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Questions tagged [martingales]

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

In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Expressing the following mathematically:

A sequence of random variables $X_0, X_1, \dots$ with finite means such that the conditional expectation of $X_{n+1}$ given $X_0, X_1, X_2, \dots, X_n$ is equal to $X_n$, i.e., $$\mathbb{E}[X_{n+1}\mid X_0, X_1, \dots, X_n] = X_n.$$

A one-dimensional random walk with steps equally likely in either direction $(p=q=\frac12)$ is an example of a martingale.

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Question about a predictable process - submartingale

Question: Assume that (Xn, Fn) is a submartingale, that (Hn)n∈N is predictable, and that each Hn is non-negative and bounded. Show that ((H · X)n, Fn) is a submartingale. Can someone help with this?
Juliane Buch
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Martingales: What does $F_{n-1}$ measurable mean compared to $F_{n}$ measurable?

My workbook gives the following property for Conditional Expectation and martingale: Measurability: If $Y_n$ is "$F_n$ measurable", then: $E[Y_n~|~F_n] = Y_n$ Let's say I have a Stochastic Process $\{A_n, n\ge 0\}$ that is "$F_{n-1}$" measurable?…
pico
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Convexity of a set of martingale measures

Let $0\leq \underline{\sigma} \leq \overline{\sigma}$ be two constant matrices in $\mathbb{S}^d$. Let $W$ be a Brownian motion under the measure $P_0$ and define $$ \mathcal{P} := \{P^\sigma \colon \sigma \in L^0(F; \mathbb{S}^d) \text{ such that }…
T-at-R
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Relationship between martingales and positive submartingales

Suppose $X_n$ is a submartingale with $X_n > 0$ for all $n. There must exist $A_n$, $M_n$ such that $A_n$ is predictable and increasing and $M_n$ is a martingale so that $X_n = A_nM_n$. How is it possible to show this?
qp212223
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Find sequence for a martingale.

Let $(X_n)_{n\geq 1}$ be independent such that $E(X_i)=m_i$, ${\rm var}(X_i)=\sigma_i^2$, $i\geq 1$. Let $S_n=\sum_{i=1}^n X_i$ and $\mathcal{F}_n=\sigma(X_i,1\leq i\leq n)$. Find sequences $(b_n)_{n\geq 1}$, $(c_n)_{n\geq 1}$ of real numbers such…
FrankZ
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Using CBS-principle to show submartingales

I need to give a proof for some lemma which states the following: If $(X_n,\mathbb{F}_n)_{n\geq 1}$ is a submartingale and $\tau$ is a stopping time, then $(X_{\tau\wedge n},\mathbb{F}_{n})_{n\geq 1}$ is also a submartingale. Now the CBS-principle…
mas2
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Can a process be shifted by some $cn$ and it retains martingale? Martingale linearity?

Can a process be shifted by some $cn$ and it retains martingale? Martingale linearity? Particularly the problem in my case is that: My process is defined by $M_n=Y_n-cn$, $c>0$, where $Y_n=f(X_1,...,X_n)$ is known as sub-martingale process. My $F_n…
mavavilj
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Showing that supermartingale is a u.i. martingale

Let $M_t$ be a non-negative supermartingale, $EM_0 < \infty$ and $M_t \to M_\infty$ a.s. I want to show that if $E M_\infty = EM_0$, then $M$ is a uniformly integrable martingale. I thought about doing so by showing that $M_t = E ( M_\infty |…
Felix P.
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State Price Density-Derivation ( Black Scholes Economy)

Good morning, I am trying to understand the state-price density in a Black-Scholes Economy. Model Setup Consider a filtered probability space $( \Omega, \mathcal{F}, \mathcal{F}_t, P ). $ Furthermore let's define a standard Brownian Motion $W_t$…
Mike
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Details of Lp martingale convergence theorem

In class we learned about Lp martingale convergence theorem. I could not figure out why |Xn-X∞|<2X* implies the convergence in Lp by the dominated convergence theorem. Hope someone could explain a bit. Lp martingale convergence theorem: Let p ∈ (1,…
TNightS
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What is the significance of IIDs for Martingale Processes

I notice that every Martingale Process refers somehow to an iid. I was wondering why do we need that and what role does the iid actually play in the process. Sometimes when the iid is not obvious to be spotted but it is there, is there a way to…
site
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CDF of Martingale

$(X_k)_{k \in \mathbb{N}}$ is iid with $\mathbf{P} [X_1 = \frac{1}{2}] = \mathbf{P} [X_1 = \frac{3}{2}] = \frac{1}{2}$. $M = (M_n)_{n \in \mathbb{N}_0}$ with $M_0 = 1$ and $M_n = \prod_{k = 1}^n X_k$ for $n > 0$ is a martingale. For $0 < a <…
Andy
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Checking for martingale and Itô's formula

Assuming that $\{ W ( t ) | t \geq 0 \}$ is a Brownian motion, I'm trying to check whether the process $$X ( t ) = W ( t ) + 4 t$$ is martingale with respect to filter $\mathcal{F}_t$. For this we should check if $\mathbb{E}[X(t)|\mathcal{F}_s] =…
Blade
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Martingale, decomposition, expectancy value

I am trying to understand a proof in my book but i can't figure out a few things. Definitions first: $(\Delta X)_n :=X_n-X_{n-1}$ with $X_{\alpha-1}:=X_\alpha$, if $\alpha>-\infty$, so $(\Delta X)_\alpha=0$ if $\alpha>-\infty$. A process…
Benjamin
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computational attempt on symmetric random walk

Theorem: Let $X_{n}$ be a Markov chain with transition probability p and let $f\left ( x,n \right )$ be a function of the state x and the time n so that $f\left ( x,n \right )=\sum_{y \in Y}{}p\left ( x,y \right )f\left ( y,n+1 \right )$. Then…
Mathematicing
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