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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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  • 9,378
7 votes
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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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  • 577
4 votes
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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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4 votes

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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4 votes
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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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  • 3,013
3 votes
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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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  • 723
3 votes

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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3 votes

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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2 votes

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 ...
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2 votes

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 ...
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2 votes

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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2 votes

Approximation on partitions in $L^2([0,1]\times \Omega)$

Jensen inequality works fine. Indeed (correcting the mistake spotted by Gordon): $$\int_0^1 (\tilde{u}^n)^2 (t) dt = \int_0^1 \sum_{i=0}^{2^n-1}\left(\int_{(i-1)2^{-n}}^{i2^{-n}}u(s)2^nds\right)^2 1_{...
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2 votes
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Hilbert space-valued $L^2$ random variables.

You may consider an H-valued random variable as a measurable function from $\Omega$ to $H$, where $H$ is equipped with its Borel $\sigma$-algebra. Then as you say, $L^2(\Omega; H)$ is the set of (...
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Malliavin calculus integration by parts?

Writing $M_t = \int_0^t g_1(t_1)dB_{t_1}$, $t\in[0,1]$, then $M$ is in the domain of Skorokhod divergence $\delta$ [see https://en.wikipedia.org/wiki/Skorokhod_integral ]. Write $F = \int_0^1 f(t) ...
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Why do we need progressive measurability to obtain adapted process after integration?

The progressive measurability of $u$ ensures that $\int_{(i-1)2^{-n}}^{i2^{-n}}u(s)\,ds$ is $\mathcal F_{i2^{-n}}$ measurable, and this implies that $\tilde u_n(t)$ is $\mathcal F_t$ measurable (...
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Is the second moment enough to characterize the Brownian motion?

For each $i$, he defines $\{W^i(t), t \in \mathbb{R}^+\}$ to be a zero-mean Gaussian process with covariance function $E[W^i(s)W^i(t)] = s \wedge t$. This means that we know that $W^i(t) - W^i(s)$ is ...
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2 votes

Further Reading on Stochastic Calculus/Analysis

I think it takes courage to identify what you are weak at and make that public. The mindset of identify weaknesses and seeking to improve is essential. If you is able to keep this mindset awake even ...
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2 votes

What does $C_b$ norm mean

\begin{align*} \|f\|_{C_{b}(K)}=\sup\{|f(x)|: x\in K\}, \end{align*} and \begin{align*} \|f\|_{C_{b}^{m}(K)}=\sum_{k=0}^{m}\|f^{(k)}\|_{C_{b}(K)}, \end{align*} where $K$ is a compact set, here we ...
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2 votes
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Proving sufficient conditions

I must be frank, I am not up to speed on the consequences of Malliavin differentiability. But it seems that under some mild conditions on $f$ it ought to be so. Suppose that there exists a point $y$ ...
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2 votes
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Textbooks on Malliavin Calculus

Malliavin's stochastic analysis is rather difficult, it's mostly approachable when you've already gotten familiar with the themes of Malliavin Calculus (it assumes familiarity across several topics, ...
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2 votes
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Commutativity relationship between the Malliavin derivative and the Skorkohod divergence operators.

Let me start by answering the second point. Nate Eldredge points to the right place in Nualart's book in a comment. Here I will just recreate the detail from there that if $G$ is a smooth $H$-valued ...
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Malliavin integration by parts using Girsanov's theorem

Consider the Hilbert space $H$, and let $$F=\varphi(I(h_1),...,I(h_n))$$ be a smooth Brownian functional (here $I(\cdot)$ denotes the Wiener integral, or the isonormal Gaussian process in Nualart's ...
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  • 3,183
2 votes
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What is the Malliavin derivative of $\int_0^T f(B(t))dB(t)$?

Assume that $f$ is good enough so that everything I'll write is well defined, in that case we have that $$D_t \int_0^T f(B(s)dB(s)=f(B(t))+\int_t^T f'(B(s))dB(s).$$ This is a particular case of ...
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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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  • 306
1 vote
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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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