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.
377 questions with no upvoted or accepted answers
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Expected value and variance of a stochastic process
I would like to ask if there is a way to find the expected value and the variance of the following process
$$
dv_t=(a-be^{\alpha v_t})dt+\sigma dW_t, \quad v_t=v_0
$$
where $a\in (-\infty,+\infty), b&...
- 323
2
votes
1
answer
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Question about the stochastic integral of martingales
Let $M=(M_t)_{t\ge 0}$ be a continuous martingale defined on some filtered probability space taking values in $\mathbb{R}$. Let $H=(H_t)_{t\ge 0}$ be some bounded progressively measurable process, i.e....
- 1,835
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0
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$\alpha$ stable processes without jumps
Levy processes with jumps can be formulated following the Levy-kinchkine representation, which provide a decomposition of the characteristic function into three factors corresponding to the diffusion (...
- 305
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0
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58
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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:
$$\tilde{b}(x,t) = - b(x,t) + \int_{\mathbb{R}} y \left( 1 + \frac{...
- 305
1
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0
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28
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Existence and moment estimation for a linear stochastic differential equation (SDE) with random coefficients
Let $W$ be one-demensional Brownian motion, and suppose $X$ satisfies the following SDE
$$
\mathrm{d}X_s=(A_sX_s+B_s)\mathrm{d}s+(C_sX_s+D_s)\mathrm{d}W_s, \quad X_0=x_0\in\mathbb{R}^n,
$$
where $A, C\...
- 125
1
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0
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52
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An application to product formula of multiple integral
It is well-known that from Nualart's book, two multiple integrals can be expanded into a sum of multiple integrals, i.e.,
$$I_n(f)I_m(g)=\sum_{i=0}^{m\wedge n}i!C_m^iC_n^iI_{m+n-2i}(f\otimes_ig),$$
...
- 57
1
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0
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45
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Adding a data-dependent term to the porous medium equation while retaining an explicit solution
I am working with the porous medium equation, which I am treating it as a type of Fokker-Planck equation given by:
$
\frac{\partial u}{\partial t} = \Delta(u^m), \quad m > 1
$
For this equation, ...
- 11
1
vote
0
answers
59
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Gaussian Hypercontractivity of Chaos based on Gaussian with value in Hilbert spaces?
The classical Gaussian hypercontractivity is stated as following: Suppose $\xi$ is a Gaussian variable and $H_n(\xi)$ is the space of n-th homogeneous Wiener chaos constructed from $\xi$, then for any ...
- 253
1
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0
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95
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A stochastic optimal control problem with filtering-like dynamics
I want to extend the following stochastic optimal control problem with randomized feedback control to the continuous time case
\begin{align}
\text{minimize}\quad \mathbb{E}_{\mathbb{H}}&\bigg[\...
- 303
1
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0
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53
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The limit ratio of two Markov Chain Probability
Suppose there are two given SDE in $\mathbb{R}^d$:
$$
\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
39
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White noise, stochastic convolution: $0$–$1$ law of a stopping time
Let $\mathscr{C}^\alpha:=B_{\infty,\infty}^{\alpha}$ be the Besov space with the usual norm and let $C_T\mathscr{C}^\alpha:=C([0,T],\mathscr{C}^\alpha)$ the space of continuous functions from $[0,T]$ ...
- 573
1
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0
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133
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A question about one Malliavin derivative calculation
Recently, I've asked here a question. While trying to find an answer on my own, I found an idea which I now will briefly describe below. I am not familiar enough with the Malliavin calculus, so my ...
- 620
1
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0
answers
99
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Expectation of $B_u \operatorname{argmax}_t B_t$
This question is a repost from math.stackexchange. The question turned out to be harder than I initially thought, so I decided to try my luck here.
Yesterday I asked a question about the joint law of ...
- 620
1
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0
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122
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Derivative with respect to initial condition for the solution of an SDE
Suppose we have an SDE (assuming the Lipschitz continuous conditions required for the existence of the solution):
\begin{align}
dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t
\end{align}
and define its ...
- 111
1
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0
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159
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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
1
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0
answers
70
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On calculating the second quantization operator $\Gamma(A)$ of the Ornstein-Uhlenbeck operator $A$
Let $A$ be a self-adjoint operator on a Hilbert space , and let $d\Gamma(A)$ be the generator of the second quantization of $A$. Consider the following theorem from Segal's "Non-Linear Quantum ...
- 90
1
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0
answers
134
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Piecewise Ornstein-Uhlenbeck process time integral
Let $X_t$ be a piecewise Ornstein-Uhlenbeck process with infinitesimal variance $\sigma^2$ and (piecewise) infinitesimal mean $\theta_1$ for $x<c$ where $c$ is a constant and $\theta_2$ for $x\geq ...
