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Questions tagged [stochastic-analysis]

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

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7 views

Unit used in continuous time process noise matrix in kalman filters, when STD is from discrete time data

I'm trying to make a process noise matrix in continuous time. But i can't seem to find a clear definition of what "unit" the matrix should contain in continuous time. From our control book we have $...
2
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0answers
32 views

Proving one form of Ito Isometry using Functional Analysis

I would like to know whether it is possible to give a proof of (one form of) Ito Isometry using a tool which I like to call "the functional analysis"-way. Let me explain the settings first. What we ...
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1answer
15 views

Questions about Quadratic Variation given by Brownian Motion

We know that for a submartingle $A(t)$, $A(t)-\langle A\rangle_t$ is a martingale where $\langle A\rangle_t$ is its quadratic variation. For processes like $W^3(t)$ ($W(t)$ being standard Brownian ...
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0answers
17 views

Uniform integrablility of Radon-Nikodym derivatives if measures are locally equivalent.

Before the proof of Girsanov's theorem, we were proving the following result in class:- Lemma- Let $Q$ and $P$ be mutually locally equivalent probability measures on $(C[0,\infty),\mathcal{B}(C[0,\...
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1answer
42 views

Prove that $\lim_{L\rightarrow\infty} P\left(\sup_{0\leq s\leq t}|B(s)|>L\right)=0$, for each $t\geq0$, where $B$ standard Brownian motion.

Let $B(t)$, $t\geq0$, be a standard Brownian motion. I would like to prove that $$\lim_{L\rightarrow\infty} P\left(\sup_{0\leq s\leq t}|B(s)|>L\right)=0,$$ for each $t\geq0$. In my class notes, ...
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0answers
20 views

How should I adapt my continuous model to the discrete data records?

Let $N(t) :N \in \mathbb{N}, t \in \mathbb{R}$ a stochastic jumping process over the time. $N(t)$ is characterized by a unknown pmf (probability mass function) and represents the number of jumps ...
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0answers
25 views

Using Ito's Lemma to compute the process followed by a function

I have the following process $$dS_t = \mu S_t dt + \sigma S_t dz_t $$ and the function $$f(S) = S^2$$ where $$\frac{\partial f}{\partial t} = 0, \frac{\partial f}{\partial S} = 2S, \frac{\partial^2 ...
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0answers
7 views

Under what conditions does the solution to a mean reverting sde satisfy $E[\sup_{[0,T]} r(t)^2]<\infty$ for all $T>0$

Consider the mean reverting square root SDE $dr(t)=\alpha(\mu-r(t))dt+\sigma \sqrt{r(t)}dW(t)$ Under what conditions on the coefficients does the solution to a mean reverting sde satisfy $E[\sup_{[0,...
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37 views
+50

If $\kappa$ is a contractive operator on $C_0(ℝ)$, is $λ(κ-\text{id})$ the generator of a Feller semigroup?

Let $E$ be a locally Hausdorff space, $$C_0(E):=\left\{f\in C(E):\left\{|f|\ge\varepsilon\right\}\text{ is compact for all }\varepsilon>0\right\},$$ $\kappa$ be a Markov kernel on $(E,\mathcal B(E))...
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0answers
37 views

Domain of attraction $F(x)=\exp(-x-\sin(x))$

I need to show that $F(x)=\exp(-x-\sin(x)),x >0$ is in no domain of attraction. That means that there exist no $a_{k} >0, b_{k} \in \mathbb{R}, k \in \mathbb{N}$ with $\lim\limits_{k \to \infty}...
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1answer
29 views

Conditional expectation of integral of Ornstein-Uhlenbeck process

Given that $X(t)$ is an Ornstein-Uhlenbeck process with $X(0) = x_0$, which is a Markov process, but not a Martingale, how could I go forward if I would like to calculate $E[\int_0^T X(s)ds | \...
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2answers
67 views

Frullani integral for $f(x)=e^{x}$ in a complex context

I should prove the Frullani integral equality $$ \int_{0}^{\infty} (1-e^{zx}) \frac{\beta}{x} e^{-\gamma x}dx = \beta \log \left(1- \frac{z}{\gamma}\right) $$ for $z \in \mathbb{C}$ with non-positive ...
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0answers
189 views

Convergence of a Stochastic Process - Am I missing something obvious?

