Questions tagged [martingales]
For question about discrete or continuous (super/sub)martingales. Often used with [probability-theory] tag.
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Applying Ito's rule to: $Y(t)=f(M(t),⟨M⟩(t))$ (a function of Martingale and its quadratic variation)
I am attempting to compute $dY(t)$ where $Y(t)=f(M(t),⟨M⟩(t))$ and $M(t)=\int_0^t\sigma(s,\omega)dB(s,\omega)$.
My attempt is that $dY(t)=\frac{\partial f(M,⟨M⟩}{\partial ⟨M⟩} d⟨M⟩ + f'(M,⟨M⟩)dM + \...
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Arbitrage opportunities in the primary market
Let $\Omega$ be the set of the set of all $2^T$ outcomes for the values of the stock price $(S_0,\dots,S_T)$. Denote by $\mathbb{P}^{\ast}$ the risk-neutral probability measure and by $V_t$ the value ...
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Expected return is independent of trading strategy
We have proven the following statement in class: (discrete case)
The discounted stock price process $\hat{S}$ and the value process $\hat{V}(\Phi)$ of any self-financing trading strategy $\Phi=\{(\...
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Proving Wald's second equation using Optional Stopping Theorem
I am working on an exercising about proving Wald's second equation, which is as follows:
Let $S_n = \xi_1 + \cdots + \xi _n$ where $\xi _i$ are independent, $\mathbb{E} \xi_1 = 0$
and $\text{Var}(\...
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Prove that $S^0=S^1$ so there is one bank account in the market without arbitrage
In a two-period market with four possible scenarios, the risk-free interest rate is $10$%. There is one stock in this market whose prices are described by process $S$:
$$S_0 = 100, S_1(\omega_1) = S_1(...
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continuous martingale and brownian motion [duplicate]
I am trying to prove that if $B(t) $is a continuous martingale and $B(t)²-t$ is also a martingale then $B(t)$ is a brownian motion.
I was able to prove that:
$B(0)=0$ almost surely.
Stationary and ...
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Show that it is a stopping time
My task is to describe the following stopping time formally and to prove that it is indeed a stopping time.
Let $A\in \mathcal{B}(\mathbb{R})$ be a fixed set. $(X_t)_{t\ge 0}$ hits A for the first ...
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Let $M$ be a continuous martingale, $r >0$, and $\tau := \inf \{t \ge 0 : \langle M \rangle_t \ge r \}$. Then $M_\tau$ is square-integrable
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $\mathcal G = (\mathcal G_t, t \ge 0)$ a filtration. Let $M$ be a real-valued continuous martingale w.r.t. $\mathcal G$ such that
$$
(\...
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Brownian motion: how is DCT for conditional expectation applied on this stopped martingale?
I'm reading a theorem at page $43$ of these notes, i.e.,
Proposition 7.12. Let $M$ be a continuous local martingale with respect to a filtration $\left(\mathcal{F}_t, t \in \mathbb{R}_{+}\right)$ ...
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Le Gall's Lemma 5.14: how to obtain $B \subset C$ a.s. from $B \subset \{X^{(n)}_a = X^{(n)}_b\}$ for all $n$?
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $\mathcal G = (\mathcal G_t, t \ge 0)$ a completed filtration. Let $M$ be a real-valued continuous local martingale w.r.t. $\mathcal G$....
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Submartingale convergence (Durrett 5.3.1)
Exercise 5.3.1 in Durrett's "Probability Theory and Examples" states
Let $X_n$, $n\ge 0$, be a submartingale with $\sup X_n < \infty$. Let $\xi_n=X_n-X_{n-1}$, and suppose $E(\sup \xi_n^+)<\...
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Stopped martingale: is this a more direct proof to this question?
I'm reading this question by @KennethNg, i.e.,
I am reading the Dubins-Schwarz theoem on Brownian motion and stochastic calculus. It says, given a continuous local martingale $M$ such that $\lim_{n\...
