All Questions
Tagged with pr.probability stochastic-differential-equations
237 questions
0
votes
1
answer
95
views
If a probability measure is stationary in both forward time and reverse time, does this imply that the measure is incompressible?
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and let $X$ be a compact metric space. Let $F \colon \Omega \times X \to X$ and $\bar{F} \colon \Omega \times X \to X$ be measurable ...
- 17k
3
votes
2
answers
380
views
Large deviation bound for O-U process
Assume $X_t$ is an Ornstein-Uhlenbeck process in the form of
$$
d X_t = -\alpha X_t dt + \sigma dB_t
$$
Is there an exponential bound (large-deviation bound) for
$$
P\left(
\max_{t\le T} |X_t| \ge z
\...
- 325
1
vote
1
answer
154
views
Is there solution to a backward stochastic differential equation with $yz$ in the generator?
Please consider the following backward stochastic differential equation:
$$ Y(s)=\xi+\int_{s}^{T}a(u)Y(u)+b(u)Y(u)Z(u)du-\int_{s}^{T}Z(u)dW(u)$$
Here $a(s)$, $b(s)$ are square-integrable stochastic ...
- 11
2
votes
0
answers
94
views
Defining weak solutions to infinitely many SDEs on the same probability space
Suppose I have an SDE of the form
$$dX_t=b(X_t)dt+\sigma (X_t)dB_t+\int_{\mathbb{R}}G_{t-}(y)N(dtdy)$$
which I can solve weakly if I cut off the last integral to range over the set $\{\mid{y}\mid > ...
- 121
8
votes
1
answer
2k
views
total variation distance between two solutions of SDE
Suppose we have two stochastic differential equations with the same initial conditions:
$$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,$$
$X_0^1=X_0^2=x_0$; $W_\cdot$ is a ...
- 6,045
3
votes
1
answer
2k
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On a reflecting Brownian motion and its boundary local time
I have a question about a reflecting Brownian motion and its boundary local time.
Bass and Hsu studied the existence of Reflecting Brownian motion and boundary local time on a bounded Lipschitz ...
- 99
3
votes
1
answer
345
views
Why control a continuous approximation of stochastic gradient descent instead of just the SGD?
In "Stochastic modified equations and adaptive stochastic gradient algorithms" (Li et. al 2015) the authors approximate stochastic gradient descent, as in
$$x_{k+1} = x_k - \eta u_k \nabla f_{\...
- 4,361
0
votes
1
answer
111
views
Conditioning on an irrelevant variable in a martingale control problem
Suppose I have two independent Brownian motions $B^1_t, B^2_t$ and $\mathbb F_t$ be the natural filtration generated by them. Let $T > 0$ be a fixed finite number. Let $q_t$ be a $[-1,1]$ valued $\...
- 984
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0
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48
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Characterization of Time-homogeneous flows for conditional expectation
Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...
- 5,405
3
votes
0
answers
569
views
Domain of the Generator of a Bessel process
Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$
\begin{align}
\rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t}
\end{align}
where $(W_{t})_{t\geq ...
1
vote
0
answers
127
views
Gradient bound for the Markov semigroup generated by the solution to an Langevin SDE
Let
$h\in C^2(\mathbb R)$ with $$h''\ge\rho\tag1$$ for some $\rho>0$ and $$\int\underbrace{e^{-h}}_{=:\:\varrho}\:{\rm d}\lambda=1$$
$\mu$ be the measure with density $\varrho$ with respect to the ...
- 167
1
vote
1
answer
305
views
Existence of a Lyapunov function for a log-concave measure
Let $d\in\mathbb N$, $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1$$ and $\mu$ denote the measure with density $e^{-f}$ with respect to the Lebesgue measure on $\...
- 167
0
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1
answer
204
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How is the Cauchy-Schwarz inequality used in the proof of Lyapunov's criterion in the book "Analysis and Geometry of Markov Diffusion Operators"
Let $(E,\mu,\Gamma)$ be a full Markov triple (see definition below), $J\in\mathcal A$ with $J\ge1$ and $g\in\mathcal A_0$. In the proof of Theorem 4.6.2 of the book "Analysis and Geometry of Markov ...
- 128k
2
votes
1
answer
391
views
Is there an Itō formula for random functions in infinite-dimensions?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
$T>0$
$I:=(0,T]$
$(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\...
CommunityBot
- 1
3
votes
0
answers
95
views
"Expanding" around the invariant measure
In the spde literature we have results of the form
$$|P_{t}F(x)-\mu(F)|\leq O(g(t)),\text{for all } x\in H, F\in S$$
where $P_t$ is a semigroup, $H$ some Hilbert space, $F\in S$ some function space, $...
4
votes
1
answer
320
views
Uniqueness of a SDE with positivity constraint
We start by fixing some notation.
If $x\in\Bbb R^N$, we denote the usual euclidean norm in $\Bbb R^N$ with $\|x\|$: we omit the reference to the space $\Bbb R^N$ or to the dimension $N$ since it ...
