# Questions tagged [malliavin-calculus]

Malliavin Calculus is a stochastic version of calculus of variations.

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### Does Girsanov theorem hold under conditional distribution?

One application of Girsanov's theorem is in stochastic control theory. Suppose I have uncontrolled process $dX_t = \sigma^0(X_t)dW^0_t + dW^1_t$ under $\mathbb P$ on $[0, T]$, where $W^0$ and $W^1$ ...
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### Malliavin derivative of adapted processes

Let $(\mathcal{F}_t)_{t\ge 0}$ be a filtration. A stochastic process $(X_t)_{t\ge 0}$ is adapted with respect to such a filtration, if $X_t$ is $\mathcal{F}_t$-measurable for all $t\ge 0$. Now ...
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### What is the Malliavin derivative of $\int_0^T f(B(t))dB(t)$?

What is the Malliavin derivative of $F=\int_0^T f(B(t))dB(t)$? I know if $f$ is deterministic, then the Malliavin derivative of $F$ is just $f(t)$. But what if $f$ depends on the path, can we say ...
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### Malliavin derivative of stopped Brownian motion

Let $B_t$ stand for the standard Brownian motion in $\mathbb{R}^d$. Denote $$T = \inf\{t| \|B_t\| = 1\}.$$ That is, $T$ is the first exit time from the unit ball. I am interested in calculating the ...
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### Martingale representation of European option.

Let stock price $S$ satisfy $$S(t)=S(0)e^{(\int_0^t\sigma(s)dB_s-\frac{1}{2}\int_0^t\sigma(s)^2ds)}$$ I want to calculate the Martingale representation $V(t)=E(F|F_t)$ of European option with strike ...
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### Malliavin derivative and conditional expectation

I had a problem when I came across a proposition in Oksendal's book on Malliavin calculus. In the book, it claims $$D_t\mathbb{E}[F|\mathcal{F}_G] = \mathbb{E}[D_tF|\mathcal{F}_G]\chi_G(t)$$ where ...
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### Textbooks on Malliavin Calculus

I am looking for book(s) to learn Malliavin Calculus from. Some books that I have come across are, Stochastic Analysis By Paul Malliavin Malliavin Calculus for Levy processes with Applications to ...
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### Hairer's proof of Norris' Lemma

I am studying the notes "Advanced stochastic analysis" by Martin Hairer for a seminar. In the sixth section, Hairer proves Norris' lemma (Lemma 6.6) giving an explicit exponent in the proof, using the ...
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### Local integration by parts formula for Call options

I face a lot of difficulties to answer questions from past exam about Malliavin calculus and its application to finance and more precisely the pricing of a european call (I'm a student in a Msc in ...
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### Construct identification between $L^2( \Omega;H)$ and $L^2(T \times \Omega)$ where $H=L^2(T, \mathcal B,\mu)$

I am reading page 31 of Nualart, "The Malliavin Calculus and Related Topics" . Here, it says that there is an identification between the Hilberts spaces $L^2( \Omega;H)$ and $L^2( T \times\Omega)$...
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### A construction of a Stratonovich type integral for fractional Brownian motion

I'm studying this article https://projecteuclid.org/download/pdf_1/euclid.twjm/1500574954 and I'm having problems understanding the proof of lemma 3. Let me recall some of the criminals involved. ...
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### Definition of isonormal Gaussian process

The definition of an isonormal Gaussian process (from Nualart's book: The Malliavin Calculus and related topics) is as follows: My question is: why we want the space $H$ to be a real separable ...
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### Difference between Ito calculus and Malliavin calculus

Is there some difference between Ito calculus and Malliavin calculus ? I can't find a comparison ito vs malliavin essay on the web . I am thankful if someone describe the difference or guide to a ...
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### Product of functions with finite chaos expansion is in $L^2(P)$

I'm reading the book "Malliavin Calculus for Lévy Processes with Applications to Finance". At one point the authors prove that the Leibniz rule holds for the Malliavin Derivative $D_t$ taken at ...