All Questions
Tagged with differential-equations stochastic-processes
27 questions
2
votes
1
answer
86
views
Smoothness of resolvent of the infinitesimal generator of an Ito diffusion acting on bounded continuous function
Let $dX_t=\sigma(X_t)\,dW_t+\mu(X_t)\,dt$ be an Ito diffusion with Lipschitz coefficients and $\sigma(x)>0$. Let $f(x)$ be a continuous and bounded and non decreasing function. Can we prove that ...
- 41
1
vote
0
answers
159
views
Solutions to ODE/SDE with singular coefficients $dX_t = -X_t/t \, dt + g\,dW_t$
I encountered a question regarding the solutions to SDEs with singular drifts. I searched the literature but had a hard time figuring out the intuition behind these analytic results assuming different ...
- 73
8
votes
2
answers
566
views
Existence of solutions to the heat equation on nonsmooth domains
Let $\Omega \subset \mathbb{R}^n$ be a compact domain and for given functions $g: \partial \Omega \times [0,T] \to \mathbb{R}$ and $h: \Omega \to \mathbb{R}$ consider the heat equation
$$
\begin{cases}...
1
vote
1
answer
247
views
How to rigorously prove that this sequence of stochastic processes converges to a deterministic process?
Assume that for each $n\in\mathbb{N}$, there's a stochastic function $f_n$ of type $\mathbb{R}^{m}\to\Delta\mathbb{R}^{m}$, and for each $x\in\mathbb{R}^{m}$, the distributions $\frac{f_n(x)-x}{\frac{...
- 353
1
vote
0
answers
75
views
Existence of solutions to $\alpha(s)=\mathbb P[Y_s>0] + \int_0^s \dot{\alpha}(t)\mathbb P[Y^{t,0}_s>0] dt$
Let $\alpha:\mathbb R_+\to\mathbb R_+$ be a "nice" function with $\alpha(0)=1$. Define the process
$$Y_t=Y_0+t+\int_0^t\frac{dW_u}{1+\alpha(u)},\quad \forall t\ge 0,$$
where $Y_0>0$ has a ...
- 1,334
1
vote
1
answer
103
views
BSDE without volatility
Let $(W_t)_{0\leq t\leq 1}$ be a standard Wiener process on $[0,1]$, and let $\mathcal{F}_t$ be the natural filtration. Consider a BSDE
$$
dX_t=f(t,X_t)dt+\sigma(t,X_t) dW_t
$$
with terminal condition ...
- 229
0
votes
1
answer
301
views
Is there a Gaussian process for the solutions of the wave equation?
Call a Gaussian process $g$ a prior for a topological space $X$ if the realizations of $g$ are (a.s.) contained in $X$ and dense.
Consider the 1D wave equation
$\frac{\partial^2}{\partial t^2}u(t,x)=...
-1
votes
1
answer
122
views
Approximation of function in general measure space
Let $\mu$ be a $\sigma$-finite measure on $R^n$ ($n\geq 1$) and $(E,d)$ be a complete metric space. For any measurable function $f: R^n\to E$ with
$$
\int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \...
- 411
1
vote
2
answers
413
views
Backward stochastic differential equation
Let $W_t$ be a standard Brownian motion. Let $T$ be the terminal date, $X_T=x$, and
$$
dX_t=f_tdt+B_tdW_t
$$
where $f_t$ and $B_t$ (yet to be determined) have to be adapted to the filtration generated ...
- 229
2
votes
1
answer
404
views
Feynman-Kac formula for lattice heat equation with non-diagonal potential
Suppose that $X$ is the continuous-time simple symmetric random walk on the lattice $\mathbb Z^d$ (i.e., a simple symmetric random walk with i.i.d. exponential jump times), and let
$$u(t,x):=\mathbf E\...
- 891
2
votes
0
answers
74
views
Floquet stochastic process
Let $X_t$ be defined by the SDE
$$
dX_t = A(t, X_t)dt + dW_t
$$
where $A(t, X_t)$ is linear in $X_t$ and periodic in $t$. Assume also that the process is stable. If $A(\cdot)$ didn't have $t$ ...
