1
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1answer
13 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 $M_{loc}^2$ ...
0
votes
0answers
41 views

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

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 ...
0
votes
0answers
17 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
37 views

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

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
1answer
30 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
53 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
33 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 ...
2
votes
1answer
31 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
55 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 ...
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
36 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 ...
2
votes
0answers
74 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 ...
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 $\omega \in \Omega$ ...
1
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1answer
32 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\{ ...
0
votes
2answers
44 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
68 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
98 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 ...
1
vote
3answers
105 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
20 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
37 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
95 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
27 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
164 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
65 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
61 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 = ...
2
votes
1answer
59 views

The Vacisek Model and the short rate process

I am trying to do some calculations related to the Vacisek model, but I think I am mixing up concepts and I'm not getting to any solution. Let me explain what the problem is. The Vacisek model ...
4
votes
1answer
46 views

Expectation of $e^{-4B_\tau}$, where $\tau$ is an extended stopping time

This is an specific example so with a bit of luck I can get some general methodology from your answers. I have this stopping time: $$ \tau = \inf\{t \geq 0; B_t < t-2 \} $$ This is a clear ...
5
votes
2answers
129 views

Submartingale example: proof

I am trying to prove if the process $M_t = e^{W_t^2-t}$ is a submartingale ($W_t$ is the Wiener Process). The proof becomes a bit difficult, to the point where I am unsure how to move forward. Let ...
1
vote
1answer
35 views

Derivative of a parameteric martingale — is it a martingale?

Let $X(t)$ be a martingale (in continuous time), and for each real $u$, let $Y_u(t)=g(u,X(t))$ for some infinitely-differentiable $g:\mathbb{R}^2\to\mathbb{R}$, and assume that $Y_u(t)$ is a ...
3
votes
1answer
29 views

Sufficient condition for time-changed quadratic covariation to vanish in probability

Let $(M_t^n)_{t \geq 0}$ be a sequence of continuous martingales of the form $M^n_t = \int_0^t X^n_s \, dB_s$ where $B_s$ is a Brownian motion. Let $\tau^n(t)$ be the time change associated to $M_t^n$ ...
2
votes
1answer
33 views

Is the following Markov Chain a martingale?

Say I have a finite, ergodic Markov chain with states ${0,1,2,3}$ and with the following transition matrix: $$\begin{bmatrix} \frac{7}{10} & \frac{3}{10} & 0 &0\\ \frac{1}{10} & ...
1
vote
2answers
25 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 ...
0
votes
0answers
56 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 ...
2
votes
1answer
44 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 ...
0
votes
1answer
51 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 ...
1
vote
1answer
79 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 ...
2
votes
1answer
42 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 ...