All Questions
Tagged with pr.probability stochastic-differential-equations
88 questions with no upvoted or accepted answers
12
votes
0
answers
1k
views
American put option pricing by "binomial trees"
I'm teaching a financial mathematics course and have found a fascinating (to me) numerical phenomenon and wonder if anyone has studied it, or knows anything similar.
I'll try and give a description ...
- 23.2k
6
votes
0
answers
88
views
Error estimates for projection onto the Wiener chaos expansion for stochastic Sobolev spaces (stochastic Rellich–Kondrachov theorem)
Let $n$ be a positive integer, $s\in \mathbb{R}$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ be a filtered probability space whose filtration supports and is generated by an $n$-...
- 5,405
5
votes
0
answers
412
views
Is it really interesting to prove well-posedness of unsolved SPDE?
Lots of nonlinear SPDE remained open for decades (especially the non-deterministic ones in higher dimensions because of the regularity of the noise) until Hairer's breakthrough (regularity structures),...
- 573
4
votes
1
answer
143
views
When does an Itô diffusion give a semigroup on $L^2$
I would like a reference for when an Itô diffusion generates a strongly continuous semigroup on $L^2(\mathbb{R}^n)$.
I have a time-homogeneous Itô diffusion of the form
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t$...
- 263
4
votes
0
answers
259
views
Malliavin calculus and geometric interpretation of $\nabla \cdot ({\nabla F(x)}{\|\nabla F(x)\|^{-2}})$, with regards to the surface $S = \{F = 0\}$
Let $F:\mathbb R^n \to \mathbb R$ be a "sufficiently regular" function. For any $k \ge 1$ and $x \in \mathbb R^n$, define
$$
\alpha_k(x) := \nabla \cdot \left(\dfrac{\nabla F(x)}{\|\nabla F(...
- 6,853
4
votes
0
answers
167
views
Occupation time of SDE
Let $b:\mathbb{R}^d\to\mathbb{R}^d$ be locally Lipschitz and assume that, for any $x\in\mathbb{R}^d$ and any $f\in C^{\infty}([0,1],\mathbb{R}^d)$, the equation
$$
X_t^{x,f}=x+\int_0^t b(X_s^{x,f})\,...
- 93
4
votes
0
answers
414
views
Definition of the Stratonovich integral in Hilbert spaces
Let
$T>0$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$\mathcal F=(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$
$B$ be a (standard, real-...
- 167
4
votes
1
answer
2k
views
Expected value of a stochastic integral expression
I am wondering if the following expression can be processed a bit analytically,
$$
E \left[ e^{aX} \int_0^X e^{bu}dW(u)\right],
$$
where $W_u$ is the normal Brownian motion (1D Wiener process), and $...
- 41
3
votes
0
answers
60
views
Comparison theorem for SDEs driven by a continuous martingale
Consider the well-known comparison theorem for SDEs, versions of which appear in several textbooks, e.g., Karatzas and Shreve, Proposition 5.2.18, or Revuz and Yor, Theorem IX.3.7.
The result states ...
- 103
3
votes
0
answers
86
views
Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set
Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE
\begin{equation}
dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,.
\end{equation}
Let $f(x,t|x_0,0)$ denote its transition density function. ...
3
votes
0
answers
122
views
Slow points of diffusion processes
Let $W$ be a standard $d$-dimensional Brownian motion, and $X$ the solution to the SDE
$$dX_t = \mu(X_t) dt + \sigma(X_t) \, dW_t,$$
with $\mu$ and $\sigma$ Lipschitz continuous.
Given a (...
- 6,223
3
votes
0
answers
145
views
Density of invariant measure of stochastic differential equation
I have a question: is it possible that an SDE has a "nice" density, but its invariant measure does not have a "nice" density? I asked this question at math.stackexchange but ...
- 931
3
votes
0
answers
235
views
Probability of a particle surviving forever
Consider a particle whose position is driven by the following equation:
$$Y_t = y + t + W_t + C\min\big(1,(Y_t+1)^+\big)\Lambda_t,\quad \mbox{for all } 0\le t<\tau_*,$$
where $y>0$, $0<C<1$...
user128095
3
votes
0
answers
570
views
Domain of the Generator of a Bessel process
Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$
\begin{align}
\rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t}
\end{align}
where $(W_{t})_{t\geq ...
3
votes
0
answers
95
views
"Expanding" around the invariant measure
In the spde literature we have results of the form
$$|P_{t}F(x)-\mu(F)|\leq O(g(t)),\text{for all } x\in H, F\in S$$
where $P_t$ is a semigroup, $H$ some Hilbert space, $F\in S$ some function space, $...
- 5,474
3
votes
0
answers
170
views
Feynman-Kac formula for *general* Sturm-Liouville operator
One way to state (omitting technical requirements) the Feynman-Kac formula that I am familiar with is as follows.
Let $u$ be a solution to the pde
$$u_t(x,t)=-\frac{\sigma^2(x,t)}2u_{xx}(x,t)-V(x,t)u(...
