Questions tagged [stochastic-calculus]

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.

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Spectral process for the Ornstein-Uhlenbeck process

The Ornstein-Uhlenbeck process $X(t)$ is a centered, Gaussian process with covariance function $$B(s,t) = e^{-\vert t-s \vert /2}$$ The spectral measure is abs. cont. w.r.t. the Lebesgue measure ...
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Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$ $M:\Omega\... 0answers 317 views Stochastic Leibniz Rule I have come up with the following Leibniz stochastic rule and I want to check that: The result is correct; The proof is right. Statement: let$f(\cdot,t):s \rightarrow f(s,t)$,$s \in \mathbb{R}^+, ... 0answers 186 views Can any stochastic differential equation be mapped onto a (generalized) Langevin equation? There might be different definitions of what a generalized Langevin equation is, but let us consider the following expression: \dot{x}_i = \frac{dx_i}{dt} = f_i(\mathbf{x}) + \sum\limits_{m=1}^{n} ... 0answers 116 views Solution to Singular Free Boundary PDE As part of my research, I have come across the following problem and I am trying to tackle it. Let (X_t)_{0 \leq t \leq T} be a mean controlled Brownian Motion with the following dynamics \begin{... 0answers 168 views Stochastic integral is a bounded operator? The Setup Let \xi_t be a process adapted to the filtration \mathfrak{F_t} of the semi-martinagale X_t, such that both are square integrable. Then is the map \begin{align} F_T: L^2(\Omega\times[... 0answers 347 views How to Prove the Stochastic Fubini Theorem? (Exercise 2.19 in Chapter IV of Revuz and Yor) Here is the theorem statement: Let B and C be two independent standard Brownian motions. If \phi is square integrable on the unit square (\phi \in L^2([0,1]^2) ), by suitable filtrations, ... 0answers 598 views Why is a predictable stochastic process called *predictable*? Let (\Omega,\mathcal A,\operatorname P) be a probability space I be an index set \mathbb F=(\mathcal F)_{t\in I} be a filtration on (\Omega,\mathcal A) X=(X_t)_{t\in I} be a stochastic ... 0answers 727 views Using Girsanov theorem to prove density of stopping time Let B be a standard Brownian motion and for a>0 and b>0, and set\sigma_{a,b} = \inf\{t\,:\, B_t + bt = a\}.$$There are at least two ways to solve the following problem (the other one ... 0answers 223 views Regarding proof of converse to Girsanovs theorem This is regarding an argument from Arbitrage Theory by Thomas Björk - Theorem 11.6, but is attempted self contained. Consider a Wiener process W on probability space (\Omega,\mathcal{F},\{\mathcal{F}... 0answers 598 views Determine if this is a Martingale I am trying to check if the process S_t is a martingale, where \mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t, S_0 = 1. We know that S_t is a local martingale because if we stop it ... 0answers 359 views Brownian motion integral Let (B_t) be a standard Brownian motion, f a continuous function and X_t = \int_0^t f(s)B_s ds. I was able to prove that (X_t) is a Gaussian process with zero mean and trying to find the ... 0answers 109 views Inverse powers of Bessel process not a martingale I am trying to show that X_t^{1-2a} is not a martingale, where the solution X_t to$$dX_t = \frac{a}{X_t} \ dt + \ dB_t, \ \ X_0=1$$for a>1/2. I am able to do this directly for a = (d-1)/2,... 0answers 95 views Are SDE's really “differential”? An SDE of the form$$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dB_t$$is really short-hand notation for an equation involving Ito integrals:$$X_t = X_0 + \int_0^t \mu(X_s,s)\,ds + \int_0^t \sigma(X_s,s)\,... 0answers 197 views The stopped running maximum of a Brownian motion LetW=(W_t:t\ge0)$denote a standard Brownian motion, and let$W^*_t:=\sup_{0\le s\le t}W_s$be its running maximum. For constant$\lambda>0$define the stopping time $$\tau:=\inf\{t\ge0:W_t\le ... 0answers 105 views What is the solution to SDEs of the form dx=f(x)(dt+\rho dW_t)? I've been thinking about the class of SDEs which have the form:$$dx(t)=f(x)(dt+\rho dW_t)$$Clearly when \rho=0 this is just an ODE:$$\frac{dx}{dt}=f(x) $$My question is, how does the solution to ... 