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Prove that:
Y(t)=exp{ω(t)−21t} is a martingale,...
Oct 21, 2024
To prove that Y(t)=exp{ω(t)−21t} is a martingale, we need to show that the expected value of Y(t) conditioned on the past is equal to Y(s) for s < t .
1. Recall the properties of Brownian motion: ω(t) is a standard Brownian motion, which means that ω(t)−ω(s) is normally distributed with mean 0 and variance t−s for s < t .
2. Use the definition of a martingale: We need to show that:
E[Y(t)∣Fs]=Y(s)
where Fs is the filtration up to time s.
3. Calculate \( \mathbb{E}[Y(t) | \mathcal{F}_s] \):
Y(t)=exp{ω(t)−21t}
can be rewritten using the increment of Brownian motion:
Y(t)=exp{ω(s)+(ω(t)−ω(s))−21t}
This simplifies to:
Y(t)=exp{ω(s)−21s}⋅exp{ω(t)−ω(s)−21(t−s)}
Thus,
Y(t)=Y(s)⋅exp{ω(t)−ω(s)−21(t−s)}
4. Find the conditional expectation:
Since ω(t)−ω(s) is independent of Fs and normally distributed with mean 0 and variance t−s:
E[exp{ω(t)−ω(s)−21(t−s)}∣Fs]=exp{−21(t−s)}⋅E[exp{ω(t)−ω(s)}]
The moment-generating function of a normal distribution gives us:
E[exp{ω(t)−ω(s)}]=exp{21(t−s)}
5. Combine the results:
Therefore,
E[Y(t)∣Fs]=Y(s)⋅exp{−21(t−s)}⋅exp{21(t−s)}=Y(s)
Since we have shown that E[Y(t)∣Fs]=Y(s), we conclude that Y(t) is indeed a martingale.
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