- 205
5
votes
1
answer
408
views
Is there a Feynman-Kac formula for vector-valued Schrödinger operators?
Given a vector function
$$f=(f_1,\ldots,f_n)\in L^2(\mathbb R,\mathbb R^n)$$
(for some $n\in\mathbb N$), let us define
$$\Delta f:=(\Delta f_1,\ldots,\Delta f_n),$$
where $\Delta$ is the Laplacian ...
- 891
2
votes
0
answers
169
views
Stochastic Approximation in Reproducing Kernel Hilbert Space
Consider an iterative algorithm with incremental updates
\begin{align}
x_{t+1} = x_t + \alpha_t \cdot [ h(x_t) + M_{t+1}],
\end{align}
where $\{x_t \}_{t \geq 0}$ is in a reproducing kernel Hilbert ...
- 1,127
2
votes
0
answers
82
views
Stochastic Approximation Algorithms Converging to Local Equilibriums
Consider the stochastic iterative updates
\begin{align}
\theta_{t+1} \leftarrow \theta_t + \alpha_t \cdot \left [ h(\theta_t) + M_t \right ],
\end{align}
where $\theta_t \in \mathrm{R}^d$, $h \colon ...
- 1,127
1
vote
1
answer
924
views
Solutions to linear SDE with many noise sources
It is well known how to solve the linear stochastic ODEs with one source of noise
$$dX_t=(a(t)X_t+c(t))dt+(b(t)X_t+d(t))dW_t$$
See, for instance, https://math.stackexchange.com/questions/1788853/...
- 749
2
votes
1
answer
599
views
Solving a matrix ODE
Consider the linear matrix differential equation
$\def\diag{\mathrm{diag}}$
\begin{align}
U(0) &= I\\
\frac{\mathrm{d}U}{\mathrm{d}t}(t) &= U(t) \phantom{.} Q(t) & & \quad(1)
\end{...
user50085
2
votes
0
answers
260
views
Adiabatic elimination of a variable in a system of nonlinear stochastic ODEs?
If this is too basic for MathOverflow... say the word and I shall move it to Math.SE
First consider this system of ODEs. Say I have two variables $u$ and $a$, following
$$
\dot u = -u + f(a)
$$
$$
\...
- 155
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
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
4
votes
0
answers
466
views
Lorenz attractor power spectrum
If considered Lorenz attractor (with classical parameters $\sigma = 10, b = \frac{8}{3},r>25$), it is often noted, that while the spectral density (Fourier transformation of corresponding ...
- 41
2
votes
1
answer
960
views
Branching Brownian Motion and the KPP equation
I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole ...
- 724
4
votes
2
answers
4k
views
Any suggestions on a rigorous stochastic differential equations book?
I have been looking through some books and they are not very rigorous. Any suggestions would be great.
4
votes
2
answers
416
views
Probability of winding number of 2D Brownian Motion
Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau \...
- 409
4
votes
1
answer
645
views
Path integrals for stochastic equations
Does there exist a rigorous mathematical proof for path integral representations given in the physics literature? See for example
http://arxiv.org/abs/hep-ph/9912209v1
For imaginary time rigorous ...
- 31
4
votes
1
answer
546
views
Total variation distance between diffusion processes with different volatility coefficient
Preamble:
This question is similar to the one in total variation distance between two solutions of SDE . The difference is that in my case the drift is the same but there are different diffusion ...
- 315
6
votes
1
answer
1k
views
How is Kolmogorov forward equation derived from the theory of semigroup of operators?
In Lamperti's Stochastic Processes, given
a time-homogeneous Markov process $X(t), t\geq 0$ with Markov transition kernel $p_t(x,E)$ and state space being a measurable space $(S, \mathcal{F})$,
a ...
- 357
8
votes
1
answer
2k
views
total variation distance between two solutions of SDE
Suppose we have two stochastic differential equations with the same initial conditions:
$$d X_t^1= b_1(t,X_t^1)dt + dW_t$$
$$d X_t^2= b_2(t,X_t^2)dt + dW_t,$$
$X_0^1=X_0^2=x_0$; $W_\cdot$ is a ...
- 931