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Proof of Doobs decomposition theorem
I have a question regarding the proof of Doobs theorem (discrete case): It says that every adapted and integrable stochastic process $Y$ can be uniquely decomposed as the sum
$Y_t = M_t + A_t$ where
$...
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Brownian motion: how is $\mathbb E [\sup_{t \ge 0} |\langle M\rangle_{t \wedge \tau_n}|^2] < \infty$ satisfied?
I'm reading a theorem at page $43$ of these notes, i.e.,
Proposition 7.12. Let $M$ be a continuous local martingale with respect to a filtration $\left(\mathcal{F}_t, t \in \mathbb{R}_{+}\right)$ ...
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Assume $M^{\tau_n}$ is uniformly integrable. Is $X^{\tau_n}$ uniformly integrable?
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $\mathcal G = (\mathcal G_t, t \ge 0)$ a filtration. Let $M$ be a real-valued continuous local martingale w.r.t. $\mathcal G$. Let $(\...
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Local martingale: how is $\sup_{s \in[0, t]}\left|M_s\right|$ measurable?
I'm reading about local martingale from this Wikipedia page, i.e.,
Let $M_t$ be a local martingale. In order to prove that it is a martingale it is sufficient to prove that $M_t^{\tau_k} \rightarrow ...
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A proof of Lévy's theorem: how to obtain the independence $X_t-X_s \perp \mathcal{F}_s$?
I'm reading about Lévy's theorem at page $42$ of these notes, i.e.,
Let $X$ be a continuous local martingale such that $X_0=0$ a.s. and $\langle X\rangle_t=t$ a.s., $\forall t \in \mathbb{R}_{+}$. ...
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20-sided dice 100 rounds game
In a game there is a 20-sided die. At the start of the game it is on the table and the 1 face is facing upright. In each of the 100 rounds you have 2 options: you can either roll the dice or you can ...
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If $B$ is a Brownian motion and $f \in \mathcal C^2(\mathbb R^n)$ harmonic, then $f(B)$ is a continuous martingale
I'm reading about Itô's formula for a multi-dimensional Brownian motion at page $39$ of these notes, i.e.,
Let $\underline{B}$ be a standard $n$-dimensional Brownian motion and $f \in \mathcal{C}^2\...
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Branching Process; understanding theorem proof
Now I'm reading Durrett's probability book and having trouble understanding the proof of one of the theorems related to Branching Process. (Theorem 5.3.8.)
Let $\xi_i^m,$ $\, i,n\geq 1$ be i.i.d. ...
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Is Brownian motion a semimartingale?
I have read an article about semimartingales on Wikipedia and it says that: "A Brownian motion is a semimartingale". However, it is hard for me to find any proof of this statement, and I ...
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Independence of random variable to sigma-algebra generated by previous ones
Let $(X_n)$ be a sequence of iid random variables such that $\mathbb E(X_n)=0, \Bbb E(X_{n+1}^2)$ $\forall n$.
In a martingales context, we denote a finite sum of these random variables $S_n:=\sum_{i=...
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Prove that $Z_{n} = X_{n} 1_{ (T>n)} + Y_{n} 1_{ (T \leq n)}$ is a supermartingale.
I want to prove that $Z_{n} = X_{n} 1_{ (T>n)} + Y_{n} 1_{ (T \leq n)}$ is a supermartingale. I know that $\{X_{n}\}, \{Y_{n}\}$ are supermartingales. T is a discrete stopping time and if $T< \...
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Likelihood Ratio Martingales
I am reading about so-called "likelihood ratio martingales" in this handout.
The example given is as follows.
Let $(X_n : n \ge 1)$ be a sequence of iid random variables (say, on a ...
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Waiting time pattern in Levin , Peres
In Levin and Peres, Markov chains and Mixing Times , under the section 17.3.2 Waiting time patterns, we are looking for the expected waiting time before the patter $101$ appears upon tossing fair ...
