All Questions
250 questions
2
votes
0
answers
221
views
Boundary behavior for Ito diffusions
The classification of boundary behavior for a time-homogeneous diffusion satisfying an Ito stochastic differential equation (SDE) is well known. According to the Feller classification, there are four ...
- 19.6k
2
votes
0
answers
385
views
Ito lemma for manifold semimartingales
I'm looking for a generalization of the usual Ito lemma to manifolds $M$, preferably not under the assumption that $M$ is embedded in $\mathbb{R}^d$. Unfortunately any reference I've found either ...
- 5,405
2
votes
1
answer
528
views
Any modern/recent version of Ito & McKean?
This's a wonderful book[1] but the latest edition I have is dated 1973. Is there recent book(s)/rewrite(s) that covers the same subjects and elucidate with more explicit arguments and details of their ...
CommunityBot
- 1
2
votes
1
answer
387
views
Weak convergence of sum of log normal random variables
Let $S_t$ be the Geometric Brownian Motion, we know that
$$dS_t=rS_tdt+\sigma S_tdW_t, t\in [0,T], S_0>0, r>0,\sigma>0$$
and the distribution of $S_t$ is known explicitly. Please see the ...
- 489
23
votes
1
answer
1k
views
Does a theory of stochastic differential algebras exist?
My question is motivated primarily by finance, where a non-technical student will learn how to approach SDEs using the symbolic manipulation of Itô calculus and the few basic rules of Brownian motion, ...
CommunityBot
- 1
2
votes
0
answers
107
views
Markov chain approximates a fractional diffusion
Let assume that
$$
dX_t=\mu(X_t)dt+\sigma(X_t)dW_t^H, X_0\in \mathbb{R}
$$
Where $\mu(.), \sigma(.)$ satisfy some conditions that guarantee $X_t$ exists, and $dW_t^H$ is a fractional Brownian motion ...
- 323
5
votes
1
answer
372
views
Reference: Stochastic Analysis on Hilbert Manifolds
I'm looking for a reference to a book which develops an It\^{o} lemma for semi-martingales with values in infinite dimensional Hilbert-Manifolds. I expect the techniques to be the same but still I ...
- 416
1
vote
0
answers
331
views
Mean and Variance of SDE
What is the mean and the variance of $y_t$, given the following SDE:
$dy_t = -x_t y_t dt + \sigma_1 dW^1_t$
$dx_t = -\sigma_2 y_t dW^2_t$
$W^1$ and $W^2$ are (possibly correlated) Wiener processes.
CommunityBot
- 1
1
vote
2
answers
2k
views
Deriving the HJB equation for exponential utility
I would like to derive the HJB equation for the following stochastic optimal control problem:
$ \Phi(t,x)=\sup_{h} E \left[\exp \left\{\gamma \int_t^T g(X_s,h(s);\gamma)\ ds \right\} \right]$
where ...
- 6,045
4
votes
1
answer
146
views
Time Integral over the (positive) Innovations of a Stochastic Process
Consider the Itô-Process
$$X(t) = X_{0} + \int_{0}^{t}\mu(s,X(s))\,ds + \int_{0}^{t}\sigma(s,X(s))\,dW(s)$$
where you can safely assume that the drift $\mu$ and the volatility $\sigma$ satisfy the ...
CommunityBot
- 1
0
votes
0
answers
70
views
If $(Φ^x)_{x∈ℝ}$ is a family of real-valued stochastic processes and $B$ is a Brownian motion, then $\int_0^tΦ^x_s\:dB_s=(\int_0^t\Phi_s\:dB_s)(x)$
Let
$T>0$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\in[0,\:T]}$ be a complete filtration on $(\Omega,\mathcal A)$
$B$ be a (standard, real-valued) $\mathcal F$...
- 167
0
votes
0
answers
153
views
Embedding a martingale by SDE
Let me reformulate my question. Let $(X_0,X_T)$ be a martingale on $\mathbb R$, then it is known that one has a SDE:
$$Z_t=Z_0+\int_0^t\sigma(s,Z_s)dB_s, \mbox{ for all } t\in [0,T]~~~~~~~~~~~~~~(\...
