Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [stochastic-analysis]

Questions about stochastic analysis or stochastic calculus, for example the Ito integral. See https://en.wikipedia.org/wiki/Stochastic_calculus

0
votes
0answers
12 views

Show that the solution of an autonomous SDE is a time-homogeneous Markov process

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space and $$\mathcal N:=\left\{N\in\mathcal A:\operatorname P[N]=0\right\}$$ $(W_t)_{t\ge0}$ be a Brownian motion on $(\Omega,\...
0
votes
0answers
11 views

Simple example for a predictable process that is not in $L(X)$ for a semimartingale $X$

I am supposed to find some simple and basic examples for semimartingales $X$ and predictable processes $H$ such that $H\notin L(X)$, that means $H$ is not integrable with respect to $X$. Do you have ...
1
vote
0answers
18 views

Ito Lemma and identifying martingale parts

Suppose that $X_t$ is a càdlàg semi-martingale with decomposition $$ X_t= X_0+ B_t + M_t. $$ I know that using the Ito lemma for any $C^2$-function $f$, $$ f(X_t)= f(X_0)\\ + \int_{0^+}^tf_x(X_{s-})...
0
votes
0answers
25 views

Applying Itô's formula to logged Ornstein-Uhlenbeck process

I have the following O-U process: $$d \log z_t =-\nu \log z_t d_t + \bar\sigma dW_t \tag{1}$$ and want to apply Itô's formula as: $$dy=df(x)=\bigg(\mu(x)f'(x)+\frac{1}{2}\sigma^2(x)f''(x) \bigg)dt + ...
7
votes
1answer
75 views

Voronoi cell volume inside the ball

I have the following problem: Let us denote a ball with center $C$ and radius $R$ in $\mathbb{R}^d$ as $B(C, R)$. Given a unit ball $B(0, 1)$ and vector $u$ has a uniform distribution inside the ...
0
votes
0answers
7 views

Distance functions for stochastic simulations in R

I have been using approximate Bayesian computation to parameterize deterministic models. However now I am using stochastic models (Tau-leap method) I don't know how to use these data in distance ...
1
vote
1answer
23 views

Weird Ito Transformation

Suppose that $X_t$ is an $\mathbb{R}$-valued semi-martingale with decomposition $X_0+A_t+M_t$, where $A_t$ is finite variation and $M_t$ is a local martingale. What would be the semi-martingale ...
0
votes
0answers
33 views
+50

Dependence on the initial datum of the strong solution of a SDE

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A)$ $W$ be a $\mathcal F$-Brownian motion ...
0
votes
0answers
9 views

Burkholder's inequality on $[s,t]$

Let $T>0$, $F$ be an adapted squared-integrable process and $p\ge 2$, then the Burkholder's inequality implies $$ E\bigg(\bigg|\int_0^T F(s)dB(s)\bigg|^p\bigg)\le C(p) E\bigg[\bigg(\int_0^T F^2(s)...
0
votes
1answer
20 views

Proving joint convexity in stochastic linear problems with fixed recourse

In Birge's stochastic optimization book, we have the following formulation where $\zeta$ is a random variable: $z(x,\zeta) = c^Tx + \min \{q^Ty|Wy = h -Tx, y\geq 0 \} \\ s.t. \ AX = b, x \geq0$ In ...
0
votes
0answers
4 views

existence of isonormal Gaussian process with non-separable Hilbert spaces

Let $H$ be a non-separable real Hilbert space. Does there exist an associated isonormal Gaussian process $W$ ? I know that the answer is yes for separable real Hilbert space.
1
vote
1answer
29 views

How can I show that the stochastic integral of a jump process w.r.t. Brownian motion is a local martingale by using this special localizing sequence?

