Questions tagged [stochastic-calculus]
Stochastic calculus provides a consistent theory of integration for stochastic processes and is used to model random systems. Its applications range from statistical physics to quantitative finance.
935
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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 ...
- 175
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Global well posedness of $\phi^4_1$
We consider the $\phi^4_1$ model: $\partial_t\phi=\Delta\phi-\phi^3+\xi$ on $[0,T] \times \mathbb{R},$ where $\xi$ is a space time white noise.
I know how to solve this equation locally on the torus, ...
- 399
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Reference request: showing that solution of an Ito SDE stays bounded with positive probability
Assume that we have a (well-posed) Ito SDE of the form $$\mathrm{d} X_t = b(X_t)\,\mathrm{d} t + \sigma(X_t)\,\mathrm{d}W_t \label{1}\tag{1},$$ where $b \colon \mathbb{R}^d \to \mathbb{R}^d$, $\sigma \...
- 700
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95
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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 $X^\varepsilon$?
- 1,302
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60
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On the measurability of stochastic integrals
Let $S(t)$ be a $C_0$-contraction semi-group, $W$ is a cylindrical Wiener process in a separate Hilbert space $U$. Assume the following conditions:
$$
\|F(t,u_1)-F(t,u_2)\|_{H}< C\|u_1-u_2\|_{H},~~...
- 31
2
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108
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Controlling the adjoint variables in a stochastically perturbed control problem
Suppose we have a deterministic control problem
$$dX_t = b(X_t, u_t) \, dt$$
on a finite timeframe with no terminal cost; i.e. the objective functional to be maximised is
$$\mathbb E \left [\int_{0}^T ...
- 4,832
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0
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79
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SPDE via fixed point argument and Young's theorem
Let $(P_r)_{r\geq 0}$ be a strongly continuous semi-group (not necessarily the heat kernel).
It is well known that we can prove local well-posedness of a few SPDE using a fixed point argument: Young's ...
- 399
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104
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Laplace transform of a stochastic process
Let $R := (R_1, R_2)$ be a two-dimensional diffusion process defined by the following SDE:
$$\mathrm{d}R_{1,t} = -\lambda_1 R_{1,t} \, \mathrm{d}t + \lambda_1 \sigma(R_{1,t}, R_{2,t}) \, \mathrm{d}W_t$...
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2
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If $u$ is harmonic, $\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x) \leq \alpha |x|+\beta,$ then $u$ is affine
We consider a harmonic function $u:\mathbb{R}^d \to \mathbb{R}$ $(\Delta u=0).$ Suppose that $$\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x)\leq \alpha |x|+\beta.$$
Therefore $u-u(...
- 399
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Application of Ito's formula to Liouville's theorem
Liouville's theorem for bounded harmonic functions could be proved using Ito's formula, martingale convergence and Blumenthal's 0-1 law.
I tried checking the classical books on Brownian motion and ...
- 399
2
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1
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135
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Is every simply connected domain regular?
Recall that a domain $D \subseteq \mathbb C$ is called regular if for each point $x \in \partial D$, we have $\mathbf P_x\lbrack \tau_D = 0\rbrack = 1$, where $\tau_D = \inf\{t > 0 : B_t \notin D\}$...
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Each diffusion SDE is associated to a *unique* family of transition kernels
I consider an SDE of the form $dX_t=b(X_t) \, dt + \sigma(X_t) \, dW_t$, with $b$ and $\sigma$ globally Lipschitz on $\mathbb{R}^n$.
How can I prove that there exists a unique family of transition ...
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Is a semimartingale that is continuous a continuous semimartingale?
Let $X$ be a centered semimartingale that has continuous sample paths almost surely. Is it then true that $X$ is a continuous semimartingale? Meaning that $X$ has a decomposition $X=M+V$ where $M$ is ...
- 1,302
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1
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156
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Is deterministic evolution preserved under weak converge of stochastic processes?
Suppose you have a sequence of continuous stochastic processes $X_N$ with $X_N(0)=0$, and that $X_N$ converge weakly on the space of continuous functions, to a stochastic process $X$. Suppose $X_N$ ...
- 1,302
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90
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How much is known about the action functional for small noise diffusions with general volatility coefficients?
Let $W$ be a d-dimensional Brownian motion, and for every $\varepsilon > 0$, let $X^\varepsilon$ be the solution to the SDE
$$dX^\varepsilon_t = b(X^\varepsilon_t) \, dt + \varepsilon \sigma (X^\...
- 4,832
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0
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97
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Bounding from below the distance between SDE started from different initial conditions
Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t$$
with $\mu, \sigma: \mathbb R \to \mathbb R$ Lipschitz ...
- 4,832
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0
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104
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Lower bound of $\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \varepsilon \mid X_t=x]$ (without observing history)
Let $X$ be the solution to some stochastic differential equation
$$dX_t =b(X_t) \, dt+a(X_t) \, dW_t,\quad \forall t>0.$$
Here $b,a: \mathbb R^d \to\mathbb R^d$ are bounded and Lipschitz and $W$ ...
