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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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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
33 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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Stochastic Processes: Find all functions $f$ from the integers to the real numbers satisfying $f(n) = \frac12 f(n+1) + \frac12 f(n-1) -1 $ [on hold]

Find all functions f from the integers to the real numbers satisfying $f(n) = \frac12 f(n+1) + \frac12 f(n-1) -1 $ the textbook has the following hint: First show that f(n) = n^2 satisfies the ...
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13 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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17 views

Expectation and random stopping moment

Assume $X_1, X_2, \ldots, X_n$ i.i.d. variables with $p=\mathbb{P}(X=1)$ and $q = \mathbb{P}(X=-1)$. Further, assume $S_n^x = x + X_1 + \ldots + X_n$, for some $x>0$. Say that $\tau_n^x$ is some ...
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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 ...
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2answers
56 views
+50

Conditional expectation for brownian motion

Consider two brownian motions $(W_t)_{t\ge 0}$ with starting point $x$ and $(W'_t)_{t\ge 0}$ with starting point $y$. Define $T:=\inf\{t\ge 0:W_t=0\}$, the first time when $W_t$ is equal to $0$. I ...
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2answers
37 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 ...
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1answer
24 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 ...
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1answer
19 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
16 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 ...
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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^...
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32 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\...
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22 views

adapted, increasing, (locally) integrable variation process is a (local) submartingale

I read a theorem that an adapted, increasing, (locally) integrable integrable, variation process is a (local) submartingale. (here increasing includes right continuity). Definition: A process is $\...
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1answer
23 views

What is the quadratic variation of $e^{-t\lambda}B_{\frac{e^{2\lambda t}-1}{2\lambda}}$?

Let us define $$X_t:=e^{-t\lambda}B_{\frac{e^{2\lambda t}-1}{2\lambda}},$$ where $B$ is a Brownian motion and $\lambda>0$ How can I calculate quadratic variation $<X>_t$? The first thing ...
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1answer
56 views

If $X$ is an Ito process, is $\mathbb E(\int X \mathrm d X)$ convex?

Consider the functional $F$, which is defined for each Ito process $$X(t) = \int_0^t \mu(s) \mathrm d s + \int_0^t \sigma(s) \mathrm d W(s)$$ as $$F(X) := \mathbb E\bigg(\int_0^T X(s) \mathrm dX(s)\...
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38 views

How to improve the convergence of a stochastic differential equation?

I have a stochastic differential equation, i.e, $$ d\rho_t= \hat{A} \rho_s dt + \hat{B} \rho_s \nu dt + \hat{C}\rho_s\omega_{1t} dt + \hat{D}\rho_s \omega_{2t}dt \quad , \quad t>s $$ Here A, ...
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19 views

Approximating left continuous process $(L_t)_{0 \leq t \leq T}$ uniformly on $[0,T]$ by step functions on the Dyadics

The question is closely related to optional quadratic variation. If I have a left-continuous adapted stochastic process $(L_t)_{0 \leq t \leq T}$ with $L_0 = 0$ on the probability space $(\Omega, \...
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1answer
128 views

Convergence of the distribution of the Langevin diffusion to its invariant measure

Let $(X_t)_{t\ge0}$ be a solution of $${\rm d}X_t=-h'(X_t){\rm d}t+\sqrt 2W_t,\tag1$$ where $(W_t)_{t\ge0}$ is a Brownian motion and $h$ is such that $X$ is the unique strong solution of $(1)$. Assume ...
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0answers
31 views

What is a stationary solution to a SPDE?

I'm reading Hairer's notes on SPDEs: http://www.hairer.org/notes/SPDEs.pdf He says on page 6 that "the stationary solution to the stochastic heat equation is Gaussian free field". He never defines ...
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0answers
17 views

Show that the composition of a sub-invariant measure with a sub-Markov kernel is a contraction on $L^p$

Let $(\Omega,\mathcal A)$ be a measurable space $\mu$ be a finite measure on $(\Omega,\mathcal A)$ $\kappa$ be a sub-Markov kernel on $(\Omega,\mathcal A)$ $p\ge1$ I'll denote the composition of $\...
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0answers
36 views

If $(X_t)_{t\ge0}$ is a Markov process with invariant measure $\mu$, does the distribution of $X_t$ weakly converge to $\mu$ as $t\to\infty$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $(X_t)_{t\ge0}$ be a time-homogeneous Markov process on $(\Omega,\mathcal A,\operatorname P)$ and $\kappa_t$ denote a regular ...
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34 views

