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

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Given ${\rm d}X_t=v_t(X_t){\rm d}t+\text{white noise}$, obtain a SDE for $v$ in a Hilbert space $H\subseteq L^2$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $U,H\subseteq L^2(\Omega,\mathbb R^d)$ be separable $\mathbb R$-Hilbert spaces Let $t\mapsto X_t\in\mathbb R^d$ be the trajectory of a ...
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How can we solve $\frac{{\rm d}^2u}{{\rm d}t^2}(t)=-c^2\lambda u(t)+\varepsilon\sqrt{\lambda}\frac{{\rm d}B}{{\rm d}t}(t)$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ How can we solve $$\frac{{\rm d}^2u}{{\rm d}t^2}(...
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1answer
53 views

Can we show $\int_0^tf(s){\rm d}B_s=-\int_0^tf'(s)B_s{\rm d}s$ for $f\in C^1(\mathbb R)$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ and $f\in C^1(\mathbb R_{\ge 0})$. Can we show that $$\...
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1answer
20 views

Probabilistic Method/Model for Traffic Flow

Context: Given a network system or a traffic system with some value related to the system. Question: Which probabilistic methods, model, distributions are used frequently to predict a event (for ...
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12 views

Stochastic wave equation

Let $D:=(0,a)$ for some $a>0$ $H:=L^2(D)$, $$e_n(x):=\sqrt{\frac 2a}\sin\frac{n\pi x}a\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ and $$\lambda_n:=\left(\frac{n\pi}a\right)^2\;\;\;\text{...
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1answer
20 views

Expected value of geometric Brownian motion

So everyone knows that the expected value of GBM is given by: $X_0 \exp(\mu t)$ My question is that what does this say about such stochastic processes? Since $X_0$ and $\mu$ are within "my control" ...
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convergence in p-th moment of Euler-Maruyama scheme

I have an SDE with global Lipschitz coefficients on $[0,T]$, i.e. $$ dv = \mu(v)dt + \sigma(v)dW $$ with $|\mu(u)-\mu(v)| \vee |\sigma(u)-\sigma(v)| \leq L |u-v|$ and $W$ being standard 1-d Brownian ...
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16 views

Is there a connection between generalized ODEs and stochastic ODEs

I'm working on a problem where I've run into a generalized ODE $$ \dot X \in D(X) $$ where $D(X)$ is a continuous, compact and convex subset of $\mathbf{R}^n$. To me, this problem seems in many ways ...
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Proof about solution to SDE with Lipschitz condition:

$X_t$ follows the Ito process as described by the following stochastic differential equation $$dX_t=b(X_t)dt+dB_t\quad , \quad X_0=x$$ and $b(X)$ satisfies Lipschitz Coditions.I want to show for ...
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1answer
23 views

Square Integrable Martingales and the Unit Process

Let $X_t$ be a continuous square-integrable martingale. Then it is true (I think, please correct me if I am wrong) that: $$\forall t \in [0,\infty), \quad \int_0^t 1_{[0,t]}(s) dX_s = X_t - X_0$$ So ...
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1answer
33 views

Non standard stochastic integral

I don't know how to deal with the following stochastic integral: $\int_0^t \frac{1}{\sqrt{t-s+1}} d W(s)$ As you can see, the variable $t$ appears both as an endpoint of the interval of integration ...
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23 views

When is a stochastic integral a martingale?

In what follows, let the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ as well as the chosen filtration $(\mathcal{F}_t)_{t \ge 0}$ be known, and let $f$ denote an arbitrary locally bounded ...
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How to compute integral of exponential martingale with respect to Poisson process?

Let $N=\{N_t:t\in\mathbb R_+\}$ be a homogeneous Poisson process with intensity $\alpha$ and $M_t=N_t-\alpha t$ the compensated process. I'd like to show that $N$ is not a natural process, i.e. that ...
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Are martingales progressively measurable? (Application to square integrable martingales)

This is an incredibly dumb question, but I'm not sure if I know the correct answer, and it doesn't seem to be stated anywhere on the internet, so here goes: Are martingales progressively ...
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18 views

Feller boundary conditions

The classification of boundary behavior for a time-homogeneous diffusion satisfying an Ito stochastic differential equation (SDE) is well known. According to the Feller classification, there are four ...
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23 views

Integrated Brownian Bridge is a Gaussian Process

Let $W(t),t \in [0,1]$ be a (Standard) Wiener Process. The Brownian Bridge $B(t), t \in [0,1]$ can be constructed via $B(t):=W(t) - t \cdot W(1)$ and is a Gaussian process with zero mean and ...
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1answer
26 views