0answers 84 views Reference request on the stochastic heat equation. I was watching a lecture of fields medalist Martin Hairer where he briefly describes a construction of a discrete model where the stochastic heat equation arises. (the youtube link starts with the ... 0answers 77 views Expected value of max of a Stochastic process Ciao all, I'm working on some stochastic processes and I'm stuck on this problem. Let S_t be the stochastic process defined by:$$ dS_t = \sigma S_t dW_t $$with initial data S_0 \in \mathbb{R}^+... 0answers 113 views If W is a standard Brownian motion, does W(1) take every real number? Let (\Omega,\mathcal{F},P) be a probability space. Let W:[0,\infty)\times \Omega\rightarrow\mathbb{R} be a standard Brownian motion. Is it true that for all x\in\mathbb{R} there exists \omega\... 0answers 236 views Are martingales progressively measurable? (Application to square integrable martingales) This is an incredibly dumb question, but I'm not sure if I know the correct answer, and it doesn't seem to be stated anywhere on the internet, so here goes: Are martingales progressively measurable?... 0answers 119 views Finding a unique strong solution I am brushing up on my stochastic approximation. I am having a hard time with the following problem. I have the equation dX_t = ln(1+ X_t^2)dt + X_tdB_t X_0 = x, with x ∈ ℝ I know that ... 0answers 209 views Radon-Nikodym on a Process wrt to filtration Given a probability space (\Omega,\mathcal{F},P). Let (X_t)_{t\geq0} be a stochastic process defined on it with cadlag paths, lets say on (\mathcal{X},\mathcal{B}(X)). Let be \mathcal{F}_{t} ... 0answers 312 views stochastic exponential uniformly integrable martingale N is a continuous local martingale and T_c:=\inf\left\{t>0:\left[N\right]_t>c\right\}, c>0 . I need to show that the stochastic exponential \mathcal{E}(-N) is a uniformly integrable ... 0answers 454 views Brownian motion on sphere proof? proving the brownian motion on the sphere equation the stratonovich form differential equation$$\partial X=n(X)\times \partial B$$the equation in ito's form becomes$$dX=n(X)\times dB+H(X)n(X)... 0answers 156 views Finding the mean of$X_t = \int_0^t sW_sdW_s$For the stochastic integral, where$W_t$is a Wiener process, I am trying to find the mean of$X_t = \int_0^t sW_sdW_s$. I have read before that any stochastic integral with$dWt$has mean zero, but I ... 0answers 533 views conversion from stratonovich SDE to Ito's form? conversion of stratonovich SDE to Ito SDE (Where$\partial$is differential in the stratonovich form and$d$is in ito's form): $$\partial X_t=\sigma(X_t,t)\partial B_t+b(t,X_t)\partial t$$. ... 0answers 80 views Characterization of point process, given the number of points For a point process with independent and identically distributed (i.i.d) inter-renewals, with distribution$p(x)$, we observed$N$points on$[0,T]$. What is the joint probability distribution ... 0answers 61 views Stochastic Integral of Particle Scattering I have a stochastic process that describes a particle moving through a field of randomly distributed particles and undergoing scattering collisions (modeled simplistically) off of them. In its ... 0answers 169 views Numerical method for SDEs I'm using a 4th order Adams predictor-corrector method to numerically solve a regular differential equation. Now I would be interested to be able to include a noisy term to the equation -as in the ... 0answers 335 views Is$X_t = tW\left(\frac{1}{t}\right)$a Martingale？If not, how could it be a Brownian Motion? As is proved,$X_t = tW\left(\frac{1}{t}\right)$is a Brownian motion. For example see Theorem 4.2 in this paper http://math.uchicago.edu/~may/REU2012/REUPapers/Leiner.pdf I'm just confused because ... 0answers 113 views Confusion about localization I am a bit confused about the following result I read: Let$(\Omega,\mathfrak{A},\mathfrak{F},\mathbb{P})$be a filtered probability space. Let$\left\{X_t\right\}_{t\in[0,T]}$be a continuous and ... 0answers 162 views Fundamental theorem for Malliavin derivative and Lebesgue integral I am interested in some kind of fundamental theorem of calculus for the Malliavin derivative: My notations are mainly taken from the Book Nualart: The Malliavin Calculus and Related Topics. Let$u\in\...
Take $H\in S$ to be a simple process defined as: $$H_t:=\sum_{i=1}^{n-1} H_i1_{(T_i,T_{i+1}]}(t),\ \ H_i\in \mathcal{F}_{T_i}, \ (T_1\leq...\leq T_n \ stopping\ times),$$ and $X$ a Martingale. I ...