# Questions tagged [malliavin-calculus]

Malliavin Calculus is a stochastic version of calculus of variations.

78 questions
Filter by
Sorted by
Tagged with
1 vote
14 views

### 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 $W^1$ ...
1 vote
47 views

• 185
106 views

### 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 ...
• 718
110 views

### 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 ...
• 180
173 views

• 37
157 views

### 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 ...
• 1,348
1 vote
113 views

• 31
255 views

### 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 ...
• 2,633
1 vote
255 views

### 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 ...
802 views

### 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 ...
325 views

### 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 ...
• 119
74 views

### 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 ...
1 vote
46 views

### 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 $L^2( T \times\Omega)$...
• 1,194
1 vote
58 views

### 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. ...
• 653
1k views

### 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 ...
1 vote
379 views

• 564
2k views

### 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 ...
• 23.7k
1 vote
95 views

### 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 ...
• 2,157
2k views

### Calculus of Variations in Probability Theory

Are there any places where the Calculus of Variations shows up (i.e. is used) in probability? It seems like it should be natural for functional optimization to appear (as it does in statistics, where ...
• 10.2k