Suppose that $Y$ is a pure jump process with $N_t$ jumps in $(0,t]$ and $E[N_t]<\infty$. Denote the jump times by $T_i$. Let $W$ be a Brownian motion. If $T_0=0$, then \begin{equation} M_t=\int_0^...
0
votes
1answer
35 views

How to show that the Wiener measure is singular with respect to a normal law [on hold]

We have a Gaussian process $X$, $X_t:=B_t - tB_1$, where $B$ is a $BM$, $t\in[0,1]$. Let $\nu$ be the law of $X$ and $\mu$ the Wiener measure. How can I show that $\mu$ is singular with respect to $...
0
votes
0answers
13 views

Relation between time-changed solutions to SDEs

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$ $B$ be a $\mathcal F$-...
0
votes
0answers
21 views

Distribution of the Itō integral process with respect to Brownian motion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ $X$ be a $\mathcal F$-progressive process on $(\Omega,\mathcal A,\...
1
vote
1answer
19 views

Predictable process is adapted?

If I start with the definition of a predictable process as a measurable mappings on the predictable $\sigma$-algebra generated by sets like $$ (s,t]\times F, \quad s<t, \quad F \in \mathcal{F}_s $$ ...
0
votes
0answers
29 views

Obtaining martingales

Suppose we have an SDE of the form $dX=Ndt+MdB$. Ito lemma lets us know that F(X) also is an Ito process. This would imply in particular that the term $f^{'}MdB$ is a martingale. I wounder if this is ...
1
vote
0answers
17 views

Stochastic process with predictable cost function

EDIT: I have found the answer on my own, so you can consider this as an exercise. ;-) Consider iid random variables $(X_n)_{n\ge 1}$ with $P(X_n=1)=p$, $P(X_n=-1)=1-p$ for $\frac{1}{2}<p<1$ and ...
1
vote
0answers
15 views

Are we able to determine the distribution of a general stochastic integral?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ $M$ be a real-valued continuous local $\mathcal F$-martingale on $(\...
0
votes
0answers
12 views

What is cylindrical Brownian Motion / Wiener Process

I have been given some reading on the Krylov-Bogoliubov Method for constructing invariant measures. An SDE in Hilbert space H is introduced as $$d(X)=b(X)dt + \sigma(X)dW $$ Where W is the ...
0
votes
0answers
11 views

Explicit solution of SDE with linear drift and constant diffusion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(B_t)_{t\ge0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $b:\mathbb R\to\mathbb R$ be linear $\sigma>0$ We ...
0
votes
0answers
12 views

How can we conclude weak convergence of Markov processes by a uniform convergence assumption on the related generators?

I'm searching results of the following form: Let $G_n,G$ be generators of related Markov processes $X^{(n)},X$ on a common state space $S$. Assume there is a collection of sets $F_n\subseteq S$ such ...
0
votes
0answers
16 views

Proof of Burkholder-Davis-Gundy inequality in the Almost Sure blog

I am currently reading the famous almost sure blog by George Lowther about stochastic calculus. I am currently reading the section about the Burkholder-Davis-Gundi-Inequality (BDG inequality). At the ...
0
votes
0answers
21 views

$ lim_{n \to \infty}\int_a^b|f(s)-G_nf(x)|^pds = 0$

Let $f \in L^p$ and define $G_nf(s) := \sum_{i=0}^{n-1}f_iI_{[t_i,t_{i+1})}(s)$, where $f_0 = 0, f_i = \frac{n}{b-a}\int_{t_{i-1}}^{t_i}f(s)ds$, $a = t_0 < t_1<...<t_n = b$, $t_{i+1}-t_i = (b-...
2
votes
1answer
16 views

Probability for more general sumprocess

Consider $(X_n)_{n\ge1}$ iid with $$P(X_1=1)=p,\quad P(X_1=-1)=1-p$$ for some $p\in(0,1)$, $a,b\in\mathbb{Z}$ with $a<0<b$, the sum process $S_n:=\sum_{i=1}^nX_i$ and the stopping time $$T_{a,...
1
vote
0answers
6 views

stochastic ordering of counting processes/vectors

Let $N_1(t)$ be a delayed renewal process and $N_2(t)$ be an ordinary renewal process such that $N_1(t)\geq_{st}N_2(t)$. Consider a renewal process $Z(t)$ with the same inter-arrivall distribution as $...
2
votes
0answers
19 views

