following picture: ...
1
vote
1answer
34 views

Every martingale is also a martingale with respect to its own filtration

I want to prove the following: Let $A_0, A_1, ..$ be a martingale with respect to the sequence $B_0, B_1, ..$. then $(A_i)_{i\geq0}$ is also a martingale with respect to itself. I have no idea how ...
2
votes
0answers
34 views

Rate of convergence of a martingale

I have a question related to convergence rates of martingales: Assume that there is a sequence of maximized likelihood ratios: $ \frac{f_{\hat{\theta}_{n}} \left ( Y_{1},Y_{2},\dots,Y_{n} \right ) ...
1
vote
1answer
61 views

Proving a property of hitting times of a simple random walk on $\mathbb{Z}$

I'm reading the course notes of a probability course about martingales currently and I'm trying to solve some of the exercises, however I'm very much stuck with the following exercise: Let $\left\{ ...
4
votes
1answer
83 views

If I bet half of my money each round in a fair gamble, what's the probability…

that I can make 10 times of what I initially have? Here's the formal description. In a fair gamble, I lose or double my wager each with probability 1/2. No matter how much money I have, I always ...
0
votes
2answers
45 views

Problem about martingale convergence in $L^p$

I'm trying to do the following exercise: I have a martingale $Z_n=A^{S_n}Q_A^{-n}$ where $A>1$, $Q_A=\frac{1}{2}(A+A^{-1})$ and $S_n=X_1+\cdots+X_n$ with $X_k$ r.v.'s iid such that ...
0
votes
0answers
38 views

Why is the Stopping Theorem interesting?

The theorem for discrete-time martingales is as follows: Let $X=(\Omega,\mathcal{F},(\mathcal{F}_n)_n,(X_n)_n,\mathrm{P})$ be a supermartingale and $\tau_1,\tau_2$ two a.s. bounded stopping times on ...
1
vote
1answer
53 views

An exponential martingale [closed]

Let $H_{t}$ be a bounded continuous and $\textbf{F}^{B}_{t}$ an adapted process. $B$ Brownian motion. Show that $M_{t}= \exp\left(-\int^{t}_{0}H_{s}dB_{s} -\frac{1}{2}\int^{t}_{0}H^{2}_{s}ds\right)$ ...
2
votes
1answer
69 views

Proving it's a martingale and more conditions.

Let $(X_{n})_{n>0}$ be a sequence of random variables in $[0, 1]$ and assuming that ($X_{0}=a) \epsilon [0, 1]$ then: $Pr\left(X_{n+1}=\frac{X_{n}}{2}|\mathcal{F}_{n}\right)=1-X_{n}$ and ...
2
votes
1answer
101 views

Rigorous Book on Stochastic Calculus

I have already taken a couse in Stochastic Calculus. Due to time constraints on many ocassions we had to skip some formalities among the proofs. I'm trying now to fill the gaps left, and I have been ...