# Questions tagged [martingales]

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

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### Why is conditional expectation of brownian motion with a negative sign?

I have been reading up on the below thread: conditional expected value of a brownian motion but I cannot understand how πΌ[π΅π βπ π‘π΅π‘+π π‘π΅π‘ | π΅π‘]=πΌ[π΅π βπ π‘π΅π‘| π΅π‘]βπΌ[π π‘π΅π‘ | π΅π‘]. ...
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### Necessary and sufficient condition for random sum of independent RVs to be a martingale

Let $M$ be a Poisson random measure on $(0,\infty)$ with intensity $\lambda dt$, where $\lambda\in(0,\infty)$. Let $(Y_n)_{n\in\mathbb{N}}$ be a sequence of independent random variables, independent ...
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### Does this setup imply that $E[(M_t-M_s)^4 \mid \mathcal{F}_s]$ is bounded?

Suppose $(M_t)_{t \geq0}$ is a martingale w.r.t. a filtration $(\mathcal{F}_t)_{t \geq0}$. Suppose that $$E[(M_t-M_s)^2 \mid \mathcal{F}_s]$$ is uniformly bounded by some constant. I want to prove ...
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### Understanding proof of Azuma's inequality

I am trying to understand the proof of Azuma's inequality, though one step isn't quite clear to me: To give some context: $V_1,V_2,\dots$ is a martingale difference sequence with respect to the random ...
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### Proving $E\bigg[\bigg(\int_a^b X_s dW_s \bigg)^2 \ \bigg\vert \ \mathcal{F}_a \bigg] =\int_a^b (X_s)^2 ds$

Suppose $(\mathcal{F}_t)_{t \geq0}$ is a filtration on a probability space and $W=(W_t)_{t \geq0}$ is a Brownian Motion with respect to this filtration. Let $(X_t)_{t \geq0}$ denote some ...
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### Proof that existence of specific process implies no numeraire strategies

Consider a discrete-time market with $n$ assets and (possibly negative) prices $(P_t)_{t\geq0}$. A numeraire strategy is a self-financing investment strategy such that the wealth process is almost ...
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### Supremum of Brownian motion increments

Let $W=(W(t))_{t\geq 0}$ be a Brownian motion. Consider the random variable $$Y(t):=\sup_{1\leq s\leq t}[W(s)-W(s-1)],$$ for some fixed instant $t\geq 0$. I am interested in this $Y(t).$ Could anyone ...
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### Discrete time positive martingale with non-trivial limit

Let $X_t$ be a positive discrete time martingale. (I.e., $X_t>0$ and $\mathbb{E}_t X_{t+1}=X_t$ for all $t\in\mathbb{Z}$.) Then by Doob's supermartingale convergence theorem, there exists some non-...
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### Filtration of sum of independent sub-martingales

Let $x_1,\ldots, x_n$ be independent continuous martingales with filtrations $\mathcal{F}^1_{t},\ldots, \mathcal{F}^n_{t}$. Let $Y_i=(x_i)^2$ be the associated sub-martingale with each $x_i$ (we can ...
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