- 1,835
3
votes
0
answers
78
views
Perscribed/Inverting Conditional Expectation
I'm having difficulty finding papers which deal with the following inversion problem.
Suppose I have a stochastic process $Y_t$ (which is described by a certain Hilbert-Space-valued SDE). I want to ...
- 14.5k
3
votes
0
answers
231
views
I've found a representation of the Itō-Stratonovich correction term and don't understand the used notion of a "trace"
Consider a Stratonovich SPDE $$X_t=X_0+\int_0^tb(s,X_s)\:{\rm d}s+\int_0^t\sigma(s,X_s)\circ{\rm d}W_s\tag 1$$ in a separable $\mathbb R$-Hilbert space $H$ with $W$ being a $Q$-Wiener process on a ...
- 167
4
votes
0
answers
414
views
Definition of the Stratonovich integral in Hilbert spaces
Let
$T>0$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$\mathcal F=(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$
$B$ be a (standard, real-...
- 167
3
votes
0
answers
186
views
When we integrate with respect to a $Q$-Wiener process on $U$, why do we restrict integrands to be operators on $Q^{1/2}U$ (instead of $U$)?
When we integrate with respect to a $Q$-Wiener process $(W_t)_{t\ge 0}$ ($Q$ being a bounded, linear, nonnegative and self-adjoint operator on a separable $\mathbb R$-Hilbert space $U$ with finite ...
- 167
6
votes
2
answers
748
views
Does there exist a stochastic time derivative?
The Setup
Suppose I have a stochastic process $f(Z_t)$ where $Z_t$ solve the $d$-dimensional SDE
$$
dZ_t = \mu(t,Z_t)dt + \sigma(t,Z_t)dW_t
$$
and $f$ is a smooth function.
My Question
Is there a ...
- 6,045
1
vote
1
answer
380
views
Approximate an exponential martingale through its kernel
Given a deterministic function $h\in L^2([0,T]; \mathbb{R})$, we can define the associated exponential martingale
\begin{align}
M_t = \exp\left[\int_{0}^{t} h_s \,dB_s - \frac{1}{2}\int_{0}^{t} h_s^2\...
CommunityBot
- 1
1
vote
1
answer
460
views
Reflected SDE with non-Lipschitz coefficients
I have an equation of the form:
$$dX_t=\mu(X_t)dt+\sigma(X_t)dZ_t+dL_t, \quad X_0=x_0\in (-\infty,a]$$
where, $L_t$ is the reflection function (as in Skorokhod, 1961). This reflection does not allow ...
- 6,045
2
votes
1
answer
880
views
Existence of solution for reflected SDE
I have an equation of the form:
$$dX_t=\mu(X_t)X_tdt+\sigma(X_t)X_tdZ_t+dL_t, \quad X_0=x_0\in (0,a]$$
where, $L_t$ is the reflection function (as in Skorokhod, 1961). This reflection does not allow ...
- 6,045
3
votes
1
answer
1k
views
Strong solution for geometric brownian motion with varying drift and volatility
I have an equation of the form:
$$dX_{t}=\mu(X_{t})X_{t}dt+\sigma(X_{t})X_tdZ_{t}$$
I know that if I wrote it as $dX_{t}=\mu(X_{t})dt+\sigma(X_{t})dZ_{t}$, I would need strong assumptions on the ...
- 315
2
votes
1
answer
356
views
Itô Formula for Hilbert space-valued Lévy processes
I know there are Itô formulas for cylindrical Brownian motions with values in a Hilbert space and Itô formulas for Lévy processes in $\mathbb{R}^d$. My question is:
does there exist an Itô formula ...
- 6,045
1
vote
1
answer
164
views
Hilbert-Space Values SDE in terms of Basis
Suppose:
$$
dX_t = a(t,X_t)dt + b(t,X_t)dW^H_t
$$
is an SDE with values in a separable Hilbert Space $H$, and $W^H_t$ is an $H$-valued cylindrical Wiener process. Then can we write the dynamics for $...