CommunityBot
- 1
0
votes
0
answers
76
views
Ornstein-Uhlenbeck type process with thresholding
(Edited) I met a univariate Ornstein-Uhlenbeck type process but with self soft-thresholding:
$$
dX(t) = - c\ \mbox{sgn}(X(t))\big[|X(t)|-c_1 t^{\mu}\big]_+ dt + \sigma dB(t), \quad X(0)=0,
$$
where $...
- 31
1
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0
answers
134
views
Moment Estimate
Let $\epsilon > 0$ be a small parameter and consider the following lemma.
Lemma. Let $B(t)$ be a bounded, continuous, $R^{n \times n}$-valued function defined on a time interval $[0,T]$ such that ...
- 31
6
votes
2
answers
2k
views
Tanaka-Meyer formula
I have a simple question about Tanaka-Meyer formula, I am having difficulty applying it. Let $X$ be a continous martingale vanishing at zero. From Tanaka-Meyer formula it holds $$d|X_t| = sgn(X_t)dX_t+...
- 6,045
4
votes
1
answer
405
views
Limit of first passage time
I have a conjecture that seems rather obvious but the proof seems elusive.
Consider a diffusion given by,
$dX_t = \mu(X_t) dt + \sigma(X_t) db_t$
where $b_t$ is a standard Brownian motion.
$\mu,\...
CommunityBot
- 1
4
votes
1
answer
302
views
Almost sure stability of a scalar, nonautonomous, nonlinear SDE
I asked this problem on MSE some while ago, but it has stubbornly resisted any attempts at solving it.
Maybe there is someone here who can either close the gap in one of the existing answers or has ...
- 1,675
3
votes
1
answer
191
views
Stopping time property
Hi, I am reading a textbook about SDE, and am very confused about the transition
$$X_T 1_{T\lt t} + E\{X_T 1_{T\geq t} | F_{t\wedge T}\}$$
$$= X_T 1_{T\lt t} + E\{X_T | F_t\} 1_{T\geq t}$$
I ...
- 138
2
votes
1
answer
280
views
Walker whose Velocity is a Brownian Bridge
Consider a continuous random walk $x (t) $, in which the velocity $v (t) = \mathrm dx/\mathrm dt $ rather than the position is described by Brownian motion, so that $v (t) = B_t $ where $B_{t+\epsilon}...
- 6,045
4
votes
1
answer
322
views
Asymptotic form of pdf of Escape Time of arithmetic fBm
I am trying to apply the Girsanov formula and Doobs optional sampling theorem to obtain an asymptotic form of first passage density of an fbm process with drift, but the answer i am getting seems ...
- 6,045
6
votes
1
answer
547
views
Diffusion processes with different diffusion coefficients and absolute continuity
I would first of all like to say that I am an analyst, and so I am familiar with probabilistic methods only on a basic level.
My initial situation is the following. Consider two stochastic ...
- 1,675
3
votes
1
answer
83
views
Filtering Mixed Discrete and Continous
Suppose I have signal process $\lambda_t$ following the dynamics
\begin{equation}
\begin{aligned}
\zeta_t&=\mu^{\zeta}(t,{\zeta}_t)dt+\sigma^{\zeta}(t,{\zeta}_t)dW^{\zeta}_t\\
\xi_t&=\mu^{\xi}(...
- 1,675
2
votes
0
answers
104
views
Stochastic stability of "open" continuous-time stochastic systems: reference request
I'm looking for results on the stability of stochastic systems, e.g. SDEs, whose coefficients depend on a different process that is not necessarily stable. I'm calling those systems "open" here, but ...
- 1,675
3
votes
1
answer
2k
views
Solution of multivariate Geometric Brownian Motion?
It is known how to solve the SDE $dX=X\,dW$ to get a closed form expression of $X(t)=\exp(W_t-\frac{t}{2})$. The question is, is there also a way to solve
\begin{equation} \begin{cases}
dX=X \, dW_1+...
- 128k
4
votes
2
answers
13k
views
how to find derivative of a stochastic process?
Consider the following equation for $X(t)$:
$$X(t)=e^{-bt}X(0)+\sigma\int_{0}^{b}e^{-b(t-s)}dW(t) \, ,$$
where $0 < b, \sigma\in\mathbb{R} $, $X(0)$ is the initial distribution of $X(t)$, ...
- 3,574
4
votes
1
answer
386
views
Existence of normal number except random numbers
For normality, see https://en.wikipedia.org/wiki/Normal_number. For random number/sequence, see https://en.wikipedia.org/wiki/Algorithmically_random_sequence.
Now, is there any number that is normal ...
- 66.3k
1
vote
0
answers
90
views
Onsager-Machlup Function of a Killed Diffusion Process
Given a diffusion process $ X_t $ on a Riemannian manifold $(M,g)$, with an infinitesimal generator $\mathcal{G}=\Delta_g/2 + b$, the Onsager-Machlup function is well-known to be: $$ \mathcal{L}(x,v) =...
- 213
1
vote
0
answers
340
views
Construction of the quadratic variation for Hilbert space valued local martingales
Let
$H$ be a separable $\mathbb R$-Hilbert space
$(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a ...
