Questions tagged [stochastic-analysis]
For questions about stochastic analysis or stochastic calculus, for example the Itô integral.
2,153
questions
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How to analyze a particular service queuing model?
If a job service system schedules queued jobs in the system at fixed time intervals (such as 1 hour) to execute (possibly multiple jobs at once), each job runs for 1 hour, and the number of jobs ...
0
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0
answers
43
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+50
numerical integration of a function satisfying a ode
I need to numerically approximate an integral of the form $$\int_0^\tau f(X_t)\:{\rm d}t,\tag1$$ where $(X_t)_{t\ge0}$ is the solution of a SDE $${\rm d}X_t=b(X_t){\rm d}t+\sigma(X_t){\rm d}W_t\tag2.$$...
- 13.4k
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Numerical solution to non-linear ODE (not stochastic DE) with one normally distributed initial condition
Are there any books, research on solving a non-linear ordinary differential equation (not a stochastic differential equation) with one normally distributed initial condition?
One simple way would be ...
- 53
1
vote
1
answer
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Proving $X_t = 1 + \int_0^t X_s \, dN_s$ is a supermartingale
Example 1.1.12. (Exponential Martingale) Suppose that $N$ is a semi-martingale on $\mathbb{R}$ with $N_0 = 0$. Consider the equation
$$
X_t = 1 + \int_0^t X_s \, dN_s.
$$
The solution is
$$
X_t = \exp ...
- 3,833
0
votes
0
answers
15
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A detail in the proof of a martingale theorem about a.s. convergence under the stopping time condition
I was reading a theorem about martingale in a textbook by Yuan Shih Chow.However,I cannot understand a detail in the proof,and I will appreciate if you could explain me in the red frame why we can ...
- 1
0
votes
0
answers
62
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How to solve SDE with Ito formula?
thank you very much for clicking on my question.
I'm working on this paper (https://www.duo.uio.no/bitstream/handle/10852/10566/pm12-05.pdf?sequence=1) (Page 3) and want to solve the following SDE:
$...
- 13
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votes
0
answers
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Reference request: showing that solution of an Ito SDE stays bounded with positive probability
Assume that we have a (well-posed) Ito SDE of the form $$\mathrm{d} X_t = b(X_t)\,\mathrm{d} t + \sigma(X_t)\,\mathrm{d}W_t $$ where $b \colon \mathbb{R}^d \to \mathbb{R}^d$, $\sigma \colon \mathbb{R}^...
- 2,622
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0
answers
22
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What is the quadratic variation process of an inhomogeneous Poisson process? [closed]
Suppose that $A(t)$ is an inhomogeneous Poisson process with a time-varying intensity process $\lambda(t)$. Note: The definition of inhomogeneous Poisson process could be found in https://en.wikipedia....
- 53
4
votes
0
answers
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Girsanov from stochastic starting points.
Consider a canonical Wiener space $(\Omega,\mathscr{F},\mathbb{P})$. Consider two SDEs:
$$dX_t = \alpha(X_t,t)dt+dW_t, \ \ \text{Law}(X_0)=\mu$$
and
$$dY_t = \beta(Y_t,t)dt+dW_t, \ \ \text{Law}(Y_0)=\...
- 17
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votes
0
answers
12
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Finding mutual information in discrete linear partial observation stochastic process
I have one basic question maybe is not to hard for you but I am a bit confused. Let our system be like this:
\begin{align}
X_{k+1} &= A_k X_k + W_k \\
Y_k &= C_k X_k + V_k
\end{align}
where $...
- 1
1
vote
0
answers
31
views
Stochastic continuity of a random process
Let $\xi = (\xi_t,t ∈ [0,1])$ be a random process such that all $xi_t, t ∈ T$, are independent in the aggregate, equally distributed and non-trivial (different from constant). Is the process $xi$ ...
- 443
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answers
53
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Confusion about extending the definition of stochastic integral to continuous local martingales
In Jean-François Le Gall's Brownian Motion, Martingales, and Stochastic Calculus, the author defines the stochastic integral for a continuous local martingale $M$ in Chapter 5, which is defined as $H \...
