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

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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}, ...
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2answers
40 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 = ...
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19 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 ...
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14 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 ...
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1answer
107 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 ...
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23 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!
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1answer
18 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 ...
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33 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 ...
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71 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 ...
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33 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 ...
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23 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 ...
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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 $\omega \in \Omega$ ...
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2answers
53 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: ...
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1answer
31 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 ...
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32 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 ) ...
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1answer
59 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\{ ...
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80 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 ...
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43 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 ...
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37 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 ...
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40 views

An exponential martingale

Let $H_{t}$ be a bounded continuous and $\textbf{F}^{B}_{t}$ an adapted process. $B$ Brownian motion. Show that $M_{t}= \exp\left(-\int^{t}_{0}H_{s}dB_{s} -\frac{1}{2}\int^{t}_{0}H^{2}_{s}ds\right)$ ...
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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 ...
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1answer
87 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 ...
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3answers
103 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 ...
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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 ...
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34 views

Second (centered) moment for martingales

Take the process ${x}_t$ following geometric Brownian motion (GBM) $$x_t=\mu x_t \,dt+\sigma x_t \,dW_t$$ with $x_0>0$ known. It has first moment equal to $$\text{E}[x_t]=x_0 e^{\mu t}$$ and second ...
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2answers
36 views

Why martingales need to be integrable?

In almost every book the definition of martingale ask for integrability of every random variable. Why this property is needed? If we remove it the property that the expected values is constant holds? ...
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1answer
19 views

Is there an example that shows that the optional stopping theorem fails for finite (unbounded) stopping times?

Is there a martingale $M=(M_t)_{t\geq 0}$ and finite stopping times $S,T$ with $S \leq T$ a.s. such that $\mathrm{E}(|M_T|)<\infty$, but $M_S \neq \mathrm{E}(M_T|\mathcal{F}_S)$ a.s.? I found a ...
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1answer
36 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 $$\mathbb{P}(\exists\,i=1,\dots,n\text{ s.t. }|X_i-X_{i-1}|>c_i) \,\leq\, \epsilon$$ ...
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1answer
29 views

Ito's process and martingale [duplicate]

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

Strict local martingale implies $\mathbb E_t[S_u]<S_t\ \forall t<u$

Is it true that if $S$ is a strict local martingale (i.e. it is a local martingale but not a true martingale) such that $S_t\ge 0\ \forall t$, then we have $$\mathbb E_t[S_u]<S_t\quad \forall ...
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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 ...
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1answer
84 views

Compute a probability in Random Walk by Martingales

Let $X_n$ be the state at time $n$ of a Markov chain with these transition probabilities : $$p_{i,i+1}=p_i\qquad,\qquad p_{i,i-1}=q_i=1-p_i$$ $(a)$ Show that $Z_n=g(X_n)\,;\,n\geq0$, is a ...
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1answer
23 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 ...
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1answer
63 views

Showing that a certain stochastic process does not have normal distributed increments

Edit: Question Resolved. See below. As a part of my bachelor thesis, I have to work through a paper about fake Brownian motion by Oleszkiewicz. In this paper he defines a stochastic process. Let ...
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1answer
25 views

Prove $X^2$ is a submartingale given X is a martingale

As stated in the question, I have an exam tomorrow and am looking to prove this result. I know: $$E(|X_n^2|) = E(X_n^2)$$ I don't know how to advance showing that this is finite. If anyone could ...
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21 views

Show Wright-Fisher Model is a martingale

Consider successive generations of a population of genes of fixed population size M. Each gene can be just one of two alleles, type a or A. Let $Y_n \in S = \{0,1,\dots,M\}$ denote the number of type ...
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26 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 ...
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1answer
161 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 ...
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2answers
62 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 $M_n$ ...
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52 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 ...
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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 = ...
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1answer
55 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 ...
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1answer
35 views

Prove a P Martingale

If: $$ \sigma_t $$ is a bounded function of both time and sample path, show that: $$ dX_t=\sigma_tX_tdW_t $$ is a P Martingale. *Does this question make sense, that is, should the question be: is ...
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1answer
66 views

Best martingale for sequence of “dozen” bets at roulette game

Jim goes the Casino to play roulette. He only makes “dozen” bets at each spin ; his probability of winning is therefore $\frac{1}{3}$ every time (to simplify, we neglect the effect of the zeros in ...
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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 ...
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124 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 ...
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1answer
24 views

Computing expectation of

I am reading a paper and got stuck on this simple equation: $$\mathbb{E}_t[e^{-cS_T}]$$ where $dS_t=\sigma W_t$ with $W_t$ standard 1 dimensional Brownian motion, $S_t=s$ and c some constant. I ...
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1answer
33 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 ...
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1answer
27 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 $M_t^n$ ...
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2answers
41 views

Making a non-Martingale process a Martingale

Stuck on this question for a very long time: was wondering if any kind soul could help me out: Suppose $B_t$ is a standard Brownian Motion under measure P. Question: Create a martingale process that ...