Malliavin Calculus is a stochastic version of calculus of variations.
Malliavin Calculus is a stochastic version of calculus of variations.
Questions tagged [malliavin-calculus]
84 questions
8
votes
1 answer
Malliavin derivative under change of measure
Let $\widetilde{B}$ be a Brownian Motion under the measure $\mathbb{P}$.
Let $\theta$ be a stochastic process fulfilling the Novikov's condition and $Z_\theta$ the relative Radon–Nikodym derivative for which it holds…
mastro
- 745
7
votes
2 answers
Further Reading on Stochastic Calculus/Analysis
I'm looking to read up more on Stochastic Analysis/Calculus (whatever it's called?) for PhD proposal.
So far, I've had 2 courses on Stochastic Calculus, mainly focusing on Finance, 1 course on probability with measure theory and some courses…
user198044
6
votes
1 answer
Interchanging Malliavin derivative with Lebesgue integral
I am reading Oksendal's book "Malliavin calculus for Levy processes with application to finance". In the proof of Lemma 4.9 (page 47), the author interchanges the Malliavin derivative $D_t$ with the Lebesgue integral $ds$.
$$D_t\int_0^T u^2(s)\,ds =…
ting
- 61
5
votes
1 answer
Malliavin derivative wrt time changed Brownian motion
The Malliavin derivative $D^W_\alpha$, $\alpha \in \mathbb{R}$, with respect to a standard Brownian motion $W_t$ is
$$
D^W_\alpha W_t = 1_{[0,t]}(\alpha).
$$
What would be the Malliavin derivative with respect to a time-changed Brownian motion…
Frido
- 398
5
votes
1 answer
Are there holes in my road map from calculus to malliavin differential geometry, Bayesian hypergraphs, and causal inference?
I've constructed a directed acyclic graph that leads from introductory subjects, such as calculus (single and multivariable) to some of my current interests including causal inference, Bayesian hypergraphs, and Malliavin differential geometry. The…
Galen
- 1,946
5
votes
1 answer
Malliavin integration by parts using Girsanov's theorem
I have been reading Nualart's notes on Malliavin calculus and I am aware of his derivation of the integration by parts formula. We consider a Hilbert space $(H,\langle \cdot,\cdot\rangle )$. If we let $(B_{t})_{t\in [0,\infty)}$ be a $d$-dimensional…
user7924249
- 624
5
votes
0 answers
Book Recommendations for Stochastic Analysis Preliminaries
I would like to ask for references that may help me in tackling some of the advanced stochastic analysis books. I am interested in a variety of different areas, namely (1) Malliavin Calculus, (2) Stochastic Differential Geometry, (3) Stochastic…
5
votes
2 answers
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 paper that include a comparison.
Thanks in…
Khosrotash
- 25,772
5
votes
2 answers
Approximation on partitions in $L^2([0,1]\times \Omega)$
I’m working on Nualart’s book “The Malliavin calculus and related topics” and in the proof of lemma 1.1.3 he mentions that the operators $P_n$ have their operator norm bounded by 1. I fail to see why, can you help me? Using Jensen’s inequality I get…
dlrlc
- 303
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Interchangeability of the malliavin derivative with a lebesgue integral
I was curious to know the most general conditions under which a Malliavin derivative $\mathscr{D}_t \int^T_t F_v d\mu(v) = \int^T_t \mathscr{D}_t F_v d\mu(v)$ commutes with a Lebesgue integral?
I was just curious to know all the assumptions.
Drew
- 109
4
votes
1 answer
A question about Malliavin calculus
An application of Malliavin calculus is to calculate the sensitivity of financial Greeks.
However, as in the theory of Malliavin calculus, to take the derivative of a random variable, we need to first specify a Hilbert space H, but I didn't see what…
user165367
- 41
4
votes
2 answers
Density of cylindrical random variables in classical Wiener space
I'm currently working on Malliavin calculus, and a theorem in my class notes is bothering me :
Denote W the Wiener space of continuous functions from $[0,1]$ to $\mathbb{R}$, and $\mu$ the associated Wiener measure. Let also the coordinate random…
Bertrand R
- 1,084
4
votes
1 answer
Malliavin derivative of a random variable
We consider a continuous stochastic stochastic process $X_t$ with the following dynamic on $[0,T]$ :
$$
dX_{t}^{x} = rX_{t}^{x}dt + \sigma X_{t}^{x}dB_t
$$
Where $X_{0}^{x}=x$ is the initial condition, $r>0$, $B_t$ is a standard Brownian motion and…
G2MWF
- 1,615
4
votes
1 answer
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 consider two adapted processes $(u_t)_{t\ge 0}$ and…
Davius
- 965
4
votes
1 answer
Integration by part formula in Malliavin Calculus
The set $S$ of smooth random variables is the set of random variables $F : \Omega \rightarrow \mathbb R$ such that there exist a function $f$ in $ \mathcal C_p^{\infty}(\mathbb R^n)$
(for some $n \geq 1$) and
elements $h_1, \cdots , h_n$ of $L^2$…
Zbigniew
- 865