Definition of $X_T$

Let $X$ be a stochastic process and $T$ a stopping time. Then one forms the random variable $X_T$. I have a quite vague question: apparently $X_T$ will be quite different if we replace $X$ by one of ...
2
votes
0answers
29 views

SDE existence and uniqueness

I am working on the following problem in a course using "Stochastic Differential Equations" by Oksendal. Consider the two coupled It SDEs. $dXt=−λX_tdt+σdB_t$ $dYt=−\sin Y_tdt+sX_t\cos Ytdt$ Where ...
2
votes
1answer
46 views

Calculating the conditional expectation of a Brownian Motion

Let $B$ be a BM, $t\in (0,1)$. Calculate $E(B_t|B_1)$. There is a hint: If we have a sequence of i.i.d. variables $(X_i)_{i\in \mathbb{N}}$ and the first moment of $X_1$ exists, then for $S_n:=\sum_{...
0
votes
1answer
14 views

Conditional expected value of Wiener process

I need to compute $\mathbb{E}(W_{2}W_{6}|W_{3})$, where $W_{i}$ is a Wiener process. How do I do this?
0
votes
1answer
35 views

Distribution of a stochastic process

Is it possible to find a distribution of $X(t)$ for a fixed t by looking at a single sample path of X? I'm kinda lost in the strong assumption, that it is not possible but then I remember that for ...
1
vote
1answer
49 views

Characteristic function of ito process

I'm given process $X(t) = \int_{0}^{t} a(s)dB(s) $ where a(s) is a square-integrable deterministic function I need to show, that $E[e^{imX(t)}] = e^{-\frac{m^2}{2}\int_{0}^{t}a^2(s)ds}$ My attempt: ...
0
votes
0answers
14 views

Common refinement for simple predictable processes

I'm trying to show that for a semimartingale $X$, the stochastic integral $\mathcal{I}_X : \mathcal{S}_{\text{ucp}} \rightarrow \mathbb{D}_{\text{ucp}}$ is cauchy continuous, since the idea is to ...
1
vote
0answers
22 views

Why does the integrand become non-positive?

I am reading a proof of the HJB equation (Stochastic Optimal Control) on the book by Oksendal (Chapter 11) and I encountered this: Given $Y_t=(s+t,X_{s+t})$ and $G$ a fixed domain in $\mathbb{R}\...
1
vote
0answers
31 views

In what sense is the product Gaussian measure on $\Bbb R^{\Bbb R}$ a Gaussian measure?

I'm reading Hairer's notes on SPDEs here. On page 17 he has remark 3.33 where he considers $\Bbb R^{\Bbb R}$ with the product $\sigma$-algebra and product Gaussian measure. He claims the Cameron ...
0
votes
0answers
43 views

Some problems on OU process

From Marc Yor's Continuous Martingale and Brownian Motion Page 38, we know that the process $X_t=e^{-\lambda t}B_{e^{2\lambda t}}$ is an OU process, where $B_{t}$ is an one-dimensional standard ...
0
votes
0answers
8 views

Stochastic Approximation: Convergence of a multivariate system

I'm currently stuck in a stochastic approximation problem in my research for a while. Suppose I'm approximating $(a_n, b_n, c_n)'$ using stochastic recursive sequence. I'm not interested in whether $...
0
votes
0answers
12 views

Inverse Process

Let $X_t$ be a (non-deterministic) special-semi-martingale. Is it possible to find a predictable process $H_t$ such that the stochastic integral of $H$ by $X$ satisfies: $$ \int_0^T H_tdX_t =1, $$ ...
0
votes
0answers
13 views

Does the sde explode?