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How to show Martingale property for sum of $S_k-E(S_k)$-summands where $S_k$ is a function of two RV's
EDIT: new formulation of the question (old version below).
In a paper I found the statement that a certain sum $M_n =Y_1+\dotsb Y_n$ is a martingale, $Y_i=f (X_k, Z_k) - E ( f(X_k, Z_k) | X_k)$. (The ...
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Is M defined as below a martingale?
Let $f\in L^2$. Let $$Z_t:=\int_0^t f(s)dB_s.$$
We know that
$$
L=\exp\left(\int_0^t f(s)dB_s-\frac{1}{2}\int_0^t f(s)^2 ds\right),
$$
is a martingale.
Now, let $\sigma$ be another random variable ...
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Application of Cesaro lemma
I would like to show that the sequence of random variables $\frac{S_n}{n}$ defined below converges to zero. The proof requires the application of Cesaro lemma and I don't understand how.
First, let ...
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Given a local martingale $M$ the running supremum $N_t=\sup_{0\leq s\leq t}|M_s|$ is locally integrable
I am trying to understand a detail of the proof of Theorem VI.82 of the book by Dellacherie and Meyer.
Theorem Let $M$ be a local martingale and let $N_t=\sup_{s\leq t}\left|M_s\right|$. Then $N$ is ...
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Example of a martingale with non-independent increments and fixed variance
I'm trying to come up with a martingale whose increments $\Delta_n = M_n - M_{n-1}$ are non-independent, and have fixed variance, $Var(\Delta_n) = k$.
Attempt 1
Initially I was thinking that $M_N = \...
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Matrix Azuma Inequality with non-zero mean
Based on theorem 7.1 of the paper User-friendly tail bounds for sums of random matrices, we have:
Consider a finite adapted $\{ \mathbf{X}_k\}_{k=0}^{\infty}$ of self-adjoint matrices in dimension $d$,...
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Can this integrability assumption be weakened in a way that the stochastic integration by parts still holds?
$\newcommand{\Ex}{\mathbb E} \newcommand{\diff}{~\mathrm d}$Recently, I have read the integration by parts formula for a continuous semi-martingale in these notes.
Theorem Let $X$ and $Y$ be ...
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Let X be a semimartingale, prove that $[X,[X,X]^c] = 0$
Let $X$ be a semimartingale and $[X,X]^c$ the continuous part of the quadratic variation of $X$. I need to prove that $[X,[X,X]^c] = 0$.
I thought it might be useful to divide the SM into its ...
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Show rigorously that Pólya urn describes a martingale
We work with the famous Pólya urn problem. At the beginning one has $r$ red balls and $b$ blue ball in the urn. After each draw we add $t$ balls of the same color in the urn.
$(X_n)_{n \in \mathbb N}...
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How to show that Ornstein-Uhlenbeck is not a martingale with $E[X(t)]=E[X(s)]$ for all $s,t\in [0,\infty)$?
If we are given that $X=\{X(t)\}_{t\in[0,\infty)}$ is an O-U-Process, which is defined as the continuous centered Gaussian process with covariance function given by
$$\Gamma(s,t)=\frac{\sigma^2}{2\...
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Show that the following sequence is a martingale.
Exercise: Let $X_i$ be a sequence of independent random variables with $\mathbb{E}[X_i] = 0$ and $\text{Var}(X_i) = \sigma_i^2$. Show that the sequence
$$ S_n = \sum_{i=1}^{n} (X_i^2 - \sigma_i^2) $$
...
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Proving that a Stochastic integral is a Martingale
I am trying to prove the following:
$$\lim_{n\rightarrow \infty} E\left(\int_{0}^{t \wedge \tau_{n}} e^{-\delta s} v(X_{s})X_{s} dW_{s}\right)=0$$
where $\tau_{n}$ is a sequence of stopping times with ...