In the paper On the Convergence of Stochastic Iterative Dynamic Programming Algorithms (Jaakkola et al. 1994) the authors claim that a statement is "easy to show". Now I am starting to suspect that it ...
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0answers
15 views

SDE with stationary Log-normal distribution

Is there a stochastic differential equation whose solution follows a stationary Log-normal distribution? I was thinking in the geometric Brownian motion $$dx = (\alpha x )dt + (\sigma x )db, \quad \...
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0answers
42 views

Proving a limit of a Markov process

Let $\{X_{t}\}$ be a continuous time Markov process with a phase space consisting of only two states from the set $S = \{1, 2 \}$. Let $1 - p_{ii}(t) = q_{i}(t) + o(t)$ as $t \rightarrow 0$, where $...
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1answer
31 views

SDE Integration

Does anyone know how to get the integration of the SDE below (Assume $\sigma \to 0$)? $$\dfrac{\mathrm dS_t}{S_t}=(r_d-r_f)\mathrm dt+\sigma(t, S_t)\mathrm dW_t$$ Thank you in advance! Image Link ...
2
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1answer
94 views

Solving second order nonhomogeneous ODE where the RHS is a random process

Context: I'm trying to characterize the metastability behavior of a digital latch. I'm modeling two cross-coupled inverters as RC circuits with negative gain. One of the inverters has a source of ...
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1answer
23 views

Solution of a second order Stochastic Differential Equation

Consider the following SDE $$dx = (-ax)dt + \sigma db\\ dy = (-by+e^{-x})dt$$ where $a,b,\sigma>0$ and $b$ is a browninan motion PROBLEM: What is the solution of this system? How can I estimate ...
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0answers
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Definition Strong Markov property in Stroock/Varadhan

Let $(\mathbb{P}_x)_{x \in \mathbb{R}^d}$ be a continuous Markov process. So the $\mathbb{P}_x$ are the laws on the space of continuous function from the positive real half line to $\mathbb{R}^d$, ...
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0answers
25 views

$n \times n$ system of stochastic differential equations

Let be $n \times n$ system of SDE $$ dX_t^{(i)} = \sum_{k = 1}^n a_{ik}(t) X_t^{(k)} dt + dW_t^{(k)}$$ where $i = 1, 2, \dots, n$ and $a_{ik}(t)$ are continuous function for $t \geq 0$ and $X_0^{(i)} =...
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0answers
6 views

Ucp convergence and Emery Topology

maybe you can help me once again. It is known that convergence in the semimartingale topology (Emery topology) implies ucp convergence. Can you think of an easy example, to show that the converse is ...
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1answer
32 views

Solving a stochastic differential equationn

Does anyone has ideas on how to solve this equation. $$dX_{t} = \left(\sqrt{1+X_{t}^{2}} + \frac{1}{2}X_{t}\right)\,dt + \sqrt{1 + X_{t}^{2}} \,dBt$$ where $Bt$ is a standard Brownian Motion. I have ...
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0answers
15 views

Non-Uniqueness of the generative model of a stochastic process

To explain my question, I will start from a simple example. Suppose a binary random variable $X \in \{1, 0\}$ is generated by the following two steps. $P(Y = 1) = \frac{1}{2}$, $P(Y = 0) = \frac{1}{...
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2answers
39 views

Expectation of a function of Ito diffusion

Given an Ito Diffusion i.e.: $$ dX(t) = \mu dt + \sigma dW(t) $$ and a function $$ k(x) = \lambda x^2 $$ and I want to find the expected value $E[k(X(t)]$ of the function - the only way I know ...
3
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0answers
38 views

Geometric Brownian Motion Price Processes in high Dimensions

This is my first post so I am open for an suggestions in formating improvement. For some reason I can not find suitable literature for the following problem What I want to do is calculate the option ...
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0answers
51 views

If $L$ is a diffusion operator with corresponding carré du champ operator $\Gamma$, then $Lf^2=2fLf+2\Gamma(f)$

Let $(E,\mathcal E,\mu)$ be a measure space and $$\mu f:=\int f\:{\rm d}\mu\;\;\;\text{for }f\in L^1(\mu)$$ $\mathcal A_0$ be a subspace of $\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\...
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1answer
67 views

Why do we need absolute continuity of $\langle M \rangle_t(\omega)$ with repect to the Lebesgue measure?