We consider the sde $\,\, d X_{t} = d B_{t} - \nabla{U(X_{t})}dt\,\, $ in $\mathbb{R}^{d}$. Is the translation invariance of the drift $\nabla{U}$ enough to ensure that the sde doesn't explode ...
0
votes
0answers
17 views

Convergence of Feller semigroups

Let $E$ be a locally compact separable metric space $(T_t)_{t\ge0},\left(T^{(n)}_t\right)_{t\ge0}$ be Feller semigroups$^1$ on $C_0(E)$ for $n\in\mathbb N$ Assume $$\left\|T^{(n)}_tf-T_tf\right\|_{...
1
vote
1answer
36 views

How to prove differential of product correlated Brownian Motions?

I was wondering how to prove/compute the differential of the product of two Brownian motions. I know how to do it in case they are independent as follows: Suppose $dX_t= \mu_t dt +\sigma_t dW_t$ and ...
0
votes
1answer
23 views

Measurability of the sum of measurable functions $(X_t)_{t \in I}$ ranging over a random index set $N$.

Assume I have a collection of real-valued measurable functions $(X_i)_{i \in I}$ on the measurable space $(\Omega,\mathcal{F})$. Let $N:\Omega \rightarrow 2^\Omega$ such that for every $\omega \in \...
1
vote
0answers
21 views

If $(X_t)_{t \in [0,\infty)}$ is RCLL adapted process, then left limit process $(X_{-t})_{t \in [0,\infty)}$ is locally bounded

Assume $X=(X_t)_{t \in [0,\infty)}$ is a RCLL adapted process. I set $X_{-t}:=\lim_{s \rightarrow t, s < t}X_s$ for $t \in (0,\infty)$ and $X_{-0}=0$. I want to show that $(X_{-t})_{t \in [0,\infty)...
0
votes
0answers
18 views

Itō diffusion as a solution to a martingale problem

Let $b,\sigma\in C_b([0,\infty)\times\mathbb R)$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ $B$ be an $\mathcal F$-...
1
vote
1answer
71 views

Proving adaptedness of $ (\sum_{0 < s \leq t} \Delta X_s 1_{|\Delta X_s| > 1})_{0 \leq t \leq T}$ for RCLL adapted process $(X_t)_{0 \leq t\leq T}$.

Let the probability space $(\Omega, \mathcal{F},P)$ be endowed with a filtration $\mathbb{F}=(\mathcal{F}_t)_{0 \leq t \leq T}$. We are given a RCLL adapted process $X=(X_t)_{0 \leq t\leq T}$. Let me ...
0
votes
1answer
16 views

Right continuous Stochastic process $A$ increasing on countable dense subset of $[0,1]$ has increasing trajectories on $[0,1]$

I am currently reading through the proof of the Doob-Meyer deocomposition and came across this statement: Assume that $(A_t)_{t \in [0,1]}$ is a right continuous stochastic process. If $D$ is a ...
0
votes
0answers
27 views

Show that X and the increments are independent

Can someone help me with the following exercise? Let $B$ be a $d$-dimensional Brownian motion starting at the $\mathcal{F}_0$-measureable $X$. Show for all $n \in \mathbb{N}$ and $0=t_0<t_1<...&...
0
votes
0answers
21 views

Product of orthogonal martingales

Let $(\Omega,\mathcal F,\mathcal F_t,P)$ be a probability space and $W(t)$ be a Brwonian Motion defined on it. Let $M(t)$ be a bounded martingale orthogonal to $W(t)$, i.e., there exists a constant $C&...
0
votes
0answers
17 views

optional integer-valued random measure v/s multivariate point process

I want to known what is the difference between an optional integer-valued random measure and a multivariate point process. By the formula $$ \mu(\omega;dt,dx)=\sum_{k\geq1}I(T_k<\infty)\epsilon_{(...
0
votes
0answers
13 views

Equivalence between Fokker Planck Equation and SDEs

So I have a model defined by the following transition states $$T_1\equiv T\left(x_1+\frac{1}{N},x_2-\frac{1}{N}|x_1,x_2\right)=rx_1x_2+\epsilon x_2$$ $$T_2\equiv T\left(x_1-\frac{1}{N},x_2+\frac{1}{N}|...