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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Existence of a sequence of deterministic measurable kernels (Skorohod Integral and Chaos Expansion)
I'm working on Di Nunno, Øksendal, and Proske's book on Malliavin Calculus and stuck in the problem below. I wrote a possible solution here and discussed it afterward.
Problem. Let $u(t)$, $0 \le t \...
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How to apply variational calculus to probability density functions with constraints?
I would like to use variational calculus to find the probability density function $p(x)$ that maximizes the tail probability, i.e., I would like to find
\begin{equation}
\max_{p(x)} \int_K^\infty p(x) ...
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Expectation of a stochastic differential [closed]
Given
\begin{equation}
df(X_{t})=\left(\mu_{t}{\frac {\partial f}{\partial x}}+{\frac {\sigma_{t}^{2}}{2}}{\frac {\partial^{2}f}{\partial x^{2}}}\right)dt+\sigma_{t}{\frac {\partial f}{\partial x}} ...
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Why is the function mapping the indices of an isonormal Gaussian process to its respective random variables linear?
First time approaching the book The Malliavin Calculus and Related Topics by Nualart and I see
Why do these calculations show that the mapping $h \rightarrow W(h)$ is linear?
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Inner product of tensor product elements (paper from M Hairer)
I'm reading a paper from M. Hairer on Malliavin calculus. I have a question regarding a formula in the middle of page 6. hereunder an extract :
Let $H$ be a Hilbert space with orthonormal basis $(e_i)...
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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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A computation of Skorohod integral
Let $\delta$ be the Skorohod integral on $L^2(\Omega\times [0, T])$, i.e. for $\xi\in L^2(\Omega\times [0, T])$
$$\delta (\xi) = \int_0^T \xi(t, \omega) \delta W_t.$$
Let $S_t = \exp\{\theta t + \...
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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 ...
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Exercise 1.3.4 in Nualart's book
Let $W$ be an isonormal process over a real separable Hilbert space. Let $D, \delta$ be the associated Malliavin derivative, divergence operators.
Exercise 1.3.4.
Let $F$ be in $\mathbb{D}^{1,2}$ (...
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What is the Malliavin derivative of the sum of two independent Brownian motions?
I know that the Malliavin derivative of $B(t)$ is $1$ if $B$ is standard Brownian motion. But what about $B_1(t)+B_2(t)$ if $B_i$ are independent Brownian motions?
This is really a functional on the ...
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Constants in Meyer inequalities
Let us first recall Meyer inequality in the Malliavin calculus framework
$$\|\delta(u)\|_{L^p}\leq C_p\|u\|_{\mathbb{D}^{1,p}},\qquad \forall u\in \mathbb{D}^{1,p},$$
where $\delta$ is the skorohod ...
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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 ...
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closability of the Malliavin derivative when $p=1$
Let $H$ be a separable Hilbert space and $W=(W(h))_{h\in H}$ be an isonormal Gaussian process on a probability space $(\Omega,\mathscr F,P)$, where $\mathscr F$ is generated by $W$.
Let $\mathscr S $ (...
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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 ...
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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) ...
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$L^2$decomposition
I want to know following statment's proof.
Let $L^2:=\{F:\textrm{wiener function} |\int_W F(\omega)^2 P(d\omega)<\infty \}$,and
$C_0:=\{\textrm{constant}\},C_n:=P^n \cap \{C_0\oplus...\oplus C_{n-1}...
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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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Identifying a generic Hilbert space $H$ with an $L^2$ space on some measure space.
This may be a stupid question, but I was wondering, if we are given an
infinite dimensional Hilbert space $H$, is it possible to find (or to
hypothesise that there's) a measure space $(M,\mathcal M,\...
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Isonormal Gaussian process associated with a Hilbert space.
We consider the isonormal Gaussian process $W=\{W(h),h\in H\}$ indexed by a separable Hilbert space $H$, defined on a complete probability space $(\Omega, \mathcal F,P)$ where $\mathcal F:=\sigma(W)$,...
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Commutativity relationship between the Malliavin derivative and the Skorkohod divergence operators.
On proposition $1.3.1$ of Nualart's book "The Malliavin Calculus and
Related Topics" it's stated the following. Let $H$ be a real separable
Hilbert space, and let $W=\{W(h),h\in H\}$ be a Gaussian ...
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How to calculate this derivative in Malliavin sense?
My goal is to obtain some weights to calculate greeks for lookback options.
It boils down to calculate the malliavin derivative of the following process :
$Y_{t} = 8(4(\int_{0}^{t}\int_{0}^{t}\frac{||...
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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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Closure of the space of step functions is Hilbert space
In Nualart's book "The Malliavin Calculus and related topics" ,
denotes by $\mathcal{E}$ the set of step functions on $[0,T]$ and says that
$\mathcal{H}=\overline{(\mathcal{E},\langle\cdot,\cdot\...
