# Questions tagged [martingales]

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

3,628 questions
Filter by
Sorted by
Tagged with
30 views

62 views

### One question about exponential martingale inequality at a paper of Ann. of Math.

I saw a strange inequality about martingales: If $\operatorname{M}\left(s,t\right)$ is an continuous $L^{2}$ martingale start at $s$, $\left[\operatorname{M}\right]\left(s,t\right)$ is its quadratic ...
71 views

• 111
176 views

• 919
1 vote
47 views

### Questions in proving $\mathbb{P}\left(T_a<\infty\right)=1$ with $T_a:=\inf \{t>0: B_t \ge a\}$

Let $\left(B_t, t \geq 0\right)$ be a one-dimensional Brownian motion starting from the origin (i.e, $\left.B_0=0\right)$. Let $\mathcal{F}_t:=\sigma\left(B_s: s \leq t\right)$ be the filtration ...
• 919
19 views

### Asymptotic behaviour of the difference process of the Polya urn

Assume we have a Polya urn with $(a,b)$ initial (time t=0) red and green balls, respectively. At each point in time we draw a ball at random and add it back together with further $S$ balls. This means ...
77 views

### Show this truncated stochastic process is a martingale

Consider a filtered space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\geq 0}, P)$. Let $(M_t)$ be a uniformly integrable (UI) martinagle w.r.t. $\{\mathcal{F}_t\}_{t\geq 0}$. Let $\tau, \nu$ be two ...
• 513
65 views

### Gambler ruin's: Probability of $k$ consecutive win before j consecutive loss

Assume that a stock has a probability of $p$ to win, a probability of $q$ to lose, and a probability of $(1-p-q)$ to remain every day. What is the probability of $k$ consecutive wins occur before $j$ ...
64 views

• 73
1 vote
44 views

### Renewal reward process's reward tail probability

Suppose we are given a dice with $K$ faces, denoted by $k=1,\dots,K$, where the probability of realizing a face $k$ is $p_k\in[0,1]$ with $\sum_{k=1,\dots,K}p_k=1$. Now, we roll the dice repetitively. ...
• 11
36 views

### Is there a closed form solution to the distribution of the running maximum for Gaussian processes?

Let $X=(X_t)_{t\geq0}$ be a zero mean Gaussian process with variance $\sigma(t)=\mathbb{E}X_tX_t^\top$ and define $$S_T=\sup_{t\leq T}X_t\quad T\in[0,\infty)$$ the running maximum of $X$. Question: ...
• 362
120 views

### On the Optimal Transport Distance in Equivalent Martingale Measures

I've been working on this idea for a while and could use some guidance. Right now I'm wondering if the formulas I am writing out make sense. Any help is very much appreciated. Problem Statement Idea ...
• 1,099
24 views

45 views

• 129
51 views

### Can we always find a martingale part in a càdlàg supermartingale?

Abstract: By Doob-Meyer decomposition theorem, for any càdlàg supermartingale $Z$, there exists a unique predictable increasing process $A$ starts from $A_0=0$ such that $Z+A$ is a local martingale ...
59 views

• 2,970
80 views

### An example such that $(X_n)$ and $(Y_n)$ are martingales with respect to their natural filtration and $(X_n +Y_n)$ is no martingale [closed]

I know that if $(X_n)$ and $(Y_n)$ are martingales with respect to their natural filtration then $(X_n +Y_n)$ does not have to be a martingale with respect to its natural filtration. However, I did ...
• 615
1 vote
37 views

• 1,167
43 views

### Question regarding a value in a one-period model

There is a script at my university (can't post it for copyright reasons) for a course on discrete time financial mathematics. I decided to give it a try and found this problem: Consider a financial ...
32 views

### Martingale inequality with Convex Function

I want to know how to prove following equation appearing in the book 'Stochastic approximation and recursive algorithms and applications' without proof. Let $\{Mn,Fn\}$ be a martingale, $q(\cdot)$ be ...
• 21
21 views

### Is any (integrable) centered, adapted stochastic process with independent increments a martingale?

I came across the following question: Assume $(X(t), t\geq 0)$ is an integrable, adapted stochastic process on a filtered probability space $(\Omega, \mathcal{F},(\mathcal{F}_t, t\geq 0), \mathbb{P})$ ...
• 67
720 views

### How to apply Ito's Formula to show that this is a martingale?

In the book Brownian Motion, Martingales and Stochastic Calculus by J.F. Le Gall, in order to give an alternatice derivation of the distribution of $L_{U_{a}}^{0}(B)$ where $L^{0}_{t}(B)$ is the Local-...
• 1,285
109 views

44 views

### Derive the 0-1 law from a functional equation of a martingale limit

I am trying to understand a proof given by Jabbour-Hattab in the paper "Martingales and Large Deviations for Binary Search Trees". In this paper, we consider the limit of a non-negative ...
• 178
1 vote