2
votes
1
answer
98
views
Help me understand this proof of "the covariance of a Gaussian measure is trace-class"
So I am reading an introductory script on stochastic analysis in Hilbert spaces and there is a step in the proof of "Gaussian measures have trace-class covariance" that I don't understand:
...
- 256
1
vote
1
answer
29
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Local Lipschitz continuity and explosion time in SDE.
I am self-studying the following material, whose source is https://sayanmuk.github.io/StochasticAnalysisManifolds.pdf
However, I am stuck with one step of the proof of the only theorem that follows ...
- 3,833
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votes
0
answers
17
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Extending Schilling's proof of Ito process approximation by simple processes for one-dimensional case to multivariate case
Below is the proof of Lemma 18.5 from Rene Schilling's Brownian motion which states that an Ito process can be approximated uniformly in probability by a simple Ito process.
Now it is stated in the ...
- 5,023
2
votes
0
answers
57
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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:
$$\mathrm{d}R_{1,t} = -\lambda_1 R_{1,t}\mathrm{d}t + \lambda_1 \sigma(R_{1,t}, R_{2,t}) \mathrm{d}W_t$$
$$\...
- 21
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0
answers
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Multi-dimensional Itô-isometry
I am looking to prove the following equation: $$\left(\int_s^t \sigma(X_r)dW_r\right)\left(\int_s^t \sigma(X_r)dW_r\right)^\top = \int_s^t \sigma(X_r)\sigma(X_r)^\top dr$$
for a function $\sigma(x)\in ...
- 13
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votes
0
answers
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views
How to estimate $\int_0^\tau f(X_t)\:{\rm d}t$ when $X$ is a diffusion process?
Say we have Markov processes $\left(X^{(i)}_t\right)_{t\ge0}$ with lifetime $\tau_i$ such that $\left(\left(X^{(i)}_t\right)_{t\ge0},\tau_i\right)_{i\in\mathbb N}$ is independent and identically ...
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Conversion of a multivariate Stratanovich SDE into a Ito SDE representing a Brownian motion on a sphere
Sorry, first question here, so I apologize if the formatting is not great.
I have a Stratanovich SDE given by
$$d\theta_{1}=\sin(\theta_{2})\circ dW_{1}-\cos(\theta_{2})\circ dW_{2}$$
$$d\theta_{2}=\...
- 21
2
votes
0
answers
46
views
Expectation Value of the Product of a Time integral and a Ito Integral
Consider a stochastic process $X_t$
\begin{equation}
dX_t = a(X_t)dt + \sigma dW_t
\end{equation}
where $W_t$ is a Wiener Process.
I know the expectation value of the product of two stochastic ...
- 21
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votes
1
answer
136
views
Is there a measure theoretic interpretation of rough path integrals?
In case of an almost surely continuous function $f$ we know that the Lebesgue Measure coincides with the Riemann Integral.
With this in mind, supposing $\mathbf{X}$ is a rough path lift of $X$, is ...
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votes
0
answers
82
views
How can we calculate the variance of this stopping time?
Let
$(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space;
$\mu,\pi$ be probability measures on $(E,\mathcal E)$ with densities $u,p$ with respect to $\lambda$ with $p>0$ and $$c:=c_0\frac ...
- 13.4k
2
votes
0
answers
39
views
Fokker-Planck equality implies sample path equivalence?
Let's say we have two densities $p_t$ and $q_t$ on $\mathbb{R}^d$ which have the same time-evolution in the Liouville (Fokker-Planck with no noise terms) sense. That is,
$ \frac{ \partial p_{t}(x) }{\...
- 123
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votes
1
answer
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views
Picard-Lindelöf Theorem for Stochastic Differential Equations proof
I am self-studying https://sayanmuk.github.io/StochasticAnalysisManifolds.pdf, the following proof can be found on the 9th page of the monograph I just provided the reference. It follows from a ...
- 3,833
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votes
1
answer
58
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Proof that horizontal subspace is complementary to vertical subspace in a principal bundle with connection
Let $F(M)$ be a principal bundle over a smooth manifold $M$, with a connection $\nabla$. Let $\pi: F(M) \to M$ be the projection. I am working to prove a theorem related to the horizontal and vertical ...