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Meaning of: A continuous martingale of finite varaition is almost surely constant
Theorem: If $\{X(t)\}_{t \in [0,\infty)}$ is a continuous martingale of finite variation then almost surely it holds that $X(t)=X(0)$ for all $t \in [0,\infty)$.
Does this mean that almost every path ...
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How the requirement of Itô's lemma is satisfied in this theorem about integration by parts?
I'm reading a theorem (about integration by parts) from page 9 of these notes, i.e.,
Theorem Let $X$ and $Y$ be continuous semi-martingales such that
$$
\mathbb E \bigg [ \int_0^t X_s^2 \mathrm d \...
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What's the precise statement of the continuous-time optional stopping theorem?
I searched high and low in a number of probability / financial mathematics textbooks and surprisingly cannot find any precise statement of the continuous time optional stopping theorem. In particular, ...
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Under which condition is a stochastic integral a Martingale
I know that for a predictable process $H(s)$ with $\mathbb{E}\left[\int_0^T H(s)d \langle M\rangle(s) \right]<\infty$ the stochastic integral w.r.t to a continuous Martingale $M$ is a Martingale. ...
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Why is the completeness of the filtration needed here?
I am studying how the usual conditions of the filtration are used.
Answers in this post In the definition of a Stopping Time, how important are the conditions on the Filtration being complete and ...
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Show that a martingale $\{X_n\}$ is bounded in $L^2$ if and only if $EX_n^2<\infty$ for each $n$ and $\sum_{n\ge1}E(X_{n+1}-X_n)^2<\infty$
A martingale $\{X_n\}$ is bounded in $L^2$ by definition if $\sup\limits_nEX_n^2<\infty$. Show that a martingale $\{X_n\}$ is bounded in $L^2$ if and only if $EX_n^2<\infty$ for each $n$ and
\...
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Quadratic Variation of a Semimartingale and jumps
Consider the process $X(t) = A(t) + M(t)$, where $\{A\}$ is a finite variation predictable process and $\{M\}$ is a local martingale.
Suppose that $\{A\}$ does not have any jump, hence it is a ...
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How $\int_1^{\infty} P[Y>t] dt \leq \frac 1 {\alpha -1}$ in this proof of martingale maximal inequality?
I have just encountered this question
Suppose $({X_n}, n \in \mathbb N)$ is a martingale. Let $n \ge 1$ and $\alpha > 1$ such that $E\left[|X_n|^\alpha\right]<\infty$. Then
$$
E\left[\max_{0\...
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Does $X_t = M_t+V_t$ have to hold everywhere in the definition of a semi-martingale?
I'm reading about continuous semi-martingale from page 8 of these notes.
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space.
Definition. A continuous semi-martingale w.r.t. a filtration $(\...
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How is this step of simplification valid in a proof of martingale maximal inequality?
I'm reading a proof of $L^p$ maximal inequality from these notes. In the proof, we have $\left\|X_n^*\right\|_p^p \le C_p^p\left\|X_n\right\|_p \left\|X_n^*\right\|_p^{p / q}$ after Hölder's ...
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How the author uses MCT to complete his proof of Doob’s martingale maximal inequalities?
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $M=(M_t, t\ge 0)$ a continuous martingale with respect to a filtration $(\mathcal F_t, t\ge 0)$. Let $T>0$ and $p > 1$. Let $X := ...
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Let $(M_t)$ be a continuous square-integrable martingale with independent increments. Is $t \mapsto \mathbb E[M_t^2]$ continuous?
I'm reading a remark at page 4 of these lecture notes.
Let $\left(\mathcal{F}_t, t \in \mathbb{R}_{+}\right)$be a filtration and $M=\left(M_t, t \in \mathbb{R}_{+}\right)$be a continuous square-...
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The measurability in a proof of Doob’s martingale maximal inequalities
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $M=(M_t, t\ge 0)$ a continuous martingale with respect to a filtration $(\mathcal F_t, t\ge 0)$. Let $T>0, \lambda >0$, and $p \...
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