- 891
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 ...
- 5,405
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
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 ...
2
votes
0
answers
85
views
Can an SDE be made to follow the flow lines of a vector field?
Let $V: \mathbb R^n \to \mathbb R^n$ be a Lipschitz vector field. Consider a one dimensional Brownian motion $W$ and the SDE
$$dX_t = V(X_t) \, dW_t,$$
where we identify $V(X_t) \in \mathbb R^n$ with ...
- 6,223
2
votes
0
answers
67
views
The unique weak solution to some SDE yields the unique strong solution?
For some filtered probability space $\big(\Omega,\mathcal F, (\mathcal F_t),\mathbb P\big)$, consider a stochastic differential equation (driven by a real-valued Brownian motion $W$) for $X=(X_t)$, ...
- 1,399
2
votes
0
answers
203
views
Time reversal of infinite-dimensional SDE
Consider the SDE $${\rm d}X_t=b(t,X_t) \, {\rm d}t+\sigma(t,X_t) \, {\rm d}W_t,\tag1$$ where $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\...
- 167
2
votes
0
answers
115
views
Equivalence of score function expressions in SDE-based generative modeling
I am studying the paper "Score-Based Generative Modeling through Stochastic Differential Equations" (arXiv:2011.13456) by Yang et al. The authors use the following loss function (Equation 7 ...
- 43
2
votes
0
answers
155
views
Can a diffusion process admit an invariant measure with a non-differentiable density?
The precise domain of the generator $A$ of an Itō diffusion on a Hilbert space $H$ (assume $H=\mathbb R^d$, if that's easier for you to work with) can usually not be determined explicitly$^1$. Usually,...
- 167
2
votes
0
answers
95
views
Local martingale for a (two-dimensional) diffusion
Let $X$ be a two-dimensional diffusion (a solution of $dX_t=f(X_t)\,dt+dB_t$, with $B$ a standard two-dimensional Brownian motion) living on some open set $\Lambda\subset \mathbb{R}^2$. Let $h:\Lambda ...
- 1,897
2
votes
0
answers
201
views
Continuity of density of SDE
Consider a stochastic differential equation in $\mathbb R^m$ with a parameter $\theta\in\mathbb R$:
\begin{equation}
dX_t^{\theta,x} = v(\theta,X_t^{\theta,x})dt+\sigma(X_t^{\theta,x})\circ dW_t,~...
- 21
2
votes
0
answers
187
views
Time derivative of relative entropy in this setting
I was reading the following article : https://arxiv.org/pdf/2005.13097.pdf and a question came up.
In page 30 in the proof of Lemma 16, when taking the time derivative of the KL divergence, there is ...
- 41
2
votes
0
answers
50
views
Continuation : Uniqueness of the solution to some SDE with discontinuous coefficient
Consider the SDE below
$$X_t=X_0+\int_0^t b(s)ds+\int_0^t\frac{dW_s}{1+m(s){\bf 1}_{\{b(s)>0\}}},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$
where $X_0>0$ is square integrable, $b:\mathbb R_+\...
- 1,334
2
votes
0
answers
146
views
Exit time for Brownian motion with stochastic barriers
I am interested in the expected exit time of a one-dimensional Brownian particle from a stochastically evolving interval as follows.
Context:
If $L_t$ and $R_t$ denote the distance to the left and ...
- 91
2
votes
0
answers
95
views
Itō formula for the solution of a SPDE in the distributional sense
Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be open
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(Y_t)_{t\ge0}$ be an $L^2(\Lambda)$-valued process on $(\Omega,\mathcal A,\...
- 167
2
votes
0
answers
215
views
What is the Onsager-Machlup function for $dX(t)=f(B(t)) dt+dB(t)$?
What is the Onsager-Machlup function for $dX(t)=f(B(t)) dt+dB(t)$?
I know that the Onsager-Machlup function for $dX(t)=f(X(t))dt+dB(t)$ is $$L(x,v)=\frac12\left[v-f(x)\right]^2+\frac12f'(x)$$
But ...
- 93
2
votes
0
answers
94
views
Defining weak solutions to infinitely many SDEs on the same probability space
Suppose I have an SDE of the form
$$dX_t=b(X_t)dt+\sigma (X_t)dB_t+\int_{\mathbb{R}}G_{t-}(y)N(dtdy)$$
which I can solve weakly if I cut off the last integral to range over the set $\{\mid{y}\mid > ...
- 121
2
votes
0
answers
104
views
Stochastic stability of "open" continuous-time stochastic systems: reference request
I'm looking for results on the stability of stochastic systems, e.g. SDEs, whose coefficients depend on a different process that is not necessarily stable. I'm calling those systems "open" here, but ...
- 1,675
2
votes
0
answers
107
views
Markov chain approximates a fractional diffusion
Let assume that
$$
dX_t=\mu(X_t)dt+\sigma(X_t)dW_t^H, X_0\in \mathbb{R}
$$
Where $\mu(.), \sigma(.)$ satisfy some conditions that guarantee $X_t$ exists, and $dW_t^H$ is a fractional Brownian motion ...
