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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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Ansatz for Merton's Problem

I've been reading about the Merton Portfolio problem and how you can use the HJB equation to solve it where $0 = (∂_t + rx∂_x) h(t, x) − \frac{λ^2}{2} \frac{(∂_xh(t, x))^2}{∂_{xx}h(t, x)}$ with ...
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0answers
11 views

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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35 views

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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1answer
27 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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1answer
19 views

Continuous mapping theorem, multivariate case, joint distribution.

I came across the following problem. Convergence in the following always means weak convergence, i.e. $X_n \rightarrow X$ if and only if $Ef(X_n) \rightarrow Ef(X)$ for all $f$ bounded, continuous ...
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21 views

Given a Markov process, are we able to construct another Markov process with the same transition semigroup but different inital law?

Let $E$ be a locally compact separable metric space $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroups on $C_0(E)$ with generator $(\mathcal D(A),A)$ $(\Omega,\mathcal A,\operatorname P)...
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1answer
31 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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1answer
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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10 views

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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34 views

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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13 views

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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12 views

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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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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0answers
31 views

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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18 views

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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1answer
26 views

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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0answers
16 views

Stochastic Paths

I am stuck in generating stochastic paths to the differential equation \begin{equation*} \begin{split} \frac{dx}{dt}&=(b-d)x\\ x(0)&=x_0 \end{split} \end{equation*} Whose stochastic ...
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1answer
21 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
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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0answers
5 views

Intuition behind the COGARCH modelling

I am working on a project and I seriously need help. I have come a across the conversion of a discrete time GARCH(1,1) model to a continuous state model. The continuous state model was given as; $$\...
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0answers
31 views

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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0answers
24 views

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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1answer
25 views

Proof for identity involving joint probability and conditional probability. [closed]

How do you prove the following identity? $$\mathbb{P}(X \in A, Y \in B) = \int_B \mathbb{P}(X \in A| Y = y)\mathbb{P}_Y(dz)$$ Additionally, what assumptions on $X$, $Y$, $A$ and $B$ are needed?
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1answer
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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0answers
41 views

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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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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18 views

Where is the problem in this logic regarding the supremum of the expectation being equal to the expectation of the supremum using control processes

This is just a question I have from stochastic control. I am totally new to analysis/stochastic processes, so I am unsure how controls are used properly/manipulated in expressions, but here's my ...
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1answer
35 views

Do there exist diffusions that do not solve any SDE?

Diffusions are continuous time stochastic processess having continuous paths and satisfying the strong Markov property. I know it is possible to characterize some diffusion processes as solutions to ...
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1answer
45 views

How to judge the solution process of an SDE to lie on the sphere?

Consider the following SDE on $\mathbf R^d$: \begin{equation}\tag{*} dX_t^i = -\frac{d-1}{2}X_t^i dt + \sum_{j=1}^d(\delta^{ij}-X_t^iX_t^j)dW_t^j, \quad i=1,2,...,d, \end{equation} where $W = (W^1,W^2,...
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1answer
30 views

Time homogeneity of Ito diffusion

Consider a time homogeneous Ito diffusion satisfying a SDE, \begin{equation}\label{1} dX_t=b(X_t)dt+\sigma(X_t)dB_t, X_s=x \end{equation} $t\geq s$. The unique solution of the SDE is denoted by $...
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0answers
26 views

Implementation of Brownian local time process

I am trying to replicate results of Grigoriu's Solution Of Boundary Value Problem By Monte Carlo Simulation. Essentially, we are looking for the local solution of a PDE and for this, we take samples ...
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1answer
48 views

Does a strong solution to a SDE imply lipschitz condition?

Consider $dX_t=b(X_t,t)dt+\sigma(X_t,t)dB_t$. I know that, $|b(x,t)-b(y,t)|+|\sigma(x,t)-\sigma(y,t)|\leq D|x-y|$ for some constant D implies the existence and uniqueness of a strong solution. ...
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1answer
37 views

Solving system of stochastic DE

I am trying to solve the following system of SDEs $$ \left[ \begin{array}{c}{d X_{1}} \\ {d X_{2}}\end{array}\right]=\left[ \begin{array}{l}{1} \\ {0}\end{array}\right] d t+\left[ \begin{array}{cc}{1} ...
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0answers
48 views

Computing infinitesimal generator for a jump process with continuous component

I am trying to show that the infinitesimal generator of the following process $$ dS_t = (\alpha S_{t-}+\beta )dt + (\gamma S_{t-}+\delta)dX_t,$$ where $X_t$ is a $(\lambda,G)$-compound Poisson ...
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31 views

(Proof) If $M$, $N$ are two orthogonal local martingales, and $S$, $T$ stopping times, then the stopped local martingales $M^S$, $N^T$ are orthogonal.

