1
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1answer
23 views

For $X_n$ iid, $S_n=\sum X_n$, $\mathscr{G}_n=\sigma(S_n,S_{n+1}, \dots)$, show $E(X_j|\mathscr{G}_n)=E(X_1|S_n)$

If $(X_n)$ is iid in $L^1$, and $S_n = \sum_{i=1}^{n} X_i$ and $\mathscr{G_n} = \sigma(S_n, S_{n+1},...)$, then show that $E[X_1|\mathscr{G_n}]=E[X_1|S_n]$, and that ...
0
votes
1answer
13 views

Showing $E[(M_T-M_S)^2|\mathscr{F}_S]=E[M_T^2-M_S^2|\mathscr{F}_S]$ for Martingale $M_n$.

Let $(M_n)$ be a martingale with $M_n \in L^2$. $S,T$ are bounded stopping times w $S\leq T$. Show $M_T, M_T$ are both in $L^2$ and that $E[(M_T-M_S)^2|\mathscr{F}_S]=E[M_T^2-M_S^2|\mathscr{F}_S]$ ...
1
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2answers
76 views

Martingality Theorem: Solving expectation of a stochastic integral

I am trying to prove that: $$ \Bbb E\left[\int_s^t\sigma e^{-k(t-u)}\sqrt{V_u}dW_u\right] =0$$ Where: $$ dV_t=k~(\theta-V_t)~dt+\sigma\sqrt{V_t}dW_t $$ I have attempted to use Ito's formula on the ...
2
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2answers
57 views

Doob's decomposition and submartingale with bounded increments

Let $(X_n)_{n \geq 0}$ be a submartingale defined on some filtered probability space $(\Omega, \mathcal{F}, ({\mathcal{F}}_n)_{n \geq 0}, \mathbb{P})$. It is a standard fact that $X_n = X_0 + M_n + ...
1
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1answer
42 views

Application of martingale convergence theorem

I am struggling with this question: Let $(X_n : n \geq 1)$ be a zero mean martingale in $L^2$. Show that, for $\lambda >0$, \begin{equation} \mathbb{P} \bigg( \max_{1 \leq k \leq n} X_k \geq ...
1
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0answers
37 views

A problem on super/sub martingale

Let $(X_n, \mathscr{F_n}), n \geq 0$ be a super martingale and $T$ an $\{F_n\}$-stopping time a.s. bounded by $N \lt \infty$. Show that $$ E[|X_T|] \leq 3 \max_{n \leq N} E[|X_n|]$$ I can prove that ...
0
votes
1answer
33 views

On “for all” in if and only if statements in probability theory and stochastic calculus

1 In my friend's Probability Theory long test there was this question: Let $(\Omega, \mathfrak{F}, P)$ be a probability space on which is defined all sub-$\sigma$-algebras, events and random ...
3
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1answer
29 views

how to find ALL $\sigma$-algebras of a given sample space?

I have a sample space $\Omega$={$\omega_{1}$,$\omega_{2}$,$\omega_{3}$,$\omega_{4}$} and I need to find ALL $\sigma$-algebra on $\Omega$. I know how to construct some $\sigma$-algebra like ...
1
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1answer
41 views

Proof of extinction probability in Galton-Watson-process using a Martingale

this problem is somewhat similar to the thread The extinction probability of Galton-Watson process from a Martingale perspective. I want to show, that for a Galton-Watson-process $Z_0,Z_1,\ldots$ with ...
3
votes
2answers
96 views

Proving integrability of a random variable involving stopping times

Let $X_1, X_2,...$ be i.i.d integrable random variables in $\mathbb{R}$ with $\mathbb{E}[X_i] =0$ and $\mathbb{P} (X_i >0) >0$. Let $x>0$, $S_0 = x$, and $S_n= x + \sum_{i=1}^{n} X_i $. For ...
1
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3answers
56 views

Prove $X_n = \frac{1}{a_n} 1_{(0,a_n]}$ a martingale

With the following premises, I want to prove, that the series of random variables $(X_n)_{n \in\mathbb{N}}$ is a Martingale: Let $\Omega := (0, 1] \subset \mathbb{R}, \mathfrak{F}$ the ...
1
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2answers
46 views

One Question on Law of Total Probability

Let $(X_n)$ with $n \in \mathbb N_0$ be a discrete martingale. Then I read the following identity which is said to be derived from the law of total probability. $$ \mathbb EX_m = \left( \sum_{n=0}^m ...
3
votes
1answer
81 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 ...
1
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1answer
23 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
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0answers
42 views

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 ...
2
votes
2answers
39 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
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1answer
21 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
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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 $\tau$ ...
0
votes
1answer
57 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
36 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
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2answers
77 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 ...
2
votes
1answer
35 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 ...
0
votes
1answer
59 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
54 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
27 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 ...
2
votes
1answer
117 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
29 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
78 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
39 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 ...
0
votes
0answers
10 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 $\omega \in \Omega$ ...
1
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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
35 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
67 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\{ ...
0
votes
2answers
46 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
39 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
55 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
114 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 ...
2
votes
3answers
121 views

What is a fair game?

Suppose $X_n$ is the fortune of a gambler after $n$ th game. Then the game is called fair (Breiman 1968) if $$E[X_{n+1} \mid X_1, \dots, X_n] = X_n \forall n$$ My question is why a fair game is not ...
0
votes
0answers
23 views

Applications of martingale to gambling

In an unfavourable or fair sequence of games (where one has to bet a minimum amount if he wants to bet) one cannot keep betting indefinitely. There must be a last bet. The intuition behind this is not ...
2
votes
1answer
40 views

Azuma inequality with probabilistic bound for the increments

Let $(X_i)$ be a super-martingale and suppose their differences are bounded ''with high probability'', that is $$\mathbb{P}(\exists\,i=1,\dots,n\text{ s.t. }|X_i-X_{i-1}|>c_i) \,\leq\, \epsilon$$ ...
1
vote
1answer
98 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
1
vote
1answer
25 views

Characterization of conditional independence

Definition: Let $\mathcal{G},\mathcal{K},\mathcal{H}$ be $\sigma $-subalgebras of $\mathcal{F}$, where $\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$ is a given probability space. We say that ...
2
votes
0answers
29 views

Finding a pre-visible process

Question: Let $W_t$ be a standard brownian motion under P with filtration $\mathscr F_t$. Let: $$ M_t=\mathbb E[W_T^2|\mathscr F_t] $$ Show that $M$ is a P martingale. This is simple enough using ...
4
votes
1answer
167 views

Lookback option with floating strike: boundary condition

I am trying to make sense of one of the boundary conditions of a look-back option with floating strike. Some notation first: let $v(t,x,y)$ denote the price at time $t$ of the option under the ...
1
vote
2answers
71 views

Uniformly integrable martingale

I have the following martingale. $M_n=\exp\left(aB_n-\frac{1}{2}a^2n\right)$ for $n\geq0$ and $a\neq0$, $B_n$ is a BM. I have to show that for $a>0$, $M_n\rightarrow0$ in probability. Is $M_n$ ...
1
vote
2answers
76 views

Bounded (from below) continuous local martingale is a supermartingale

Suppose $M(t)$ is a continuous local martingale. That is, there exists a sequence of stopping times $T_n$ which almost surely increase to $\infty$, and such that $M(t\wedge T_n)$ is a martingale for ...
0
votes
0answers
24 views

Decomposition of noisy measurements

What can be a good intuition behind decomposing a sequence $\{Y_n\}$ of noisy measurements (i.e. random variables) into two parts: one unpredictable and the other depending on the past. $$Y_n = ...