- 1,009
2
votes
0
answers
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views
Exact sampling from a SDE
On p. 157 of the book Applied Stochastic Differential Equations, there is an algorithm presented which claims to allow "exact" sampling from an SDE$^1$ $${\rm d}x=f(x){\rm d}t+{\rm d}\beta\...
- 13.4k
2
votes
1
answer
49
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Does BDG inequality hold even when the expectations are infinite? [closed]
When reading literature about the Burkholder-Davis-Gundy inequality, integrability is often glossed over.
The BDG inequality says for continuous local martingale that
$$E[[M]_t^{p/2}]\lesssim E[(\sup_{...
- 137
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votes
1
answer
63
views
Intuition behind $Q_t=\sum \langle M^{\alpha},M^{\alpha}\rangle_t+\sum |A^{\alpha}|^3_t+|A^{\alpha}|_t+t$
Consider the semimartingale $Z$, which by Doob decomposition can be written as $Z=M+A$, where M is a martingale and A is the process of total bounded variation.
I am trying to make sense of the ...
- 3,833
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votes
0
answers
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Stuck in a question. Could you please give me a hint?. [duplicate]
I have been stuck in this question for a while. I am not able to prove the first part. I have thought along the following possible directions:
I tried showing that $\mu^{*}(G^{c})=0$. However, I ...
0
votes
0
answers
19
views
Existence of optimal controls under the strong formulation
I'm reading a textbook that gives both a "weak" and "strong" formulation of optimal control (first is a control function, second is the whole space, filtration, Brownian motion, ...
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votes
0
answers
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Convergence of the conditional expectation in Kalman-Bucy filter for small noise
I am reading different papers on parameter estimation in the Kalman-Bucy filter scheme for small noise ("ON FREQUENCY ESTIMATION FOR PARTIALLY OBSERVED SYSTEM WITH SMALL NOISES IN STATE AND ...
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5
votes
0
answers
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views
Invariant measure for wrapped diffusion
Consider the diffusion $${\rm d}X_t=b(X_t){\rm d}t+\sigma(X_t){\rm d}W_t\tag1$$ on $\mathbb R^d$. Denote the solution starting at $x\in\mathbb R^d$ by $X^x$. Let $$\kappa_t(x,B):=\operatorname P\left[...
- 13.4k
5
votes
2
answers
90
views
Pathwise differentiability of stochastic integrals
My question: Is there a necessary and/or sufficient condition we can place on suitable continuous $f : [0,\infty) \rightarrow \mathbb{R}$ which allows us to determine whether the process
$$X_t := \...
- 941
0
votes
0
answers
31
views
Hölder continuity in the Kolmogorov-Chentsov theorem
Good morning,
I'd like to ask a question about details concerning a part of the proof of the Kolmogorov-Chentsov continuity theorem of stochatic processes (SP), w. l. o. g. we prove it on $[0,1]$.
...
- 59
0
votes
1
answer
45
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Derivative with respect to the initial datum of a stochastic process
Suppose I have a stochastic differential equation of the form
$$dX_t=b(t,X_t)dt + \sigma(t,X_t)dW_t \quad X_0=x_0 \in \mathbb R^n.$$
My professor said that in some cases one wants to compute the ...
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1
vote
2
answers
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views
predictable $\Rightarrow$ left continous? progressive $\Rightarrow$ adapted and left continous?
Assume we have a real valued continous time stochastic process $X:=(X_t)_{t\in [0,T]}$ $(T>0)$ defined on a complete, filtered probability space $(\Omega, \mathscr{F},\mathrm{P},(\mathscr{F}_t)_{t\...
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votes
1
answer
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Understanding Itô's Lemma proof in Chung Williams
I am studying the following ito lemma proof, however I am having some trouble understading, and it has been a week without figurigh out certain steps. Here is the theorem statement and full proof, ...
- 3,833
3
votes
1
answer
33
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Existence and simulation of affine jump-diffusion
In page 1349 or Section 2.1 of "Duffie, D., Pan, J., & Singleton, K. (2000). Transform Analysis and Asset Pricing for Affine Jump-Diffusions. Econometrica, 68(6), 1343-1376" an affine ...
