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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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Uniqueness of a solution for SDE

I'm trying to show the uniqueness of the following SDE $dX_t=\mu\left(t,X_t\right)X_{t}\ dt+\sigma\left(t,X_t\right)X_{t}\ dB_t$ where $B$ is a Brownian motion and $\mu, \sigma : \left[0,T\right] \...
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Existence of a weak solution to this SDE?

I am looking at an SDE of the form $d{X_t} = \left( {{1_A}({X_t}) - {1_{{A^c}}}({X_t})} \right)d{W_t}$ such that ${X_0} = 0$, $A \in \mathcal{B}(\mathbb{R})$ and ${A^c}$ has a lebesgue measure of zero....
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integration by parts formula

Below is from Liptser, Shiryaev "Theory of martingales", page 200: I have a question: How from eq. 3.5 and 3.8 they got the eq. 3.9? Since $G$ is of finite variation $\mathcal{E}^{-1}(G)$ is of ...
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Infinitesimal generator of diffusion process in one dimension

I consider a diffusion process $$ dX_t = b(X_t) dt + \sigma(X_t) dB_t. $$ From the general theory, we know that if $f\in C^2(\mathbb{R})$ and has a compact support, then the infinitesimal generator ...
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26 views

Decomposition of a cadlag and of local bounded variation function $Z:\mathbb{R}_+\rightarrow\mathbb{R}$

Let $Z:\mathbb{R}_+\rightarrow\mathbb{R}$ be cadlag and of local bounded variation with $Z(0)=0$ and $V_Z(t)$ denotes the value of the total variation of $Z$ on $[0,t]$ for all $t\in\mathbb{R}_+$. I ...
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30 views

Variance of a Laplace Transform

I have a function $F(s)$ which is the Laplace transform of $f(t)$ (which is in itself a normally distributed random process), but I don't know what $f(t)$ is (this comes from solving a differential ...
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29 views

Quadratic Variation and Brownian Motion

Let $(X_n,F_n)$ be a martingale with $X_n \in L^2(\Omega,F,\mathbb{P})$. The quadratic Variation $(<X>_n)_n$ of the process $(X_n)_n$ is defined as $$ <X>_n := \sum\nolimits_{i=1}^{n}(\...
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71 views

How to I use Ito's formula to compute quadratic variation?

Let B be Brownian motion. Use Itos formula to compute the quadratic variation of $\left[X_t^i\right]$ for $\left[X_t^1\right]=e^{B_t}$, $\left[X_t^2\right]=\ln(B_t^2+1)$ and $\left[X_t^1\right]=\sin^...
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42 views

Expected value of general diffusion

lets take measurable $b,\sigma:\mathbb{R}^+\times \mathbb{R}\to\mathbb{R}$ and consider the SDE $$dX_t=b(t,X_t)dt+\sigma(t,X_t)dB_t$$ with $X_0=x$. How can i use Itô's Lemma to show $$E_x[X_t-x]=t\...
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Proof that the integral of an adapted process has a zero quadratic variation

Take an adapted process to a given non tricky filtration and dependent on two "variables", omega (event, not really a variable) and $t$ (time). By definition adaptation means only measurability with ...
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Expectation of the product of Brownian processes (higher powers)

I have recently sat an exam that had elements of stochastic calculus, but I am now feeling like I might have gone wrong in some questions of it like the following. I am trying to evaluate $\mathbb{E}(...
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Double Wiener-Ito Integral of a Seperable Deterministic Function in $L^2([a,b]^2)$

Suppose $f,g \in L^2[a,b]$. Show that the double Wiener-Ito integral of the function $f(t)g(s)$ is given by $$ I_2(fg) := \int_a^b\int_a^bf(t)g(s)dB(t)dB(s)=I(f)I(g)-\int_a^bf(s)g(s)ds $$ where $B_t$ ...
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29 views

Discrete Exponential Martingale - Properties

This question is about the discrete exponential martingale. Let $(Y_n)_n$ be a sequence of independent and identically distributed random variables with $m_{Y}(t) :=\mathbb{E}\left[e^{t Y_{1}}\right]&...
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Martingality of a Doleans-Dade exponential local martingale.

A paper I read recently seems to make the following statement: if $\gamma_t$ is a progressively measurable process, and that $\exp\left(\int_0^T\gamma_s dW_s\right) \in L^p$ for some $p>1$, then ...
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Continuity in mean of a stochastic process

If $X$ is a stochastic process, a.s. continuos and such that $\forall t \geq 0, X_t \in L^1_\omega$, is its mean function $t \rightarrow E[X_t]$ continuos? I can show it if $X \in L^1_\omega L^{\...
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Proof: the $\max{(B_t,0)}$ is submartingale without using convex function with Jensen's inequality

Proof: the $\max(B_t,0)$ is submartingale without using convex function with Jensen's inequality To prove it using convex function and jensen's inequality, we know the max function is convex and it ...
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Is a vector of independent Brownian motions a multivariate Brownian motion?