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1
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Stationary Distribution of Langevin Dynamics driven by Lévy Process
Let $f\geq 0$ be a Lipschitz function and let $(L_t)_{t\geq 0}$ be an $\alpha$-stable Lévy process ($0<\alpha<2$, possibly multivariate). Consider the process given by $$dX_t=-\nabla f(X_t)dt+\...
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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 ...
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114
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Martingale regularization
Consider a submartingale $X,$ then for almost every $\omega \in \Omega,$ for every $v \in \mathbb{R},\lim_{u \in \mathbb{{Q},u \uparrow v}}X_u(\omega)$ exist in $\mathbb{R}.$
I was wondering if there ...
- 399
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1
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137
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Gaussian Poincare inequality in $1$ dimensions together with localization issue
Let $d\mu$ be a Gaussian measure on $\mathbb{R}$ with the center $a \in \mathbb{R}$ and variance $1$.
Let $B(a,r) \subset \mathbb{R}$ be the interval $[a-r,a+r]$.
Then, for any smooth mapping $f : \...
- 2,749
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1
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178
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Characteristic function of quadratic variation of compound Poisson process
If I have a compound Poisson process whose characteristic function is known, is there a way to calculate the joint characteristic function of this process and its quadratic variation process?
If not ...
-1
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1
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137
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joint density of two relevant random variables
It seems that for most of the examples to derive the joint density of two or more random variables, the random variables themselves need to be independent. Is it possible to get the joint density of ...
3
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94
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Dealing with noise that is white in time, colored in space numerically
I am broadly working on a dynamic process where we want to see how a field $\rho(r)$ changes in space in time with thermal noise. The system is biased around a thermodynamic saddle point dictated by $...
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1
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90
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Volterra Processes (integration wrt Brownian motion): reference request
I need some references about Volterra processes $Y=(Y_t)_{t\geq0}$ defined as
$$ Y_t:=\int_{0}^{t} g(t,s)dB_s, \ \ t\geq 0,$$
where $B=\left(B_t\right)_{t\geq0}$ is a brownian motion and $g$ satisfies
...
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3
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262
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Strong blow up limits for SDE
Note: This is a strengthening of the following result, motivated by the need for strong convergence in applications.
Let $W$ be a one dimensional standard Brownian motion, and let $X$ be the solution ...
- 4,832
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139
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Eigenvalues/eigenfunctions of a diffusion generator
Consider the following symmetric second order diffusion operator, defined, for $\phi \in \mathcal{C}^{2,1}_c\left(\mathbb{R}\times \mathbb{R}_+\right)$, by:
$$L\phi := \lambda_1 \partial_{R_1}(R_1 \...
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278
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Does the entropy of a SDE with nondegenerate noise always increase?
Let $W$ be a standard Brownian motion, and let $X$ be the solution to the one dimensional SDE
$$dX_t = \sigma(t, X_t) \, dW_t$$
with initial condition $X_0 = x_0$ a.s. for some $x_0 \in \mathbb R$. We ...
- 4,832
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1
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326
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Convergence of the quadratic variation process
Suppose we are given a sequence of stochastic processes $X^n, n\in\mathbb{N},$ with finite quadratic variations and a stochastic process $X$ such that for every $t\geq0$
$$
\lim_{n\to\infty}\mathbb{E}(...
- 99
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0
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73
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Calculation of the difference of two Brownian bridges
I was told that the difference of two independent brownian bridge process is $\sqrt{2}$ times a brownian bridge process, i.e.,
$$B_{1t} - B_{2t} = \sqrt{2}B_t$$
where $B_{1t}$ and $B_{2t}$ are ...
4
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A notion of SDE via the martingale representation theorem
$\newcommand{\d}{\mathrm{d}}$It is well-known that differentiating stochastic processes with respect to time is usually impossible in the usual sense. For instance, a Brownian motion $W$ on a ...
- 10.7k
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On a real smooth version of white noise distribution theory
In white noise analysis, one starts with a real Gelfand triple $\mathcal{N}\subset \mathcal{H} \subset \mathcal{N}^{*}$ and produces out of it, using complexifications along the way, the complex ...
- 473
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1
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141
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Local martingale with increasing process
Here is a problem in stochastic calculus:
If $M_t$ is a continuous process and $A$ an increasing process, then $M$ is a local martingale with increasing process $A$ if and only if, for every $f\in C^2$...
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0
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105
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Modulus of "set"-continuity for Wiener Field
My question concerns some "set-wise" continuity properties of Gaussian random fields, more specifically of Wiener fields (see definition here: https://encyclopediaofmath.org/wiki/...
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When is the probability measure on the "direct product" via the Kolmogorov extension theorem supported on the "direct sum"?