- 6,045
3
votes
1
answer
159
views
Differentiability of a simple value function driven by a diffusion
Consider a diffusion given by,
$d X_t = \mu(X_t) dt + \sigma(X_t) dB_t$
$X_0 = x$.
Suppose the functions $\mu$ and $\sigma$ are as follows -
$f(x) = \mu(x) = \sigma(x) = \begin{cases} 2 & \...
- 6,045
4
votes
1
answer
521
views
Stochastic process with discontinuous drift
While studying a portfolio optimization problem, I came across the process
$$dX(t) = X(t)\,\Big(\,\big(\mu - \alpha\,1_{\{X(t)\,\geq\,C\}}\big)\,dt + \sigma\,dW(t) \Big)$$
which has a discountinuous ...
- 6,045
1
vote
1
answer
133
views
Reference for convergence of Hilbert-space valued SDEs
I'm fairly familiar with the literature dealing with convergence of SDEs in $\mathbb{R}^d$ but recently I've needed to use extended results dealing with convergence of SDEs in Hilbert Spaces. However ...
- 6,045
6
votes
0
answers
245
views
Second order calculus and rough paths
In Emery's book "Stochastic calculus in manifolds", he shows how to make sense of integrals of the form
$$ \int \langle\Theta_t, \mathbf{d} X_t\rangle,$$
where $X$ is a semimartingale on a manifold $M$...
- 9,853
3
votes
1
answer
628
views
Asymptotic behavior of an integral of OU process
Let $X=(X_t)_{t\ge 0}$ be a stochastic process (Ornstein-Uhlenbeck process) determined by
$$dX_t=-aX_tdt+\sigma dW_t,$$
where $X_0=0$, $a>0$ and $\sigma>0$ are constants, and $W=(W_t)_{t\ge 0}$...
- 6,045
1
vote
0
answers
124
views
Derive a SPDE of evolutionary type for $u$ from ${\rm d}X(t)=u(t,X(t)){\rm d}t+\xi(t,X(t)){\rm d}W(t)$
Let
$U$ and $V$ be separable $\mathbb R$-Hilbert spaces
$\iota:U\to V$ be a Hilbert-Schmidt embedding
$Q:=\iota\iota^\ast$
$(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$
$(\Omega,\mathcal A,\...
- 167
3
votes
1
answer
1k
views
Calculate Moments of SDE
I have posted a similar question on math.stackexchange (https://math.stackexchange.com/questions/1848492/calculate-mean-of-sde), but didn't find anyone who could help.
I'm interested in the one-...
CommunityBot
- 1
5
votes
1
answer
828
views
Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$
I am considering the transition semigroup $P_t$ associated with the Ito diffusion process
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$
where the coefficients are assumed to be Lipschitz continuous.
I hope to ...
CommunityBot
- 1
6
votes
0
answers
774
views
Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term
Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs.
I'm reading Stochastic Differential Equations in ...
CommunityBot
- 1
4
votes
1
answer
610
views
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 ...
CommunityBot
- 1
2
votes
0
answers
98
views
Non-existence for a sort of probability measures
We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$.
$W_{t}$ is standard Wiener.
This solution is ...
- 19.6k
3
votes
0
answers
276
views
Processes with the same finite dimensional distributions as the solutions to SDEs
Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...
- 131
1
vote
0
answers
118
views
Full version of Soucaliuc's research announcement "Réflexion entre deux diffusions conjuguées"
Florin Soucaliuc published the following research announcement in 2002 containing some results from his thesis on reflected diffusion processes:
[1] F. Soucaliuc, Réflexion entre deux diffusions ...
- 43
0
votes
1
answer
360
views
Weak existence for modified Tanaka SDE
Tanaka's theorem (wikipedia) implies that $X_t = |B_t|$ is a weak solution to the SDE
$dX_t = dW_t + dL_t^0(X_t)$,
where $W_t$ is a Brownian motion and $L_t^0(X_t)$ is the local time of $X_t$ at $0$....