- 11
1
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0
answers
89
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Heat kernel and estimates
In the article by Hairer-Labbe (A simple construction of the continuum
parabolic Anderson model on $\mathbb{R}^2$), they used the following "well known" fact (picture below) in holder spaces....
- 573
1
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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
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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
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0
answers
93
views
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 ...
- 573
1
vote
0
answers
108
views
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 $W$ ...
- 1,399
1
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0
answers
237
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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 ...
- 11
1
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0
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190
views
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:
$$L\phi := \lambda_1 \partial_{R_1}(R_1 \...
- 63
1
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0
answers
115
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Modulus of "set"-continuity for Wiener Field
My question concerns some "set-wise" continuity properties of Gaussian random fields, more specifically of Wiener fields (see definition here: https://encyclopediaofmath.org/wiki/...
- 73
1
vote
0
answers
100
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Reference request: $d X_t = b(X_t) d t + f (p_t(X_t)) d W_t$ where $p_t$ is the p.d.f. of $X_t$
Let $b:\mathbb R^d \to \mathbb R^d$ and $\sigma:\mathbb R^d \to \mathcal M_{ d\times q} (\mathbb R)$ be Lipschitz. Let $(W_t, t\ge 0)$ be the standard $q$-dimensional Brownian motion. Then
$$
d X_t = ...
- 657
1
vote
0
answers
121
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Stratonovich version of Girsanov
One version of Girsanov says that, that if $\mu_0$ is the law of a Brownian motion as a Borel measure on the space of continuous functions and we define the density
$$\frac{d\mu}{d\mu_0}:=\exp\left(\...
- 1,904
1
vote
0
answers
63
views
Patching together weak solutions of SDE's at random time points
Suppose we are given a sequence of drift coefficients $b^n : \mathbb R \to \mathbb R$ and we know that the following SDE has a weak solution, unique in law on $[0,\infty)$
$$dX_t^n(\mu) = b^n(X_t^n(\...
- 677
1
vote
1
answer
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For fixed $f \in L^2$ and $T>0$, choose $g$ so that $ \mathbb{E}^x[g(T-\tau)\chi_{X_\tau=1}]=-\mathbb{E}^x[f(X_T)\chi_{\tau \ge T}]$
Let $f \in L^2(0,1)$ and $T>0$ be fixed. How can I choose $g \in L^2(0,T)$ such that
\begin{align*}
0\equiv \mathbb{E}^x\left[f\left(X_T\right) \chi_{\tau \geqslant T}+g(T-\tau) \chi_{X_\tau=1}\...
- 19
1
vote
0
answers
53
views
Showing that the natural scale function is a martingale under specific conditions
My question is related to this post How to find the "natural scale function" for more general stochastic processes?. Indeed, I am trying to solve an exercise in which I have to show that if ...
- 11
1
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0
answers
58
views
Elliptic principal eigenfunction analysis for Langevin dynamics with a varying source term
Consider the Kolmogorov forward equation for a Langevin dynamic:
$$\DeclareMathOperator{\Div}{div}
\begin{cases}
\dfrac{\partial}{\partial t} f = \Delta f + \Div(f\nabla V)\\
\\
\displaystyle\int_{\...
- 11
1
vote
0
answers
83
views
Markov property of jump type diffusions
Consider the following jump-type SDE with two random Poisson measures $N_1$, $N_2$ and a Brownian motion $B_t$:
$dX_t= b(X_t)dt + \sigma(X_t)dB_t + \int{}F_1(X_t,u)N_1(dt,du) + \int{}F_2(X_t,u)N_2(dt,...
- 111
1
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0
answers
156
views
Fokker-Planck equation for a 3D Bessel bridge
Consider a 3D Bessel bridge $\rho_t$ connecting $(x,t)=(0,0)$ and $(x,t)=(0,T)$, whose SDE is given by
$$d\rho_t = \left(\frac{1}{\rho_t} - \frac{\rho_t}{T-t}\right)dt + dB_t,$$
where $B_t$ is a ...
- 19
1
vote
0
answers
157
views
The stochastic parallel transport as a limit of piecewise geodesic parallel transports
Let $(M,g)$ be a Riemannian manifold, and $E \to M$ be a vector bundle endowed with a connection $\nabla$. If $c:[0,1] \to M$ is a continuous curve, and if $\Delta = \{t_1, \dots, t_m\} \subset [0,1]$,...
- 5,407
1
vote
0
answers
47
views
How do we need to argue in this step of the Itō-Lévy-Khintchine decomposition?
Let
$E$ be a $\mathbb R$-Banach space;
$(\Omega,\mathcal A,\operatorname P)$ be a probability space;
$(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$;
$(X_t)_{t\ge0}$ be an $E$-...
- 167
1
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0
answers
57
views
Continuation : Does the density of a stopped drifted Brownian motion vanish at zero?