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Malliavin derivative of a gaussian
Let $W$ be an $H$-isonormal Gaussian process and $H$ is a real separable Hilbert Space.
Set $$X=f\big(W(h_1),\ldots, W(h_n)\big) $$ for $f$ an infinite differentiable with their partial derivatives ...
user575807
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Apparent inconsistency in Skorohod integration
Let $W$ be a standard $1$-dimensional Wiener process. Fix time points $t$ and $\tau$, with $t<\tau$, and consider the Itô integral
$$\int_{t}^{\tau}W(t)~\mathrm{d}W(s).$$
Of course, this is a ...
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What does $C_b$ norm mean
Recently I have been reading a paper where the norm $||f||_{C_b}$ and $||f||_{C_b^m}$ appear without definition. So I would like to know what is the default definition in the mathematics community for ...
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Why does the Malliavin derivative of a Markovian semigroup being strong Feller imply the semigroup strong Feller?
I am reading "Ergodicity Results for the Stochastic Navier–Stokes Equations: An Introduction" by Arnaud Debussche
In it, he claims that for $\varphi \in B_b(H)$, a bounded measurable function on a ...
user223391
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How to empirically verify convergence results with stochastic differential equations (with fractional Brownian motions)
Let $ \frac{1}{2} < H < 1$ and let $B^H_t$ be a fractional Brownian motion with Hurst parameter $H$. Then the following stochastic differential equation
$$\mathrm{d}Y_t = 5Y_t\mathrm{d}B^H_t, \...
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Hilbert space-valued $L^2$ random variables.
Easy question, but can't seem to find it online.
Informally, let $\Omega$ denote a probability space and $H$ a Hilbert space. Then what exactly does $L^2(\Omega ; H)$ mean? I presume it is a Hilbert ...
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Malliavin calculus integration by parts?
$f,g,h: \mathbf R\to\mathbf R$. $h$ is differentiable while $f$ and $g$ are integrable. $B_t$ is a Brownian motion. We know that
$$\mathbf E\bigg[h\Big(\int_0^1 f(t)dB_t\Big)\int_0^1g(t)dB_t\bigg]=\...
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On the definition of Iterated Itô integral. A question concerning the book on Malliavin calculus from Nualart
In page 23 of the book "The Malliavin Calculus and Related Topics" from Nualart one reads:
$\ \ $ Let $f_n:T^n\to\mathbb{R}$ be a symmetric and square integrable function. For these functions the ...
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Why do we need progressive measurability to obtain adapted process after integration?
In the page 16 of the book "The Malliavin Calculus and Related Topics" from Nualart one reads:
Why do we need $u$ to be progressively measurable to ensure that $\tilde u$ is adapted?
Is there an ...
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Is the second moment enough to characterize the Brownian motion?
In the page 14 of the book "The Malliavin Calculus and Related Topics" from Nualart one reads:
I understand that if $Z_t \sim N(0,t)$ then
$$\Bbb{E}[Z_t^{2k}] = \frac{(2k)!}{2^k k!}t^k$$
However, I ...
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On the $L^2$-norm of symmetrized functions. A question concerning the book on Malliavin calculus from Nualart
In the page 10 of the book "The Malliavin Calculus and Related Topics" from Nualart one reads:
Letting $f=g$ in property $\rm (iii)$ obtains $$E(I_m(f)^2)=m!\|\tilde{f}\|^2_{L^2(T^m)}\leq m!\|f\|^...
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Non sigma aditivity of the random measures $W$ employed in the construction of the white noise.
In the page 8 of the book "The Malliavin Calculus and Related Topics" from Nualart one reads:
to simplify our ideas, take $\mu$ to be the lebesgue measure on $\Bbb{R}_+$.
In this case then we can ...
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Are the Hermite polynomials a complete orthonormal set? A question concerning the book on Malliavin calculus from Nualart
In the pages 6 - 7 of the book "The Malliavin Calculus and Related Topics" from Nualart one reads:
Theorem 1.1.1 states the following:
So, it is clear that $H_n(W_1(x)) = H_n(x)$ is a complete ...
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Proving sufficient conditions
I am wondering if a "reverse necessary condition" implies sufficiency. Suppose your objective is to solve
$$
\min_{x\in X} f(x)
$$
where $X$ is a convex and compact subset of $\mathbb{R}^n$ and $f(\...
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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 ...
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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 ...
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Malliavin derivative with respect to part of samples.
My question came from M. Hairer's paper "Homogenization of periodic linear degenerate PDEs" (page 2469), I think it could be regarded as a basic exercise of Malliavin calculus thought. I will put is ...
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