Let me restrict to the case of Hilbert spaces, which seem simplest.
Let $\{H_n\}$ be a sequence of (possibly infinite dimensional) Hilbert spaces and $\{ \mu_n \}$ be a sequence of Borel probability ...
- 2,749
2
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1
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Interacting particle system: how are the particles independent conditionally to the knowledge of their initial positions?
$\newcommand{\Ex}{\mathbb E}\newcommand{\diff}{\ \mathrm d}$Let
$(\Omega, \mathcal F, \mathbb P)$ be a probability space.
$B=(B^1, \ldots, B^N)$ independent one-dimensional Brownian motions.
$X=(X_0^...
- 851
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0
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228
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Ito lemma for SDEs on a Lie group
I'm trying to generalize the theorem described in this paper https://arxiv.org/abs/2001.01098 to the case of a semisimple compact matrix Lie group.
In doing so i'm trying to define a formula ...
- 263
1
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2
answers
195
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Converse Cameron-Martin theorem for shifts by adapted processes
Let $W$ be a standard one dimensional Brownian motion, $\mathcal F_t$ its natural filtration, and $\mathbb P$ be the induced Wiener measure on $\Omega := C[0, 1]$.
Given a $C[0, 1] $ valued random ...
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1
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130
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Are there any known results on the probability distributions of perpetuities with power law discount rates?
Currently I am working on studying stochastic integrals of the form: $$Z_\infty = \int_0^\infty e^{-f(t)}\mathop{d}S_t$$
where $S_t$ is a Compound-Poisson process with Exponentially-distributed ...
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2
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Convergence in sup norm of elementary integrals to the Itô integral process
Let $W$ be a standard one dimensional Brownian motion, and $X$ a continuous process adapted to $W$ such that $\int_0^T X^2 \, ds < \infty$ almost surely for some $T > 0$.
Define for any sequence ...
- 4,832
4
votes
1
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134
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Finite number of ergodic random Dirac measures
Let $\Omega$ be a Polish locally compact space and $(\Omega, \mathscr{F}, \mathbb{P})$ be a probability space. Consider a measurable map
\begin{align*}
\theta\colon T\times \Omega &\to \Omega\\
(t,...
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1
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90
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How to obtain this differential relation about moments of a stochastic process?
$\newcommand{\Ex}{\mathbb E}$ I'm reading an argument in the proof of Proposition 3.8. in the paper Nonlinear self-stabilizing processes - I Existence, invariant probability, propagation of chaos.
...
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1
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640
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How is the Gronwall lemma used in this paper?
Let $(X_t, t \ge 0)$ be a $\mathbb R^d$-valued stochastic process. Let $\lambda>0$. Assume we have $\mathbb E [|X_0|^2] < \infty$ and
$$
\mathbb E [|X_t|^2] - \mathbb E [|X_0|^2] \le -2 \lambda \...
- 851
2
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0
answers
241
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KL Divergence between the solution to two SDEs
What is the KL divergence between the laws of solutions to SDEs? That is, let
\begin{align*}
dX^1&=b_1(X^1,t) \, dt+\sigma(X^1,t) \, dB\\
dX^2&=b_2(X^2,t) \, dt+\sigma(X^2,t) \, dB
\end{align*}...
- 21
1
vote
1
answer
109
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Asymptotic expansion on the following integral of exponential function
I wish to obtain the asymptotic for the following integral
$$
\int_{r: \|r\|\le 1} \exp(M\cdot a^Tr) \, dr,
$$
where $a$ is a given vector in $\mathbb{R}^d$, $\|\cdot\|$ is a general norm function and ...
- 21
1
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0
answers
95
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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
$$
d X_t = ...
- 647
1
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1
answer
143
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The space of linear operators between Hilbert spaces has martingale type 2
I am trying to prove whether the space $L(H,K)$ has martingale type 2 for Hilbert spaces $H,K$. It is known that Hilbert spaces have martingale type 2, so I was wondering whether the space of bounded ...
- 113
3
votes
1
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322
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Reference request for a Riemannian Fokker-Planck equation
The original post is in StackExchange but no one has answered it yet. I personally think it is more related to the research area so I put it in MathOverflow. Below is the question in the original post:...
- 187
4
votes
1
answer
168
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Small noise limits with irregular drift
Let $W$ be a standard $d$-dimensional Brownian motion.
Suppose $b: \mathbb R^d \to \mathbb R^d$ is measurable and bounded. Consider, for every $\varepsilon > 0$, the solution $X^\varepsilon$ on $[0,...
- 4,832
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1
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100
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How is $\mathbb E[ \int_0^T H_s^2 \mathrm d s] < \infty$ important for this claim?
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space. Let $H=(H_t, t\ge 0)$ be a stochastic process with continuous trajectories. Fix $T>0$. For $n \ge 1$, we define
$$
H_{s,n} := \sum_{i=1}...
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