- 380
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votes
0
answers
43
views
Uniform convergence in distribution implies convergence of moments
I am reading a paper in which the author wants to prove the convergence of the moments. He transforms the object of interest $\varepsilon^{-1} (\vartheta_\varepsilon^*-\vartheta_0)$ into
\begin{align*}...
- 61
2
votes
0
answers
66
views
Time-dependent transition probabilities
I am studying stochastic processes using An Introduction to Stochastic Modeling by Pinsky and Karlin. I stuck on this question 3.4.18 in Chapter 3. I would really appreciate if someone could help me ...
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2
votes
1
answer
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Stochastic integral (integer powers of white noise)
It is known how to calculate stochastic integrals of the kind, e.g., $\int_0^T W_t \, dW_t$ or $\int_0^T W_t^2 \,dW_t$, where $W_t$ is the Wiener process, aka Brownian motion.
Question: How about the ...
- 1,983
5
votes
0
answers
56
views
Form of invariant measures for SDsE on the toroidal domain $[0,1)^d$
Consider the SDE $${\rm d}X_t=b(X_t){\rm d}t+\sigma(X_t){\rm d}W_t\tag1$$ with Lipschitz continous $b:\mathbb R^d\to\mathbb R^d,\sigma:\mathbb R^{d\times d}\to\mathbb R$ and a $d$-dimensional Brownian ...
- 13.4k
1
vote
0
answers
111
views
Invariant measure for the Euler-Maruyama discretization of an Itō diffusion
Consider the following Itō diffusion: $${\rm d}X_t=b(X_t){\rm d}t+\sigma(X_t){\rm d}W_t\tag1$$ and the corresponding Euler-Maruyama discretization $$Y_i:=Y_{i-1}+\Delta tb(Y_{i-1})+\sqrt{\Delta t}\...
- 13.4k
2
votes
0
answers
49
views
Can we show that a (random) time discretized Markov process is still Markov?
Let
$(E,\mathcal E)$ be a measurable space;
$(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$;
$(\Omega,\mathcal A,\operatorname P)$ be a probability space;
$(\mathcal F_t)_{t\ge0}$ be a ...
- 13.4k
0
votes
0
answers
34
views
Is there a continuous-time Markov process whose generator domain is not contained $C^2$?
Let $(X_t)_{t\ge0}$ be a Hilbert space $H$ (take $H=\mathbb R^d$, for simplicity) valued time-homogeneous Markov process with transition semigroup $(\kappa_t)_{t\ge0}$. The latter can be considered as ...
- 13.4k
1
vote
1
answer
89
views
Grönwall inequality for semimartingales
I'm trying to prove the following version of the Grönwall inequality: Suppose that
$0 \leq A_t \leq \alpha + \int_0^t A_{s-}dC_s$
for a non-decreasing cadlag process $C$. Show that
$A_t \leq \alpha e^{...
- 295
1
vote
0
answers
47
views
Finding the roots of an equation / continuous function
Consider the equation
$$\sum_{i=1,\ldots,n}p_{i}x_{i}^{1-r} = \sum_{j=1,\ldots,m}q_{j}y_{j}^{1-r}\tag{1}$$
where the "moving part" is $r$, while the $x_{i}, y_{j}, p_{i}, q_{j} \in \mathbb{R}...
- 321
0
votes
0
answers
25
views
White noise with non-constant variance
Is there a name for white noise that has non-constant variance? I have some examples from experimental data where the variance of the white noise increases with time. However, I am not sure how to ...
- 11
1
vote
0
answers
53
views
Regularity of continuous martingales
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 ...
- 546
1
vote
1
answer
28
views
Find parameters for process to become gaussian
Find parameters $ a,b,c $ for process $ aW_t^2+cW_{bt^2+a} $ to make it gaussian.
The process is gaussian if
$$ E \left [ \exp \left ( i \sum_{l=1}^{k} s_l Y_{t_l} \right ) \right ] =
\exp \left ( -\...
