# Tag Info

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### Difference between Ito calculus and Malliavin calculus

The Ito calculus extends the methods of classical calculus to stochastic functions of random variables. The Malliavin calculus extends the classical calculus of variations to stochastic functions. ...
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### Further Reading on Stochastic Calculus/Analysis

just to offer two cents on this (health warning is that this is from an ex-trader rather than quant, and based on personal experience through self-study rather than eg as part of a formal PhD ...
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### Density of cylindrical random variables in classical Wiener space

This proof is the proverbial "long and winding road". It can be made somewhat less tedious by the use of Dynkin's multiplicative system theorem, which I state and discuss in this answer. Note that ...
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### A question about Malliavin calculus

Applying the Malliavin calculus in order to calculate sensitivies is a little bit difficult, since the Malliavin derivative is not as simple to use as the ordinary derivative operator. Essentially (...
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### Martingale representation of European option.

Note that your Stock Market satisfies the following SDE $$dS(t) = \sigma(t)S(t)dB(t).$$ Hence it is a martingale with respect to $\mathcal{F}_t$ (the natural filtration of your Brownian motion $B$). ...
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### Apparent inconsistency in Skorohod integration

The first term is a regular Itô integral, [...] That's not true, the term $$\int_0^{\tau} W(t) \, dW(s)$$ is not a regular Itô integral since the integrand $W(t)$ is clearly not $\mathcal{F}_s$-...
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### Approximation on partitions in $L^2([0,1]\times \Omega)$

I believe that there are some typos. It should be \begin{align*} \tilde u^n(t)=\sum_{i=1}^{\color{red}{2^n}}2^n\left(\int_{(i-1)2^{-n}}^{i2^{-n}}u(s)ds\right)1_{]\color{red}{(i-1)2^{-n}, i2^{-n}}]}(t)...
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### Malliavin derivative of a Lebesgue integral.

I assume that $X_t$ is progressively measurable with respect to the corresponding Wiener process $W_t$ (otherwise the formula doesn't hold). $X_t$ being $\sigma(W_s,s\le t)$-measurable implies that it ...
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### Malliavin Derivative

This is a consequence from the Clarke-Ocone Theorem, and uses Malliavin derivative. See also the Clarke-Ocone formula paragraphe here. If you want a technical reference, see this introductory course, ...
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### Elementary Malliavin Derivative question about definition.

Since the $(I_n)$ form an orthogonal family, the $\mathrm{L}^2(P)$ norm of $F$ is given by $$\|F\|^2=\sum_{n\ge0}\|I_n(f_n)\|^2=\sum_{n\ge0}n!\|f_n\|_{\mathrm{L}^2([0,T]^n)}^2,$$ which is finite ...
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### What is the Malliavin derivative of the sum of two independent Brownian motions?

If you assume that $\Omega=C_0([0,1];\mathbb R^2)$ and you let $H:=L^2[0,1]\oplus L^2[0,1]$ you can see that in Nualart's notation (see section 1.2. of The Malliavin Calculus and Related Topics) you ...
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### Interchanging Malliavin derivative with Lebesgue integral

We'll prove this in two steps. First, we pull the derivative operator $\mathfrak{D}$ inside the (Lebesgue) integral, and then apply a "Chain Rule". Since the book by Øksendal et al. is referenced, I ...

### Malliavin derivative under change of measure

(I repost here my partial answer from mathoverflow, https://mathoverflow.net/questions/229044/malliavin-derivative-under-change-of-measure/229874#229874) Here an answer for the case with determinist ...

### Calculus of Variations in Probability Theory

I found a neat example. It answers the question: Given constraints on its expected values, which distribution has maximum entropy? The answer can be shown to be the exponential family, using the ...
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• 3,183
1 vote

### Apparent inconsistency in Skorohod integration

I think you should use distinct notations for the Itô integral and the Skorohod Integral. By Proposition 2.6 of Malliavin Calculus for Lévy Processes with Applications to Finance (the book you refered)...
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1 vote
Accepted

### On the $L^2$-norm of symmetrized functions. A question concerning the book on Malliavin calculus from Nualart

Yes, take $f = 1_{[a,b]\times [c,d]}$. with $a > d$ so that the function does not intersect the diagonal. The symmetrization of \$f = \frac{1}{2}1_{[a,b]\times [c,d]} + \frac{1}{2}1_{[c,d]\times [...
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