CommunityBot
- 1
2
votes
0
answers
204
views
Onsager-Machlup function for special matrix-valued diffusion process
Potentially useful background info
For standard vector-valued diffusion processes the following result is well-known:
Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by
\begin{align*}
...
- 83
1
vote
1
answer
3k
views
using Feynman-Kac formula
I've been learning about Feynman-Kac recently and I understand the underlying ideas. I am stuck however in actually computing explicit solutions for specific problems. For example, suppose I have the ...
- 509
6
votes
1
answer
2k
views
Intuition about Skorohod integral
I'm teaching myself Malliavin calculus and Skorohod integrals and with this kind of math I find myself following the logic through but lacking solid intuition about what is going on.
In particular ...
- 10.3k
0
votes
1
answer
379
views
What is the derivative of this integral?
I have asked this question here
https://math.stackexchange.com/questions/1536018/how-to-find-derivative-of-this-intergral
but still has no response.
Might I ask it here ?
Let $\alpha(t)\in\{0,1\}: ...
CommunityBot
- 1
0
votes
0
answers
77
views
Law of motion when initial condition is perturbed
We know how to find the law of motion (Ito process) of the value function:
$$V_t(x)=E\Big{[}\int^{T}_te^{-r (s-t)}f(s,X_s)ds+e^{-r (T-t)}g(T, X_{T})|\mathcal{F}_t\Big{]}$$
such that
$$dX_t=\mu(t,X_t)...
- 1,533
2
votes
1
answer
3k
views
Time Change of a Brownian motion
We know that for if $X$ is a stochastic integral of the form below -
$X_t = \int_0^t v(s,\omega) db(s,\omega)$.
then we can use time change formula to claim that
$X_t = W_{\alpha(t)}$ where $W$ is ...
- 3,085
2
votes
0
answers
288
views
The existence of stationary measures for certain Markov process
My question is that:For a discrete-time random process $\{x_{t}\}_{t=1}^{\infty}$ and $x_{t} \in \Omega$ where $\Omega$ is a general state space(If $\Omega$ is a discrete space, it is a discrete-time ...
- 121
5
votes
2
answers
919
views
Analytic Solution to SDEs
Are there any example of SDEs with constant diffusion terms, other than the Ornstein Uhlenbeck process, which have exact solutions? I'm thinking of something of the form:
\begin{equation}
dX_t = f(...
1
vote
2
answers
119
views
SDEs: Bounding the variance of a solution
I've been thinking about something that would seem intuitive, but I haven't really been able to dig a direct answer to. This is a rough draft of it.
Let
$$X_t = \mu_{X,t} \mathrm{d}t + \sigma_{X,t} \...
- 29.8k
2
votes
2
answers
733
views
Existence of strong solution to SDEs with non-Lipschitzian drift
Consider the SDE:
$$dX_t=b(X_t)dt+dW_t\quad X_0=x$$
If $b$ is bounded Borel function, using Zvonkin's Transform, one can prove there exists a unique strong solution.
I want to know if we assume $b$ ...
- 778
7
votes
1
answer
4k
views
Change of time variable in Wiener process
I'm following a solution of an SDE from here
http://www.math.ethz.ch/~delbaen/ftp/preprints/CEV.pdf
Start with the SDE
$$
dX_t = \delta dt + 2\sqrt{X_t} dW_t
$$
consider a deterministic time change
$...
- 29.8k
5
votes
1
answer
820
views
Onsager-Machlup function and most probable path of a diffusion process
Let $X_{t}$ be a real, one-dimensional diffusion process satisfying the stochastic differential equation
\begin{equation}
dX_{t} = f(X_{t})dt + dW_{t},
\end{equation}
where $f \in C_{b}^{2}(R)$ is a ...
- 83
1
vote
1
answer
238
views
Perturbation of a Bessel process of dimension 2
Bessel process of dimension 2 is defined to be solution of
$$
dX_t=dB_t+\frac{1}{2X_t}dt,\quad X_0=x_0>0
$$
where $B$ is a standard 1-dimensional Brownian motion.
$X$ can be viewed as the norm of a ...
- 487