Brownian noise perturbing a differential equation

The following one-degree-of-freedom oscillator is given; $$\ddot{x}+kx=w(t),$$ where, $k>0$ and $w(.)$ is a Brownian noise perturbing the system. Assume we want to study boundedness of the ...
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36 views

Computation of an Ito Integral [closed]

I have the following Ito Integral $$\begin{align}\int_{0}^{t}B_se^{-\sigma B_S}dB_s&& \end{align} $$ Here, $\sigma \gt 0$. Can someone please show me how to compute this Ito integral? ...
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1answer
27 views

Proving an SDE has a unique strong solution

I have the stochastic differential equation $$dX_t = \ln(1+ X_t^2) \, dt + X_t \, dB_t$$ In this equation, $X_0 = x$, and $x \in\mathbb R$. How can we show that this equation has a unique strong ...
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Bound moments of SDE

I have a rather messy SDE with solution $x(t)$ and I would like to bound its moments. Now using Ito I obtained approximately the following form: $$ \frac{d}{dt} \mathbb E |x(t)|^k \leq -C \mathbb E |...
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1answer
76 views

Solving an Itô Integral

Can someone please show me how to solve this Itô Integral? $$\begin{align}\int_{1}^{t}\frac{dB_s}{B_s^2 + B_s^4} && \end{align} $$
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1answer
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Finding the limit $\lim_{t\to ∞} \mathbb{E}[R_t]$ of an SDE

I have the SDE $$dR_t = (1 - \beta R_t)dt + \sigma dB_t$$ In this equation, $R_0 = r$ in which $r > 0$ Can someone please help me find the $\lim_{t\to ∞} \mathbb{E}[R_t]$? Thus far I have ...
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1answer
63 views

Differentiability of a simple value function driven by a diffusion

Consider a diffusion given by, $d X_t = \mu(X_t) dt + \sigma(X_t) dB_t$ $X_0 = x$. Suppose the functions $\mu$ and $\sigma$ are as follows - $f(x) = \mu(x) = \sigma(x) = \begin{cases} 2 & \...
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1answer
42 views

Computing the expectation value of a stochastic process

I have a stochastic differential equation for which I have solved the process X$_t$. The SDE is as follows: $$ dX_t = \left( r\mu X_t + \frac{r(r-1)} 2 \sigma^2 X_t \right) \, dt + r\sigma X_t\,dB_t, ...
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Why is a Brownian Motion completly specified by its increments?

Without any formal base frame I want to know why a Brownian Motion is completly specified by its increments? Even though I think here is no need to give a Reference you can see the following. ...
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1answer
35 views

How to deduce the expectation of a stochastic equation [closed]

I am having a difficult time deducing the expectation, $\mathbb{E}[R_t]$, of the following stochastic equation: $$dR_t = (1 - \beta R_t)dt + \sigma dB_t$$ $R_0 = r$, with $r > 0$. Please help me ...
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Finding a unique strong solution

I am brushing up on my stochastic approximation. I am having a hard time with the following problem. I have the equation dX$_t$ = ln(1+ X$_t^2$)dt + X$_t$dB$_t$ X$_0$ = x, with x ∈ ℝ I know that ...
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1answer
31 views

Finding Stochastic processes

I have the following differential equation dX$_t$ = (r$\mu$X$_t$ + $\frac{r(r-1)}{2}σ^2X_t$)dt + rσX$_t$dB$_t$, X$_0$ = x, with x > 0. Here, r>0. I am having trouble figuring out how to find the ...
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1answer
35 views

The hitting time $T-\tau^{l}$ has the same distribution as $\min\{\tau^{f},T\}$ regarding an Poisson Process.

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq T}$ is a Filtration, with $T < \infty$. On that prbability space we want to ...
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1answer
30 views

Difference between stationarity and independence properties for Brownian motion

What is the difference between the stationarity and independence properties of proving that a stochastic process $W(t)$ is Brownian motion? I only understand that for stationarity, we're trying to ...
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Why are we allowed to multiply the differential form of an SDE by a function?