I am currently reading the first chapter on the general theory of stochastic processes in "Limit Theorems for Stochastic Processes" by Jacod and Shiryaev. On page 41 they state the following: 4.13 ...
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1answer
72 views

Solving SDE using Itô's lemma

I'm trying to solve stochastic differential equation: $$d X(t)=X(t) d t+d W(t)$$ Here, W(t) denotes a Brownian Motion. Comparing with Itô's lemma (with $f\left(W_{t}, t\right) = X(t)$), $$ d X(t)=\...
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0answers
22 views

What does non-degeneracy of of the diffusion coefficient in the context of a SDE mean?

In the introduction of a paper I was reading the author writes without elaborating that " In the case of a non-degenerate diffusion coefficient, Stroock and Varadhan , proved the existence of a ...
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31 views

Backwards Euler issues in Stochastic Differential Equations

I recently read a paper which I think has an incorrect numerical simulation. I say 'think' because the issue was centered about solving a stochastic ordinary differential equation (SDE) and I have ...
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2answers
79 views

Definition of a family of probability measures for Ito diffusions

I have a question concerning the definition of a family of probability measures for the solutions to an Ito diffusion $$X_t^x = x + \int_0^tb(X_s^x)ds + \int_0^t \sigma(X_s^x)dB_s$$ as it's given in ...
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3answers
75 views

How do I evaluate the following combination of random variables? Is it martingale?

I'm about to analyse the following expression $$Z_n:=\prod_{k=1}^n \left(\frac{\frac{Y_k}{\prod_{i=1}^k X_i}}{\sum_{j=1}^k \frac{Y_j}{\prod_{i=1}^j X_i}} \right),$$ where $Y_j$ for all $j\in \mathbb{...
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1answer
73 views

Is this random Lebesgue-integral well-defined?

Let $$ X : [0,T] \times \Omega \rightarrow \mathbb{R} $$ be an almost-surely continuous stochastic process. Then how is the random Lebesgue-integral $$ \omega \mapsto \int_{0}^{T} X_t(\omega ) dt \...
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0answers
28 views

Multidimensional Correlated Geometric Brownian Motion, finding exact form of the matrices

My goal is to understand the dimensions of the matrices involved, so I am initially writing things as column vectors, and defining all the dimensions. I am working with the following setup: ...
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0answers
7 views

The formula of CCI (Commodity Channel Index) indicator in stock market and the mathematical proof of it.

I am looking for the formula of CCI oscillator and the mathematical proof. If you can advise me some references, I am really thankful! Thank you so much!
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36 views

How to solve $\mathop{dX_{t}} = \alpha X_{t} \mathop{dt} + \beta \mathop{dW_{t}}$ holds, where $X_{0} = x?$ [duplicate]

I'm learning about stochastic processes, and I want to solve $$\mathop{dX_{t}} = \alpha X_{t} \mathop{dt} + \beta \mathop{dW_{t}}$$ where $X_{0} = x$. I think that the solution uses Ito's Lemma; ...
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0answers
27 views

Joint density of a system of SDEs coupled through noise

I have a system of two (one-dimensional, Ito) stochastic differential equations, one describing the evolution of $X_t$ and the other the evolution of $Y_t$. The two SDEs are coupled through the noise,...
2
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1answer
40 views

Independent Increments of Time Integral of Brownian Motion

I am wondering if $\int_0^tW(s)ds$ is independent of $\int_t^TW(s)ds$, where $W$ is a standard brownian motion/wiener process, and for $0 \leq t \leq T$ Writing them as limits of Lebesgue Integrals, ...
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0answers
14 views

For a continuous Gaussian process starting at the origin, can the covariance be bounded by left endpoint?

Consider a continuous real valued Gaussian process $X_t:[0,T]\to \mathbb{R}$ with covariance $R(s,t)$. We know that $X_0=0$ a.s. We want to consider $[a,b]\subset [0,T]$ with $a\gt 0$. We know by ...
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0answers
17 views

Does being $C^\alpha$ Holder for $\alpha>1/2$ imply positively correlated increments?

Consider a centered Gaussian process $X_t$ with covariance $R(s,t)=E(X_s X_t)$. It is known that if $X_t$ is fractional Brownian motion with $H>1/2$ then $X_t$ is has positively correlated ...
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0answers
25 views

Prove bounded set implies $L^2$ integrable function?

On a complete and filtered probability space, $(\Omega, \mathcal{F}, \mathcal{P})$, I would like to show that a function $f(t, \omega_1, \omega_2): \mathbb{R} \times \mathbb{R}^2 \to A \in \mathbb{R}$ ...
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1answer
26 views

Estimation of integral of stochastic process(Krylov estimation)

Let $X_n$ be a sequence of Ito diffusions $$dX_n(t)=b_n(t) \, dt+\sigma_n(t) \, dW(t), \qquad 0\leq t\leq T$$ with $b_n$ uniformly bounded and $\sigma_n$ uniformly elliptic. Then Krylov's estimation ...