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How to upper bound the difference between these two Gaussian-like densities?
@IosifPinelis You are right! In my context, $h$ is the discretization size step of the Euler scheme, so $h$ is bounded by a fixed $T>0$.
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How to upper bound the difference between these two Gaussian-like densities?
@IosifPinelis $h$ is a constant...
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How to upper bound the difference between these two Gaussian-like densities?
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Is this map (from the space of probability densities to the Wasserstein space) Lipschitz?
Thank you so much for your help!
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Is this map (from the space of probability densities to the Wasserstein space) Lipschitz?
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Is this map (from the space of probability densities to the Wasserstein space) Lipschitz?
@Dirk Thank you so much for your comment. I have edited the question.
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Is this map (from the space of probability densities to the Wasserstein space) Lipschitz?
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Reference request: $d X_t = b(X_t) d t + f (p_t(X_t)) d W_t$ where $p_t$ is the p.d.f. of $X_t$
@ThomasKojar Thank you so much for your suggestion! I will check it out.
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How is $\mathbb E[ \int_0^T H_s^2 \mathrm d s] < \infty$ important for this claim?
I got it. Thank you so much for your help!
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How is $\mathbb E[ \int_0^T H_s^2 \mathrm d s] < \infty$ important for this claim?
Ah I meant the claim "The hypothesis of boundedness can be weakened to the hypothesis $$ \mathbb{E}\left[\int_0^T H_s^2 d s\right]<\infty $$
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How is $\mathbb E[ \int_0^T H_s^2 \mathrm d s] < \infty$ important for this claim?
I have just found the same statement with the same hypothesis (equation (8.20)) in this note. Could you please have a look at it?
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How is $\mathbb E[ \int_0^T H_s^2 \mathrm d s] < \infty$ important for this claim?
I got that $E\int_0^1 H_t^2 \, dt = \sum_{k=1}^\infty p_k (2^k)^2 \int_0^1 \big [ \sum_{j=0}^{2^k} \big (1- 8^k \big |t-\frac{\color{red}{j}}{2^k} \big| \big)_+ \big ]^2 dt$. Could you explain how you simplify it to $\sum_{k=1}^\infty p_k\; (2^k)^2\,2^k \int_0^1 \Big(1-8^k\Big|t-\frac {\color{red}{1}}{2^k}\Big|\Big)_+^2\,dt =\sum_{k=1}^\infty p_k\; (2^k)^2\,2^k \frac23\,\frac1{8^k}$?