Given a filtered probability space $(\Omega, \mathcal{F}, \mathcal{F}_{t\geq 0}, P)$: If $B_1, B_2, \dots, B_m $ are all real $\mathcal{F}_t$ Brownian motions, jointly independent. Is the resulting ...
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19 views

Continuous function as difference of convex functions

Can every continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ be written as the difference of two convex functions? If not, can every twice continuously differentiable function $f:\mathbb{R} \...
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Motivation of Milstein scheme

Milstein scheme is motivated by improving the convergence speed sigma terms of Euler scheme. where sigma and mu is globally Lipschitz continuous$$t_i=\frac{T}{n}i\quad dX_t=\mu(X_t)dt+\sigma(X_t)dW_t$$...
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Meaning of $\int _0 ^T X_t dt$ when $(X_t)_t$ is a process

I am studying stochastic calculus (Ito integrals, to be precise) , and I am not sure if I got some things right. For instance, we have defined $\Lambda_B ^2 (a,b)$ as the space of progressively ...
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M/M/1 queue with two types of customers, distribution of the total number of customers

I have problems to derive the following: "A gas station offers two services. For each service customers arrive according to a Poisson process. On average 20 customers per hour for service 1 and 5 ...
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Mean and Variance in the Jacobi Stochastic Volatility model

I would like to compute $ E[X_{T}]$ and $Var[X_{T}]$ in the Jacobi model, where the Dynamics are given as \begin{align} dY_{t}&=\kappa(\theta-Y_{t})dt+\sigma\sqrt{Q(Y_{t})}dW_{1t}\\ dX_{t}&...
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30 views

Stopped uniformly integrable process in discrete time is uniformly integrable?

I'm studying for work the book: "Stochastic Calculus and Application" by Choen and Elliot 2 ed. In section $4.2$ (pg. 91) it states the discrete version of the Optional Stopping Theorem for bounded ...
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Why does calculating the quadratic variation of a Brownian motion in this way not work?

This seems like a simple question, but I am stumped. I know the proofs for quadratic variation and cross variation, etc..., but for some reason can't understand why the following doesn't make sense to ...
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22 views

stochastically independence

I never took stochastic courses and need a proof for this task to continue my work at another problem. Can somebody help me out? Let $X$ and $Y$ be stochastically independent real discrete random ...
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39 views

Fokker-Planck equation

I'm struggling to proof the Fokker-Planck equation. Let $b:[0,T]\times \mathbb{R}^N\to\mathbb{R}^N$ and $\sigma:[0,T]\times \mathbb{R}^N\to\mathbb{R}^{N\times d}$ two measurable functions. Let $X=\...
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How can I show that $E[\sup_{0 \le t \le T} N_t^2]<\infty$ for all $T>0$

Given a Stochastic differential equation $dN_t=\sqrt{2\mu N}dW_t$ starting with a deterministic initial value $N_O$. How can I show that $E[\sup_{0 \le t \le T} N_t^2]<\infty$ for all $T>0$? I ...
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Using Girsanov Theorem Backwards?/ Obtaining Radon-Nikodym Derivative

On page 112/133 of Den Hollanders book on Large Deviations he wants to calculate the R.N derivative between two path measures : one is the path measure of the solution to an SDE $dX_t=H(X_t)dt+dW_t$ ...
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Process of sum of flip coin is not uniformly integrable?

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $\left(Z_{n}\right)_{n \in \mathbb{N}}$ be a sequence of i.i.d. random variables with $\mathbb{P}\left(Z_{n}=1\right)=\mathbb{P}\left(...
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1answer
30 views

Simulation of SDE

I need to write down a code to simulate an SDE of the following type: $$ dX_t = - ( \eta_t X_t + \chi_t ) dt + \sigma dW_t, \ X_0=v_0 \quad t \in [0,T] $$ where $\eta_t$ is a deterministic function, ...
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1answer
20 views

Equality of the stochastic integral under two probability measures

This questions is very short. Under the Girsanov Theorem assumptions we have two equivalent probability measures $\mathbb P$ and $\mathbb Q$ and a measurable space $(\Omega,\mathcal F)$, right? We ...
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1answer
43 views

Stochastic Question: $d \int B_s ds = ?$ [closed]

Stochastic Question: $d \int_0^t B_s ds = ?$ $B_s$ is the standard Brownian motion at time $s$. This is an Ito integral. Operator $d$ is defined in the standard Ito sense. For those who understands ...
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Find the compenstor of the standard ito integral

Let $B_t$ be a Brownian motion and let $\{\mathcal F_t : a<t<b\}$ be a filtration such that for each $t$ we have that $B_t$ is $\mathcal F_t$ measurable and for and $s<t$, the random variable ...
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1answer
39 views

Differential/derivative of time integral of a stochastic process, where the stochastic process depends on upper limit

For a standard Wiener Process/Brownian Motion, $W$, for the usual integrals $\int_0^t\sigma(u)dW(u)$ and $\int_0^tW(u)du$, I know how to manipulate them using Ito's Lemma/normal calculus rues like ...
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44 views

Gaussianity of a stochastic process

I am given the process $X_t = B_t -\int_0^t \frac{B_u}{u}du$ How can I show that it is gaussian, given a standard continuos Brownian motion $B$? As I know that $sB_{1/s} \rightarrow 0$ as $ s \...
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Bootstrap-Resampling: Why are the Bootstrap-Resamples independent again?