- 167
1
vote
0
answers
249
views
Ito's formula for jump diffusions
Suppose I have $dP_t^i = (r^i + h_i^{\mathbb{P}})P_t^i dt - P_{t-}^i dH_t^i$ where $H_i(t) = \mathbb{1}_{\tau_t \leq t}$ denotes a default indicator process of i. $\tau_i$ is the default time and $h_i$...
- 138
3
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1
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110
views
Sequence of diffusions
Can every càdlàg semi-martingale be written as a sequence of diffusions? That is, is the set of continuous semi-martingales dense in some Skorohod space?
- 724
2
votes
1
answer
212
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Modified square root process
I am dealing with the following stochastic differential equation (SDE)
$ \begin{cases} dS_t &= \mu S_t dt + \sigma_1 S_tdW^1_t\\dG_t &= kS_t(\alpha - G_t)dt + \sigma_2\sqrt{G_tS_t}dW^2_t
\end{...
CommunityBot
- 1
2
votes
0
answers
96
views
Smoothness of Value function for SDE with discontinuous coefficients
Let $\mu: \mathbb{R}\to \mathbb{R}$, $f: \mathbb{R}\to \mathbb{R}$, and $r: \mathbb{R}\to [1, \infty)$ be bounded measurable functions (which may be discontinuous).
I'm interested in the function $v:\...
- 19.6k
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
$$
\dot u = -u + f(a)
$$
$$
\...
- 19.6k
2
votes
0
answers
221
views
Boundary behavior for Ito diffusions
The classification of boundary behavior for a time-homogeneous diffusion satisfying an Ito stochastic differential equation (SDE) is well known. According to the Feller classification, there are four ...
- 19.6k
1
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0
answers
57
views
Matching Numbers in Ito McKean
Matching numbers are the basics Ito and McKean use to build out a bunch of stuff, like singular points and shunts. The four maching numbers $e_1, e_2, e_3, e_4$ are defined as
$e_1 = \lim_{b \...
- 19.6k
2
votes
1
answer
387
views
Weak convergence of sum of log normal random variables
Let $S_t$ be the Geometric Brownian Motion, we know that
$$dS_t=rS_tdt+\sigma S_tdW_t, t\in [0,T], S_0>0, r>0,\sigma>0$$
and the distribution of $S_t$ is known explicitly. Please see the ...
- 489
23
votes
1
answer
1k
views
Does a theory of stochastic differential algebras exist?
My question is motivated primarily by finance, where a non-technical student will learn how to approach SDEs using the symbolic manipulation of Itô calculus and the few basic rules of Brownian motion, ...
CommunityBot
- 1
1
vote
1
answer
924
views
Solutions to linear SDE with many noise sources
It is well known how to solve the linear stochastic ODEs with one source of noise
$$dX_t=(a(t)X_t+c(t))dt+(b(t)X_t+d(t))dW_t$$
See, for instance, https://math.stackexchange.com/questions/1788853/...
CommunityBot
- 1
2
votes
0
answers
107
views
Markov chain approximates a fractional diffusion
Let assume that
$$
dX_t=\mu(X_t)dt+\sigma(X_t)dW_t^H, X_0\in \mathbb{R}
$$
Where $\mu(.), \sigma(.)$ satisfy some conditions that guarantee $X_t$ exists, and $dW_t^H$ is a fractional Brownian motion ...
- 323
3
votes
2
answers
1k
views
Is the "hybrid" Black-Scholes Hull-White model arbitrage free?
Given a "hybrid" Black-Scholes Hull White (BSHW) model. That is, the stock price is modelled by a Black Scholes SDE:
\begin{equation} dS(t) = \mu(t)S(t)dt + \sigma_{S}(t)S(t)dW^{\mathbb{P}}_{S}(t)
\...
- 1,533
3
votes
0
answers
170
views
Feynman-Kac formula for *general* Sturm-Liouville operator
One way to state (omitting technical requirements) the Feynman-Kac formula that I am familiar with is as follows.
Let $u$ be a solution to the pde
$$u_t(x,t)=-\frac{\sigma^2(x,t)}2u_{xx}(x,t)-V(x,t)u(...
- 891
0
votes
0
answers
57
views
Parametric distribution where the parameter follows a diffusion process
I'm looking for a distribution $P_{\theta}$ with pdf $f (t,\theta)$ over $\mathbb{R}^{+}$ such that there exists functions $\mu(\theta)$ and $\sigma(\theta)$ such that for all $t>0$:
$$\mu(\theta)\...
- 1,902
1
vote
0
answers
260
views
Transforming reaction-diffusion equations to random walk processes
I have a two species reaction-diffusion system which is a Turing-type (activator-inhibitor) equation. I am trying to transform my reaction-diffusion system into a system of multiple walkers on a ...
- 6,045
2
votes
1
answer
2k
views
Linking Wasserstein and total variation distances
I seek to bound the total-variation distance between two probability measures $p_1$ and $p_2$. It is extremely easy to build a parameter space where $p_1$ and $p_2$ are the marginals of some joint ...
- 9
0
votes
0
answers
153
views
Embedding a martingale by SDE
Let me reformulate my question. Let $(X_0,X_T)$ be a martingale on $\mathbb R$, then it is known that one has a SDE:
$$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