I am trying to understand the proof of proposition 3.2.6 in Stochastic Calculus and Brownian Motion by Karatzas and Shreve. For $X$ bounded they use Lemma 3.2.4 in the same book and eventually claim(...
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1answer
87 views

Extending the domain of the Dirichlet form associated with a symmetric Markov semigroup

Let $(E,\mathcal E)$ be a measurable space $\mathcal M_b(E,\mathcal E):=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$ $(P_t)_{t\ge0}$ be a Markov semigroup ...
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3answers
31 views

Probability question with intuitive answer - odds two travelers overlap at the same time?

Let's say Person A sets out from Point X at 8:00 AM on Day 1 and travels for 12 hours until he reaches Point Y at 8:00 PM. Person B sets out from Point Y at 8:00 AM on Day 2 and travels for 12 hours ...
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2answers
109 views

Show that the carré du champ operator is nonnegative

Let $(E,\mathcal E)$ be a measurable space $\mathcal M_b(E,\mathcal E):=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$ $(\kappa_t)_{t\ge0}$ be a Markov ...
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1answer
65 views

Ornstein Uhlenbeck process hits zero with probability 1 in finite time

I am looking for a reference or a proof which shows that $P(\tau_0^Y<\infty)=1$ for an ornstein uhlenbeck process $Y$ given by $$ dY(t)=-\frac{1}{2}\alpha Y(t)dt+ \frac{1}{2} \sigma dW(t),Y(0)=y&...
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0answers
72 views

Show that $C_c^∞(\mathbb R)$ is a core of the generator of the Feller semigroup induced by the strong solution of an SDE with Lipschitz coefficients

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $b,\sigma:\mathbb R\to\mathbb R$ be Lipschitz continuous and $$Lf:=bf'+\frac12\sigma^2f''\;\;\;\text{for }f\in C^2(\mathbb R)$$ $W$ be ...
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0answers
11 views

Brownian motion absorbed at the boundaries

Why the solution of the following SDE $$dX_t=\sqrt{X_t(X_t-1)}dB_t$$ is considered a Brownian motion which is absorbed at the boundaries?
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1answer
67 views

Is the transition semigroup of the solution of an SDE with Lipschitz coefficients strongly continuous on $C_b$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $b,\sigma:\mathbb R\to\mathbb R$ be Lipschitz continuous (and hence at most of linear growth) and $$Lf:=bf'+\frac12\sigma^2f''\;\;\;\...
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1answer
60 views

Show $\text P\left[|X^x_t|<r\right]\xrightarrow{|x|\to\infty}0$ for strong solutions of SDEs

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $b,\sigma:\mathbb R\to\mathbb R$ be Lipschitz continuous $(X_t^x)_{t\ge0}$ be a continuous process on $(\Omega,\mathcal A,\...
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1answer
35 views

Estimate for the distance from the initial value of a strong solution of an SDE

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $b,\sigma:\mathbb R\to\mathbb R$ be Lipschitz continuous $(X_t^x)_{t\ge0}$ be a continuous process on $(\Omega,\mathcal A,\...
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1answer
30 views

Drawing Balls of Different Colors from an Urn

Let consider an urn model containing $n$ balls such that $m_g$ balls are green, $m_r$ are red and $m_b$ are blue and we have $n = m_g + m_r +m_b$. Especially this means that the $m_g$ green balls are ...
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1answer
43 views

Derivation of the Fokker-Planck equation

Let $b\in C^1(\mathbb R)$ be Lipschitz continuous $\sigma\in C^2(\mathbb R)$ be Lipschitz continuous with $\sigma(\mathbb R)\subseteq\mathbb R\setminus\left\{0\right\}$ and $\sigma''$ being bounded $...
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0answers
19 views

What is $E[X_1]$ where $dX_t=(1-X_t^2)^{-1}dB_t,$ $X_0=1$?