Integrability in Dynkin's formula

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $W$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $b,\sigma:\mathbb R\to\mathbb R$ be Borel measurable with $$...
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1answer
36 views

How to decrease number of noises in stochastic differential equation

I have a Quantum SDE containing both white and color noises (open quantum system). $$ \dot\rho(t) = A\rho_s + (\nu_{1t}\hat{c}^\dagger \hat{X}^-_1 + \omega_{1t} \hat{X}^+_1 \hat{c})\rho_s +(\nu_{2t} \...
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32 views

If $(X^x_t)$ is the stochastic flow generated by a SDE and $(X_t)$ is the strong solution with $X_0=ξ$, is $X_t=X^ξ_t$ for all $t$ a.s.?

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

Existence of a regular version of the conditional distribution in a specific application

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ $(X_t^x)_{(t,\:x)\in[0,\:\infty)\times\mathbb R}$ be a continuous ...
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1answer
59 views

Maximal inequality for the Itō integral

Let $W$ be a Brownian motion and $X$ be a predictable process with $$\operatorname E\left[\int_0^t|X_s|^2\:{\rm d}s\right]<\infty\;\;\;\text{for all }t\ge0.$$ Now, let $p\ge2$. How can we show ...
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50 views

Is the strong solution of a SDE adapted to the filtration generated by the driving Brownian motion?

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$ $\xi$ be an $\...
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24 views

If we want $\int_0^⋅XdM$ to be a martingale, do we need to assume $E[\int_0^tX_sd[M]_s]<∞$ for all $t$ or even $E[\int_0^∞X_sd[M]_s]<∞$?

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

Find constants $a$ and $b$ such that $X(t)$ is a Brownian Motion

Let $B(t)$ be a Brownian Motion. Find all constants $A$ and $b$ such that $X(t)=\int_0^t(a+b\frac{u}{t})dB(u)$ is also a Brownian Motion. First we know that if $f \in L^2[a,b]$ then $\int_a^bfdB(u)$...
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1answer
48 views

Invariant measure for Itō diffusion

Let $f\in C^2(\mathbb R)$ be positive and $h\ge 0$. Assume that $g:=f'/f$ is Lipschitz continuous and let $U$ be a strong solution of $${\rm d}U_t=\frac h2g(U_t){\rm d}t+\sqrt h{\rm d}W_t$$ ($W$ being ...
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0answers
26 views

Markov property of continuous time stochastic processes

Suppose $X_t$ is a continuous time stochastic process and $B_t$ is a Brownian motion. Suppose there is a deterministic function $f:\mathbb{R}\rightarrow \mathbb{R}$. What conditions that $f$ needs to ...
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0answers
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Stability of parameters in sde

From lecture notes on SDE's. Consider the Stratonovich equation $dX_t=rX_tdt+\sigma X_t\circ dB_t$. It has initial condition $X_0=x$. What are the conditions for the parameters $(\sigma,r)$, for $\...
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White noise is not a signed measure for fixed $\omega$

I'm going over some materials on stochastic analysis, and stuck with a problem on Gaussian white noise: Let $(\mathbb{R}^d,\mathcal{B},m)$ be the Borel measurable space on $\mathbb{R}^d$. A white ...
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How has Stochastic calculus changed the way we think and/or design systems?

I am an outsider to the field of Stochastic calculus, and I was reading about the historical developments in the field: from Louis Bachelier, Einstein, to Ito, and other pioneers in the field. I see ...
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conditional expectation of variance process

Suppose that each asset-i satisfies the sde: $dS_t^i=S_t\sigma_t^i\sqrt{v^i}dB_t^i$, $i=1,2$ where, $u^i=\exp{\int_{0}^{t} c^ie^{-k(t-s)}dW_s^i }$, $i=1,2$, $\sigma^i$ deterministic, $c^i\in \mathbb{...
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0answers
83 views

Reference request in optimal stopping

I am given the following task. Distributed over a trading day, I am supposed to buy a certain quantity of a good. The price of this good changes during the day. The goal is to buy the required ...
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1answer
21 views

Why is $P^b(B_{T-T_b} \in (-\infty,b))=1/2$

Why is $P^b(B_{T-T_b} \in (-\infty,b))=1/2$ on the set $\{T_b<t\} $ where $T_b=\inf\{t \ge 0 :B_t=b\}$ and $T=t 1_{\{T_b<t\}}+\infty 1_{\{T_b \ge 0\}}$. I am trying to understand Proposition 2....
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The property of coercivity in stochastic analysis

Given an SDE $$ dX_{t}=b(t,X_{t})dt+\sigma (t,X_{t}) dW_{t} $$ With $W$ the Wiener process. I have seen some results that under some assumptions on the coefficients $b,\sigma$ such as : i) ...
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1answer
23 views

Why is $T:=\inf\{t\geq0:B_t\leq at^p-b\}$ a stopping time?