Suppose for example that we have the following SDE: $$dX_{t} = a(X_{t})\,dt + b(X_{t}) \,dB_{t}. $$ What rigorous justification is there for then saying, for example, that we can multiply both sides ...
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1answer
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Notation in the stochastic derivatives in the mean square sense

The stochastic limit $X$ in the mean square sense is given the definition: For a row (sequence?) of stochastic variables $X_n$ if $\displaystyle\lim_{n\to\infty}E\{(X_n-X)^2\}$ = 0 and we write $\...
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1answer
48 views

Probability of an Ornstein-Uhlenbeck process

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration, with $\tau < \infty$. The following definition is ...
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Calculate mean of SDE

This might be a stupid question but I'm going to ask it anyway because I can't find a way to do it. I'm trying to find the expectation (and -- if possible -- higher moments) of the solution of the ...
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Applying Ito formula to Ito process

I would like to simplify the expression $\left(\phi(s_{1})\cdot(X_{s_{1}}-X_{s_{2}})+\phi(s_{2})\cdot(X_{s_{2}}-X_{s_{3}})+\ldots+\phi(s_{n-1})\cdot(X_{s_{n-1}}-X_{s_{n}})\right)^{2}$ where $X_{t}$ ...
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2answers
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Nonuniqueness of Stochastic Differential Equation

Let $B_t$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)\ne 0$ are real valued continuous functions where $|\mu(t,x)|+|\sigma(t,x)|$ is NOT Lipschitz continuous, and $$dX_t = \mu(t,X(t)...
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If $(W_t)_{t\ge 0}$ is a $L^2(D)$-valued Wiener process, then $W_t(x)$ is normally distributed

Let $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $D\subseteq\mathbb R^d$ be a domain $U:=L^2(D)$ and $\langle\;\cdot\;,\;\cdot\;\rangle:=\langle\;\cdot\;,\;\cdot\;\rangle_U$...
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Return probability of a SRW in an even number of steps

I am looking for some references for the following problem. Consider a graph $G$ and a simple continuous time random walk $(X_t)_{t\geqslant 0}$ on this graph. Consider the family of events $(e_t)...
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122 views

Calculation of $\ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right)$ where $S$ are stocks

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration of an incomplete finance market with stocks $S_j(t)$ for $...
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Doob Meyer decomposition in an exercize

I have to find the Doob Meyer decomposition for the following process: $Y_t=e^{(1+B_t^2)}$ I think that the method is to derive with the Ito's formula the process and I've obtained: $dY_t=2B_te^{(...
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1answer
40 views

Integration with respect to a Poisson random measure

Let $N$ be a Poisson random measure (PRM) on a Polish space, $\left(X,\mathcal{B}(X)\right)$, and let $\tilde{\nu}$ be its mean measure. Then, let $f$ be any non negative and bounded function on $X$. ...
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1answer
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For a one-dimensional Brownian motion $B_t$ $Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}.$

A one-dimensional Brownian motion $B_t$ has exponential moments of all orders, i.e. $$Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}. (2.6)$$ This is given as a corollary to ...
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Time-changed Brownian Motion

Let $B_t$ be a standard Brownian motion and let $\tau_{-1}= \inf \{ t \geq 0: B_t(\omega) = -1\}$. By the Continuous Time Stopping Theorem we know that \begin{align} Z_t = B_{t \wedge \tau_{-1}} \...
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1answer
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Sharpen Doob's Maximal Inequality

Let $B_t$ be a Brownian motion, $B_T^* = \sup_{0\leq t \leq T} B_t$ and $\lambda > 0$. Applying Doob's maximal inequality gives: \begin{align} P(B_T^* \geq \lambda)\leq \frac{\mathbb{E}[B_T^p]}{\...
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+100

How can we prove that $\langle\int_0^t\Phi_s{\rm d}W_s,x\rangle_H=\sum_{n\in\mathbb N}\int_0^t\langle\sqrt{λ_n}\Phi_se_n,x\rangle_H{\rm d}B_s^{(n)}$?

Let$^1$ $U$ and $H$ be separable $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U,H)$ be nonnegative and symmetric operator on $U$ with finite trace $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $...
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Can Local Martingales be characterized only using their FV process and BM?

Prove or Disprove: A process $(X_t)_{t \ge 0}$ is a (continuous) local martingale if and only if it can be represented in the form: $$\int_0^t \xi dB = \large B_{\int_0^t \xi_s^2 ds} $$ where the ...
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1answer
33 views

Thinning a Renewal Process - Poisson Generalization

If we have a Poisson point process with rate $\lambda$ and we keep each of its point with probability $p$, we obtain another Poisson point process with rate $\lambda p$. Does this result holds for a ...
2
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0answers
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Uniqueness of the trajectories of the solution of an SDE

Consider an SDE \begin{equation} dX_t=f(X_t,t)dt+b(X_t,t)dW_t \end{equation} Suppose firstly that the coefficient are Lipschitz continuous. So by the theorem of existence and unicity I have that exist ...
2
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0answers
51 views

I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
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1answer
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Vasicek equation [closed]

Vasicek interest rate stochastic differential equation is $$dR(t)=(\alpha-\beta R(t))dt+\sigma dW(t)$$ where $\alpha , \beta$ are positive constants. I need to use Ito-Doeblin formula to compute $...