All Questions
405 questions
0
votes
1
answer
73
views
Nonparametric estimation in diffusion
Fan and Wang
In the above paper, the Authors provide estimators for the squared spot volatility process $\left(\sigma^{2}_{t}\right)_{t\geq 0}$.
My question is how to find estimators for the process ...
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 ...
- 167
2
votes
1
answer
219
views
The (infinite) invariant measure of an SPDE
Consider a 1-dimensional stochastic heat equation on $[0, 1]$, with boundary conditions of Neumann's type:
\begin{equation}\left\{
\begin{aligned}
&\partial_t u(t, x) = \frac{1}{2}\partial_x^2 u(...
- 199
4
votes
1
answer
405
views
Limit of first passage time
I have a conjecture that seems rather obvious but the proof seems elusive.
Consider a diffusion given by,
$dX_t = \mu(X_t) dt + \sigma(X_t) db_t$
where $b_t$ is a standard Brownian motion.
$\mu,\...
- 553
1
vote
1
answer
503
views
Existence of strong solution in SDEs and continuity in the time variable
I recently come across some literature in stochastic analysis that uses the following result:
Consider the one-dimensional SDE
$$dX_t= a(t, X_t) \, dt + b(t, X_t) \, dW_t, $$
where $a, b: \mathbb{R}^...
- 357
5
votes
2
answers
437
views
A Stochastic Taylor Expansion/Asymptotics
Question:
Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and
$$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$
$(\mu,\sigma)$ obeys the linear growth ...
- 2,239
1
vote
1
answer
375
views
Computing transition operators for Markov processes
Is there a way to compute transition operators for Markov processes? To ask something much more tractable, suppose I have an Ito diffusion $$dX_t \ = \ \sigma(X_t) dB_t \ + \ b(X_t) dt$$
(or given by ...
- 138
3
votes
1
answer
217
views
Optimal control / Portoflio optimization: Maximize expected utility of total consumption
I came across a portoflio optimization problem, where I need to solve for optimal investment and consumption processes, such that the expected utility of total consumption and terminal wealth is ...
- 169
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 ...
- 503
7
votes
0
answers
304
views
Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories
Let
$d\in\left\{2,3\right\}$
$\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$
$\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...
- 167
1
vote
1
answer
762
views
Feller property for Ito diffusion with Lipschitz coefficients
Consider the following Ito diffusion $X_t$ satisfying
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x\in \mathbb{R}^n,$$
with Lipschitz coefficients $b,\sigma$.
It can be shown that if $g$ is bounded ...
- 503
4
votes
0
answers
112
views
Feynman-Kac formula and time-ordering for vector bundles
Let $M$ be a compact Riemannian manifold and let $\mathrm{d}\mathbb{W}^{yx;T}(\gamma)$ denote the Brownian Bridge measure, i.e. the Wiener measure on the paths that travel from $x$ to $y$ in time $T$ (...
- 9,853
1
vote
0
answers
102
views
What is meant by local time of BM on the boundary $\partial D$?
I'm familiar with local time $L_t^a$ at level $a$ for a 1-D Brownian motion $B$. I'm reading this paper which talks about a 2D Brownian motion $B$ in a bounded domain $D$ that gets reflected when it ...
- 282
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
2
votes
1
answer
2k
views
Linking Wasserstein and total variation distances
I seek to bound the total-variation distance between two probability measures $p_1$ and $p_2$. It is extremely easy to build a parameter space where $p_1$ and $p_2$ are the marginals of some joint ...
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, ...
- 231
3
votes
1
answer
525
views
Malliavin differentiability of solutions to SDEs
In Bass's book on Diffusions and Elliptic Operators, the author gives a brief introduction into Malliavin Calculus. He calls a functional $F:C([0,1],\mathbb{R})\rightarrow \mathbb{R}$ $L^p-$smooth if ...
- 213
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 ...
- 141
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 $0$....
- 43
4
votes
1
answer
388
views
Hitting time of an Ornstein-Ulhenbeck process
If we consider a nice Ornstein-Uhlenbeck process
$d x (t) = - \gamma x(t) \,dt + \sigma \,d w (t)$
with $x(0) = x_0 \in (-L,L)$.
Here $\gamma, \sigma$ are positive constants and $w(t)$ is a Wiener ...
- 375
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 ...
2
votes
1
answer
594
views
General solution to system of stochastic linear differential equations
Assume we are given the system of linear stochastic differential equations
$$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot dt+\...
- 749
2
votes
0
answers
204
views
Onsager-Machlup function for special matrix-valued diffusion process
Potentially useful background info
For standard vector-valued diffusion processes the following result is well-known:
Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by
\begin{align*}
...
- 83
6
votes
1
answer
2k
views
Intuition about Skorohod integral
I'm teaching myself Malliavin calculus and Skorohod integrals and with this kind of math I find myself following the logic through but lacking solid intuition about what is going on.
In particular ...
- 181
0
votes
1
answer
379
views
What is the derivative of this integral?
I have asked this question here
https://math.stackexchange.com/questions/1536018/how-to-find-derivative-of-this-intergral
but still has no response.
Might I ask it here ?
Let $\alpha(t)\in\{0,1\}: ...
- 131
2
votes
0
answers
98
views
Non-existence for a sort of probability measures
We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$.
$W_{t}$ is standard Wiener.
This solution is ...
- 257
1
vote
2
answers
1k
views
Understanding the limits of the Ito Process Defintion
I would like to understand what kind of stochastic process are Ito Processes. According to Kuo[p. 102] an Ito Process is a stochastic process of the form
$$dX_t=g(t)dt+f(t)dW(t),$$
where $W(t)$ is a ...