How can I prove that $T:=\inf\{t\geq0:B_t\leq at^p-b\}$ is a stopping time w.r.t. a natural filtration of $B$, where $B$ is a $BM$, $p>1/2$ and $a,b>0$? I can introduce a new random process, $...
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0answers
34 views

If $Y$ is a Markov chain and $h>0$, why is $(Y_{\lfloor t/h\rfloor})_{t\ge0}$ not a Markov process?

Let $\left(Y^{(n)}_k\right)_{k\in\mathbb N_0}$ be a Markov chain for $n\in\mathbb N$, $(h_n)_{n\in\mathbb N}\subseteq(0,\infty)$ with $h_n\xrightarrow{n\to\infty}\infty$ and $$X^{(n)}_t:=Y^{(n)}_{\...
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0answers
37 views

A construction of a Stratonovich type integral for fractional Brownian motion

I'm studying this article https://projecteuclid.org/download/pdf_1/euclid.twjm/1500574954 and I'm having problems understanding the proof of lemma 3. Let me recall some of the criminals involved. ...
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0answers
31 views

Random time change for a Poisson process and convergence with respect to the Skorohod topology

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\left(Y^{(n)}_k\right)_{k\in\mathbb N_0}$ be a time-homogeneous Markov chain on $(\Omega,\mathcal A,\operatorname P)$ and $$X^{(n)}_t=...
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0answers
24 views

Interpolation of martingale

Consider a non-constant martingale $(M_n)_{n\ge 0}$ with the filtration $(\mathcal{F}_n)_{n\ge 0}$, where $\mathcal{F}_n=\sigma(M_m:m\le n)$. Define for $t\in\mathbb{R}_{\ge 0}$: $$X_t:=M_{\lfloor ...
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0answers
44 views

Covariance of two Ito / Diffusion processes

Let $B_t$ denote the standard Brownian motion process. $X_t$ and $Y_t$ are Ito diffusions with the following SDEs: \begin{align} dX_t &= \mu(t,X_t) \; dt + \sigma(t,X_t) \; dB_t \\ dY_t &= \mu(...
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1answer
22 views

Left continuous adapted process X is optional

I was reading through "Limit Theorems for Stochastic Processes" by Jacod and Shiryaev and came across 1.24 Proposition: Every process $(X_t)_{t \geq 0}$ that is left continuous and adapted is ...
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1answer
17 views

Why $P^\mu[X_{S+t} \in \Gamma \mid \mathcal{F}_{S+}]=P^\mu[X_{S+t} \in \Gamma \mid X_{S}]=0 \text{ ,} P^\mu\text{-a.s. on } \{S=\infty\}$

For any progressively measurable process $X$ and any optional time $S$ of $\{\mathcal{F}_t\}$ why do we have that $$P^\mu[X_{S+t} \in \Gamma \mid \mathcal{F}_{S+}]=P^\mu[X_{S+t} \in \Gamma \mid X_{S}]=...
2
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0answers
110 views

Convergence of discrete-time Markov chain to Feller processes

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(X_t)_{t\ge0}$ be a Feller process on $(\Omega,\mathcal A,\operatorname P)$ $(h_d)_{d\in\mathbb N}\subseteq(0,\infty)$ with $$h_d\...
1
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1answer
34 views

Conditional expectation of exponent to the power of BM, representation theorem

Let $T > 0$ and $M(t) = E[e^{W(T)}|F(t)]$ where $\{F(t) : t \geq 0\}$ is a natural filtration generated by W and $t \leq T$ I need to show that M(t) is a martingale and I also need to find a unique ...
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1answer
40 views

A problem on equivalent definitions of Markov property

Suppose that $X, (\Omega,\mathcal{F}),\{P^x\}_{x \in \mathbb{R}^d}$ is a Markov family with shift operators $\{\theta_s\}_{s \ge 0}$ and for every $x \in \mathbb{R}^d,s \ge 0, G \in \mathcal{F}_s$ ...