I have a stochastic equation $$dX_t=(1-X_t^2)^{-1}dB_t,$$ $X_0=1$ and $B$ is a Brownian motion. I know there exists a strong solution $X$. How can I compute $$E[X_1]?$$ I thought about trying showing ...
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0answers
22 views

comparison of two stochastic processes

Let $(\Omega, \mathcal{F}, P, \mathcal{F}_t)$ be a probability space under "usual conditions" and two semimartingales X and Y such that $$X=W_t+I_{(h\leq1)}h(\omega,t,X,u)*(\mu-\nu)_t+I_{(h\leq1)}(h(\...
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0answers
20 views

integrability of system of SDE relative to Langevin dynamics

Similar to the system in this question and a follow up to here, consider the system (S) defined as \begin{align} {\rm d}X_t&=V_t\,{\rm d}t+\sigma_x\,{\rm d}W_t^1\\ {\rm d}V_t&=F_t\,{\rm d}t+\...
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1answer
42 views

Stochastic Processes: Find the function f(n) , n = 0,1,2,… that satisfies f(0) = 0, f(n) = 1/3f(n-1) + 1/3f(n+1) +1/3f(n+2), n >= 1

Find the function $$f(n),\ n \in \mathbb{N},$$ that satisfies $$f(0) = 0,$$ $$f(n) = \dfrac{f(n-1)}{3} + \dfrac{f(n+1)}{3} +\dfrac{f(n+2)}{3},$$ and $$\lim_{n\rightarrow +\infty} f(n) = 1.$$
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0answers
58 views

Show that if $(κ_t)_{t≥0}$ is the transition semigroup of a strong solution to an SDE, $t↦(κ_tf)(x)$ is continuous for all $x$ and suitable $f:ℝ→ℝ$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $b,\sigma:\mathbb R\to\mathbb R$ be Lipschitz continuous and $$Lf:=bf'+\frac12\sigma^2f''\;\;\;\text{for }f\in C^2(\mathbb R)$$ $W$ be ...
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0answers
25 views

Example of ANY stochastic process (SDE), with reversible distribution

Can anyone provide an example (as simple as they like) of a process $X_t$ on $\mathbb{R}$ solution to $dX=\sigma (X,t)dt+b(X,t)dW$. Where $W$ is a Brownian Motion, and $\sigma$ and $b$ can be any ...
1
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2answers
64 views

Supremum of Brownian motion

I am trying to understand the proof in "Roger and Williams" for the Lemma Lemma: Let $B_t$ be a Brownian motion, then$$P(\sup_t B_t=+\infty,\inf_t B_t=-\infty)=1$$ Let $Z:=\sup_t B_t$, they started ...
1
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1answer
25 views

Where is an error in my deduction? (question about martingales)

Suppose we have a filtration $\{\mathcal{F_{t}},t\geq 0\}$ and a stochastic process $\{ X_{t},t\geq 0\}$ which is adapted to this filtration and also integrable. All we need for this process to be a ...
0
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1answer
26 views

Progressively Measurable on a Topological Space

In Baldi's book, Stochastic Calculus, Proposition 2.1 states that a right continuous adapted process, taking values in a topological space $E$ will be progressively measurable. The proof starts out ...
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0answers
33 views

Fokker-Planck equation for a Markov semigroup with densities

Let $(E,\mathcal E)$ be a measurable space $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$ and $$\kappa_tf:=\int\kappa_t(\;\cdot\;,{\rm d}y)f(y)$$ for $$f\in F_0:=\left\{f:E\to\mathbb ...
1
vote
0answers
32 views

Finding a strong solution to $X_t := (A_tX_t+a_t)dt+(S_tX_t+\sigma_t))dB_t$

I have an SDE $$X_t := (A_tX_t+a_t)dt+(S_tX_t+\sigma_t))dB_t,$$ $X_0=x_0$ and $A,a,\sigma,S$ are continuous stochastic processes, $B$ is a BM. Now if I define: $$Y_t:=e^{(\int_0^tA_sds+\int_0^...
0
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0answers
33 views

Standard Brownian motion almost surely not $0$. [duplicate]

Consider a continuous standard brownian motion $(B_t)_{t\ge 0}$. I want to show, that $$\mathcal{L}(\{t\ge 0: B_t=0\})=0$$ I also have a hint that I need to show that $B:\mathbb{R}_+\times\Omega\...