- 323
2
votes
0
answers
52
views
Minimizer of a class of SDEs
Setup
Let $\mathscr{H}$ be a separable Hilbert space, $\mathcal{X}\triangleq \langle \Omega,\mathscr{F},\mathscr{F}_t,\mathbb{P}\rangle$ be a stochastic base and $X_t$ be an $H$-valued stochastic ...
- 5,405
2
votes
0
answers
221
views
Boundary behavior for Ito diffusions
The classification of boundary behavior for a time-homogeneous diffusion satisfying an Ito stochastic differential equation (SDE) is well known. According to the Feller classification, there are four ...
2
votes
0
answers
260
views
Adiabatic elimination of a variable in a system of nonlinear stochastic ODEs?
If this is too basic for MathOverflow... say the word and I shall move it to Math.SE
First consider this system of ODEs. Say I have two variables $u$ and $a$, following
$$
\dot u = -u + f(a)
$$
$$
\...
- 155
2
votes
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:\...
- 21
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
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
2
votes
0
answers
413
views
On the infinitesimal generator of a 1-dimensional stochastic heat equation: core and explicit form
Denote $E = C([0, 1])$. I am consider a 1-dimensional stochastic heat equation on $h$:
$$\partial_tu(t, x) = \partial_x^2u(t, x) - V'(u(t, x)) + \dot{W}(t, x), \quad\text{ for all } (t, x) \in (0, \...
- 199
1
vote
0
answers
58
views
Drift of reverse SDE with Lévy processes ($\alpha$ stable distributions)
Given an SDE with a Lévy process with a drift $b(x,t)$ the reverse SDE will have a drift, $\tilde{b}(x,t)$, given by the relation:
$$\tilde{b}(x,t) = - b(x,t) + \int_{\mathbb{R}} y \left( 1 + \frac{...
- 305
1
vote
0
answers
95
views
A stochastic optimal control problem with filtering-like dynamics
I want to extend the following stochastic optimal control problem with randomized feedback control to the continuous time case
\begin{align}
\text{minimize}\quad \mathbb{E}_{\mathbb{H}}&\bigg[\...
- 303
1
vote
0
answers
53
views
The limit ratio of two Markov Chain Probability
Suppose there are two given SDE in $\mathbb{R}^d$:
$$
\begin{align}
\left\{
\begin{aligned}
dX_t&=\begin{bmatrix}-\nabla V(X_t)+2\beta^{-1}v_F^\theta(X_t)\end{bmatrix}dt+\sqrt{2\beta^{-1}}dW_t,&...
- 303
1
vote
0
answers
193
views
Marcus-SDE to Itô-SDE
In the field of stochastic calculus, everyone knows the Itô and Stratonovich integrals, as well as the conversion from Stratonovich to Itô SDEs.
The Stratonovich integration has the particularity of ...
- 11
1
vote
0
answers
134
views
Generating realizations from $n$-dimensional geometric Brownian motion where the variables are constrained to sum to 1
Is there a way to simulate an $N$-dimensional geometric Brownian motion i.e. variable $$x_i, i \in [1, N] $$ is diffusing in log-space such that $$\log (x_i)$$ follows a Brownian motion with a given ...
- 21
1
vote
0
answers
193
views
Stochastic volatility model question
Let suppose that $S_t$ is a process defined as:
$$ \begin{cases}dS_t = \mu S_t\,dt+m(v_t)\,dW^1_t\\ dv_t = \mu_v(v_t)\,dt + \sigma_v(v_t)\,dW^2_t\end{cases}$$
where the two Brownian motions have ...
- 393
1
vote
0
answers
102
views
Freidlin Wentzell for stochastic differential inclusions
Consider the SDI
$$dX^\varepsilon(t)\in b(X^\varepsilon(t))\,dt + \varepsilon \sigma(X^\varepsilon(t)) \, dB(t).$$
Is there any Freidlin-Wentzell large deviations principle for $X^\varepsilon$?
- 1,904
1
vote
0
answers
237
views
Characteristic function of stochastic integral of a pure jump Lévy process with respect to another pure jump Lévy process
(I am cross-posting this question here from MSE: https://math.stackexchange.com/questions/4725734/characteristic-function-of-stochastic-integral-of-a-pure-jump-l%c3%a9vy-process-with. I apologize if ...
- 11
1
vote
0
answers
100
views
Reference request: $d X_t = b(X_t) d t + f (p_t(X_t)) d W_t$ where $p_t$ is the p.d.f. of $X_t$
Let $b:\mathbb R^d \to \mathbb R^d$ and $\sigma:\mathbb R^d \to \mathcal M_{ d\times q} (\mathbb R)$ be Lipschitz. Let $(W_t, t\ge 0)$ be the standard $q$-dimensional Brownian motion. Then
$$
d X_t = ...
- 657