Let
$$Y_t:=1+\int_0^t b_sds + W_t,\quad\forall t\ge 0,$$
where $(b_t)_{t\ge 0}$ is a bounded adapted process and $(W_t)_{t\ge 0}$ is a standard Brownian motion. Denote $\tau:=\{t\ge 0: Y_t\le 0\}$ and ...
- 1,334
1
vote
1
answer
183
views
Let $(X, W)$ be a weak solution to a SDE. Is $W$ a Brownian motion w.r.t. $\sigma(X_s : s \le t)$?
Let $(X, W)$, $(\Omega, \mathcal{F}, \mathbb{P})$, $\{\mathcal{F}_t\}$ be a weak solution to an SDE.
Per definition $W$ is an $\mathcal{F}_t$-Brownian motion and both $X$, $W$ are adapted to $\mathcal{...
- 11
1
vote
1
answer
171
views
Does the convergence of drifted Brownian motion imply the convergence of expectation?
Let $(f_{\epsilon})_{\epsilon>0}$ be a family of non-increasing and continuous functions on $\mathbb R_+$ s.t. $f_{\epsilon}(0)=1$ and $f_{\epsilon}(\infty)=0$. Assume that $\epsilon\mapsto f_\...
user420828
1
vote
0
answers
124
views
On the Lipschitz constant of $\Gamma$
Let $b: \mathbb R_+\times\mathbb R\times \mathbb R\to\mathbb R$ be a function as nice as possible, and $C^1([0,T])$ be the space of continuously differentiable functions $\alpha:[0,T]\to\mathbb R$ ...
- 1,334
1
vote
0
answers
328
views
Preservation of variance for log-normal variables under change of measure
Aim: to show that changing a probability measure via the application of a Radon-Nikodym derivative preserves variance of a log-normally distributed random variable (for the case when variance is non-...
- 141
1
vote
0
answers
100
views
Ito formula for fractional BM + drift and supremum bound
Let $W^H$ be a fBm with Hurst parameter $H$ and let $\mathcal{H}$ be its Cameron-Martin space. Then by Girsanov theorem we know that if $\mathbb{P}$ is an fBm measure, it holds that there exists a ...
- 111
1
vote
0
answers
91
views
When enlarging a filtration makes a stochastic processes into a solution to an SDE
Let $n$ be a positive integer and let $(Y_t)_{t\in [0,1]}$ on $\mathbb{R}^n$ be a stochastic process defined on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,1]},\mathbb{P}...
- 5,405
1
vote
0
answers
41
views
Dependency of first hittimg time on coefficients of SDE
Let $b: \mathbb R_+\times\mathbb R\times [0,1]\to [\underline b,\overline b]$ and $a: \mathbb R_+\times\mathbb R\times [0,1]\to [\underline a,\overline a]$ be Lipschitz, where $\overline b>\...
- 1,334
1
vote
0
answers
240
views
Where to submit a new proof of the continuous martingale convergence theorem?
There were various proofs of the discrete martingale convergence theorem, but as far as I know there is only one proof of the continuous version of this theorem using the up-crossing lemma.
I wrote a ...
- 11
1
vote
0
answers
177
views
A question on Gaussian small ball probability
Consider the random variable $$ G = \sum_{j=1}^{\infty} \lambda_j Z_j^2 $$
where $Z_j \sim_{\substack{i.i.d}} N(0,1)$ and $\lambda_j$ some non increasing sequence of positive numbers with $\sum_{j=1}^{...
- 119
1
vote
0
answers
240
views
Convergence of the Ito integral along a filtration
Let $W$ be a standard Brownian motion, and $\mathcal F_t$ its natural filtration. Let $X$ be an $\mathcal F_t$-predictable process.
Question: Fix $b > a > 0$. Is it true that for all sequences $...
- 6,205
1
vote
0
answers
46
views
How to show a space is an invariant core for a strongly continuous semigroup?
This question comes from a paper 2015(Kolokoltsov) Theorem 4.1.
In the end of the proof i), “Applying to $T_t$ the procedure applied above to $T_t^h$ shows that $T_t$ defines also a strongly ...
- 165
1
vote
0
answers
464
views
Reference request: Introduction to stochastic control theory
I’m looking for a nice readable introductory text to stochastic control theory. Background wise, I know some general stochastic analysis and deterministic optimal control theory.
Some criterion I’m ...
1
vote
0
answers
206
views
The quadratic variation of $\int_0^t\int_T^Sg(s,x) \, dW_s^x \, dx$
Consider the process $W^x_t$ which is a Brownian motion for every $x\geq 0$ such that
$$d\langle W_t^x,W_t^y\rangle=Q(x,y)\,dt$$
where $Q$ is some non-negative definite function. Now consider the ...
- 77
1
vote
0
answers
177
views
Adiabatic elimination of "fast"/"velocity" variable
My question comes from section IV, part A of the paper titled Stochastic resonance. Specifically, their equation (4.1) states that, if we start with a Langevin equation of the form $$m\ddot{x} = -m\...
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