- 211
4
votes
1
answer
294
views
Exponential of approximate quadratic variation of Brownian motion
Let $X_t$ be a Brownian motion or a Brownian Bridge on a (\edit: compact) Riemannian manifold. Let $T>0$ be given.
The question is: Does there exists a constant $C>0$ such that for all ...
- 9,853
1
vote
0
answers
66
views
$X_t = B_t^q$, $X_t = (\sin B_t)^q$, $X_t = B_t^q (\sin B_t)^r$, $dM_t = R_t\,M_t\,dB_t$ [closed]
What are the SDE's satisfied by the following processes?
$X_t = B_t^q$
$X_t = (\sin B_t)^q$
$X_t = B_t^q (\sin B_t)^r$
Assume $B_t$ is a standard Brownian motion with $B_0 > 0$ and the equations ...
- 11
0
votes
0
answers
77
views
Law of motion when initial condition is perturbed
We know how to find the law of motion (Ito process) of the value function:
$$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)...
- 355
7
votes
1
answer
816
views
Solve SDE $dX_t=(c+\sigma_\zeta W'_t)X_tdt + \sigma_\epsilon dW_t$
I am trying to solve the following SDE
$$dX_t=(c+\sigma_\zeta W'_tX_t)dt + \sigma_\epsilon dW_t$$
$c\in \mathbb{R}$ is a constant, $X_t$ is a stochastic process, $\sigma_\zeta,\sigma_\epsilon \in \...
- 211
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 ...
- 553
3
votes
1
answer
604
views
Brownian bridge on a Lie group as a stochastic differential equation
Brownian motion $g_t$ on a compact Lie group satisfies the stochastic differential equation
$$dg_t = dB_t \circ g_t$$
where $B_t$ is Brownian motion on the Lie algebra and $\circ$ denotes ...
- 73
3
votes
0
answers
240
views
Using compactness method to prove the existence of a pathwise solution to an SPDE
For given initial data $u_0\in H^k$ for some $k$, I want to prove the existence of solution to some PDE with multiplicative white noise. I modify the SPDE by regularizing it and then use the ...
3
votes
0
answers
235
views
Strong solution to an SDE with a discontinuous diffusion term
I am having an SDE for which I would be in trouble if there were no strong solution.
The SDE is -
$ dX = \mu(x) dt + \sigma_1 (x) db_{1t} + \sigma_2(x) db_{2t}$
where $b_1$ and $b_2$ are two ...
- 553
1
vote
0
answers
155
views
Convergence of approximate quadratic variation in $L^p$
For a diffusion $X_t$, I can set
$$[X]^N_t = \sum_{j=1}^N \bigl(X_{t\frac{j}{N}}-X_{t\frac{j-1}{N}}\bigr)^2$$
Then it is well-known that the process $[X]^N_t$ tends to the quadratic variation $[X]_t$ ...
- 9,853
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
1
vote
1
answer
739
views
Joint law of a standard Brownian motion and its local time at a nonzero level
Let $B_t$ be the standard Brownian motion and $L_t^a$ be the local time at level $a$. It is known that the joint-density of $(L_t^0,B_t)$ is
$$
P\left(B_t\in d y, L_t^0\in d v\right) = \frac{|y|+v}{\...
- 1,649
4
votes
1
answer
610
views
Stochastic differential equation associated with an optimal control problem
We know how to find the stochastic differential equation (Hamilton-Jacobi-Bellman equation, HJB) of the control problem where a process $X_t$ is controlled up until it is stopped at a stopping time $\...
- 355
1
vote
2
answers
119
views
SDEs: Bounding the variance of a solution
I've been thinking about something that would seem intuitive, but I haven't really been able to dig a direct answer to. This is a rough draft of it.
Let
$$X_t = \mu_{X,t} \mathrm{d}t + \sigma_{X,t} \...
- 133
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 $b$ ...
- 515
6
votes
1
answer
392
views
Does $E^{x,t}(f(X_T))$ solve a PDE if $f$ is not continuous?
Many books [see below for references] explore the connections between partial differential equations and expectation values.
Assume $X$ is a diffusion with generator $A$, then they conclude, that ...
- 237
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
$...
- 173
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(...
- 379
1
vote
1
answer
208
views
Finding a stochastic differential equation as limit of a discrete stochastic equation
I'm dealing with the following problem:
Choose $Z_0 \in [0,1]$ and define a process governed by the following discrete stochastic equation:
$Z_{k+1}-Z_k=P_k(1-2Z_k)$
where $P_k=0$ with probability $...
- 11
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
11
votes
1
answer
498
views
Does Brownian motion immediately visit both sides of a Jordan curve?
Let $C$ be a Jordan curve in $\mathbb{R}^2$. By the Jordan curve theorem, $\mathbb{R}^2 \smallsetminus C$ is uniquely partitioned into two connected regions $A$ and $B$ (the interior and exterior).
...
- 113
1
vote
0
answers
119
views
When the completed filtration of a process increases slowly
If $\mathcal{F}_t$ is the filtration of the evaluation process on $C_T$ (continuous function on $[0,T]$). Can we find some law of continuous process $\mathbb{P}$ so that for $t\leq T$
$$\mathcal{F}^{\...
- 19
3
votes
1
answer
105
views
Density for Translated Process
Let $M$ be a (compact) Riemannian manifold. Let $v$ be a smooth vector field on $M$ with flow $\Theta_t$. Let $L$ be an elliptic second order differential operator on $M$ that generates the Ito ...
- 9,853