# Questions tagged [martingale]

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### What does expectations of a Random variable change with respect to an indicator function [closed]

What does E[Y 1A] mean where E is expectations, Y is a Random Variable, 1A is indicator function with respect to another function A. And how to prove the below property? E [Y 1A] = E [E[Y | Fn] 1A]
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### Different risk neutral measure

I don't understand in the following example how there can be a single risk neutral measure. The risk free asset price $B$ at time $t = 1$ is $1+R$. An other asset $S$ at time $t=1$ can take two values:...
1 vote
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### Martingale property of the CEV model

I am a bit confused about the martingale property of the CEV model. Given $dS(t)=σS(t)^βdW(t)$, is $S$ a martingale for values of $β<1$?
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### replicating self-financing portfolio for risk neutral measure

Let the price process $S_{t}, 0 \leq t \leq T$, be a diffusion, and savings account be $\beta_{t}$ such that the Equivalent Martingale Measure $Q$ exists. Let $C_{T}=g\left(X_{T}\right)$ be the claim ...
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### Derivation of option pricing PIDE: Why does the drift need to be zero?

I started studying PIDE methods for option pricing and am struggling to understand or find the necessary theory that shows why the PIDE is obtained by the condition that the drift term has to be zero. ...
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1 vote
142 views

### Hermite polynomials as martingales [closed]

Let $\left\{W_{t}: t \geq 0\right\}$ be a standard B.M. on the filtered probability space $\left(\Omega, \mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t \geq 0}, \mathbb{P}\right)$. Define the Hermite ...
1 vote
271 views

### Martingale problem on biased random walk

I am struggling to understand the martingale property of exponential of a biased random walk. For example, in the following problem how do I verify whether the following is a martingale, submartingale ...
838 views

### Proving that a stochastic process is a martingale using Ito's Lemma

Assume a Wiener process W and a bounded F-adjusted stochastic process a. Show that the following process is a martingale on F $$X(t)=(\int_{0}^{t}a(s)dW(s))^{2}-\int_{0}^{t}a^{2}(s)ds,\ t\geq0$$ Can ...
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### If arbitrage can happen exactly at one moment, is it really arbitrage?

There are many "interpretations" of what no-arbitrage means in mathematical finance, the most well known is no free lunch with vanishing risk: If $S=\left(S_{t}\right)_{t=0}^{T}$ is a ...
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