For the simple Bootstrap method, we consider n i.i.d. random variables with common distribution function F, which is estimated by ^F (parametric or non parametric). For the procedure we then generate ...
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1answer
49 views

Cox-Ingersoll-Ross Model

CIR model foresee (on the basis of structure of similar model) the following system: $\left\{\begin{matrix} \dot A(t,T)-a \gamma B(t,T)=0, A(T,T)=0\\ \dot B(t,T)-aB(t,T)-\frac{\sigma^2}{2}(B(t,T)...
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Interpolating process that follows geometric Brownian motion

So I have a set of time series data that follows (at least assumed) GBM but there are missing data. To interpolate, I'm thinking about simulation and create some sample paths. I have two options in ...
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1answer
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Deriving an equation and boundary condition from an SDE

Let $\dot{X_{t}} = b(X_{t}) + \sigma(X_{t})\dot{W_{t}}$ be a stochastic differential equation, where $W_{t}$ is a Wiener process. Also, let $X_{0} = x \in \mathbb{R}$. Define $$u(x) = \mathbb{...
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If $X$, $Y$ are continuous semimartingales and we have that $XdY = YdX$ can I conclude that $X = Y$?

If not true in general, are there any (mild) conditions on $X$ and $Y$ under which this is true? Sorry if this has already been asked!
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Hitting time expectation squared for Brownian motion

Let $\dot{X_{t}} = b(X_{t}) + \sigma(X_{t})\dot{W_{t}}$ where $X_{0} = x \in \mathbb{R}$ and $W_{t}$ is a Wiener process. Let $\tau = \min\{t \mid X_{t} \not \in G\}$, where $G = (M, N) \subset \...
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1answer
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Why the convergence of $\sum_{k=0}^{n-1}B_{t_{i}}(B_{t_{i+1}}-B_{t_i})$ depend on the partition pointwise but not in $L^2$?

Let $(B_t)$ a Brownian motion. Let $0=t_0<t_1<...<t_n=T$ a partition on $[0,T]$. I know that $$\lim_{n\to \infty }\sum_{k=0}^{n-1}B_{t_{i}}(B_{t_{i+1}}-B_{t_i})$$ doesn't depend on the ...
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Uniqueness of Solution to Stochastic Integral Equation

Suppose that $N$ is an $(\mathcal{F}_{t})$-continuous local martingale, with $N_{0}=1$, $N_{t}\gt0$ a.s. for $t\geq0$ and $N$ satisfies: $$ N_{t}=1+\lambda\int_{0}^{t} N_{s}dB_{s} $$ Applying Ito's ...
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GBM within brownian bridge

I noticed that a brownian bridge between say a and b amy start at B0 and B1 where B1 can be greater than B0. I wonder if this path can be a GBM. If so, what’s the SDE like?
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60 views

Ito's Differential Problem

Let $W_t$ be the standard Brownian motion. Is the random process a martingale? - $Y_t = exp(\int_0^t sdW_s)$ (Find $dY_t$ using Ito formula in its differential form) Base on what I have learned we ...
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scaling invariance of brownian local time

I am studying Brownian local time processes and several references mentioned the scaling invariance of local time. For example, page 10 of this reference (https://hal.archives-ouvertes.fr/hal-00091335/...
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Continuous Modification of Stochastic Process Indexed with Compact Space

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a (complete) probability space, let $D$ be a compact, separable, metrizable topological space and let $$ S : \Omega \times D \rightarrow \mathbb{R}$$ be a $(\...
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1answer
44 views

Finite sequence of random step processes such that $\lim_{n\to\infty}E(\int_{0}^{\infty}|f(t)-f_n(t)|^2dt)=0$ for $f(t)=e^{-t^2/4}$

Let $$f(t)=e^{-t^2/4}, \ \ \ t \ge 0$$ I want to show that $f$ is in $M^2$ where $M^2$ denotes the class of stochastic processes $f(t),t\ge0$ such that $$E\left(\int_0^\infty|f(t)|^2dt\right)&...
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37 views

Density function related to Brownian motion

I am dealing with a question listed below. I am trying to use the running maximum of Brownian motion to deal with the problem, but it does not work out. Let $ \tau_{M}=\inf\{t;W(t)=M\},M>0,$ and ...
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50 views

Using Ito's lemma to determine $dY(t)$ when $Y=\sin(t+B_t), \ \ \ t\ge0$

Let $(\Omega, \mathcal F, P)$ be a probability space and $\{B_t\}_{t\ge0}$ a Brownian motion. Furthermore let $\{F_t\}_{t\ge0}$ be the natural filtration of $B$. Let $$Y(t)=\sin(t+B_t), \ \ \ t\ge0$$ ...