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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Density of invariant measure of stochastic differential equation

I have a question: is it possible that an SDE has a "nice" density, but its invariant measure does not have a "nice" density? More precisely, consider a stochastic differential ...
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Example of function having finite quadratic variation except Brownian motion [duplicate]

I am looking for an example of function having finite quadratic variation except Brownian motion. As a I know that for Brownian motion B(t) have finite quadratic variation and quadratic variation of B=...
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26 views

What does the term $dN_t/N_t$ mean in a Stochastic differential equation?

The following is an excerpt from page 62, chapter 5 in Oksendal's textbook on Stochastic Differential Equations: $$dN_t = r N_t dt + \alpha N_t dB_t$$ or, $$ \dfrac{dN_t}{N_t} = r dt + \alpha dB_t$$ ...
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25 views

Ito integral, interchange limit and expectation

The Ito integral with respect to a standard Brownian motion is defined by $$ I_t = \int^t_0 g_s \,dW_s = \lim_{n \to \infty} \sum^{n-1}_{k=0} g_{t_k} (W_{t_{k+1}} - W_{t_k}), $$ where $g_t$ is a ...
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46 views

Norm in the space of square integrable martingales [closed]

I was learning stochastic integration and encountered two different norms used in the space of square integrable martingales They are as follows: 1.Let M be a square integrable martingale, then $|M|_t$...
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48 views

Random level reaching of Wiener process

The question: $W$ denotes a Wiener process in filtration $\mathcal{F}$, where $X$ is an $\mathcal{F}_{0}$-measurable a, Exponential b, Cauchy distributed random variable. Let us define $$\tau=\inf\...
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51 views

Only positive or only negative

How can I calculate the following probability where $W$ denotes a standard Wiener process? $$\mathbb{P}\left(\left|W_{t}\right|>0:\forall t\in\left[1,2\right]\right)=?$$ So what is the probability, ...
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43 views

Gaussian distribution of integral with respect to brownian motion [closed]

Let $f$ be a continuous function and $B_t$ be a standard Brownian motion. Define $$X_t=\int_0^t f(s) dB(s)$$ I want to how that $X_t$ follows a normal distribution with variance $\langle X,X\rangle_t$...
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29 views

Discrete stochastic process to stochastic differential equation

I was studying about stochastic processes and got stuck at a question. Let's say, I have a stochastic process like the following, $X_i(t+1)=\epsilon(X_i + X_j)$ $X_j(t+1)=(1-\epsilon)(X_i + X_j)$ ...
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Optimization facing uncertainty: Average (of Optimal) vs Optimal (over distribution)

I face a lot of problems in the following format: "Make a choice $a$ that optimizes $r(a)$ over states of the world $x$, where states are distributed $F(x)$." In some cases, the solution is: ...
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1answer
40 views

Expectation of a local martingale and uniform integrability

Let $N_t$ be a local martingale and $\tau_n$ be the localizing sequence. As per the book I am following then we have the following I dont understand how uniform integrability will make the ...
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38 views

Solution of general linear SDE system

Introduction I have a multidimensional, inhomogeneous SDE system: $$ d\underline{X}(t) = \left[\underline{\underline{A}}(t) \cdot \underline{X}(t) + \underline{a}(t) \right] \cdot dt + \sum_{m = 1}^{M}...
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Harmonical behaviour of solution of nonlinear stochastic differential equation

There is a stochastic differential equation what i want to solve $$dX = \left[ \left[ A(t) - k \cdot \left(X-X_{0} \right) \right] \cdot X + a(t) \right] \cdot dt + \sum_{m = 1}^{M} \left[B_m(t) \...
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34 views

Discrete-time dynamics can be modelled by the stochastic differential equation

I'm reading the following paper, in which the stochastic gradient descent consists of performing partial computations of the form $$x^{k+1}=x^k-\eta_k\nabla f_{i_k}(x^{k-1})\qquad\mbox{ where }\eta_k&...
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22 views

Differential of a a function of a semimartingale [closed]

Let $X_t$ be a continuous semimartingale and $U_t$=$X_t-1/2[X,X]_t$ where [] is the quadratic variation.Is $$dU_t=dX_t-d[X,X]_t/2?$$ If yes how can I prove this from the Ito's Lemma.In particular what ...
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41 views

Cumulative Geometric Brownian Motion

If I accumulate the realisations of a Geometric Brownian Motion, with drift=$\mu$, volatility=$\sigma$ and initial value $x_0$, what will the resulting process be? Some mathematical justifications and/...
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weak convergence of probabilities

I'm getting confused with the following: Let $\mu, \mu_n$ probabilities on $\mathbb{R}$. Suppose that $\mu_n \to \mu$ weakly. Then Can I say that $\mu_n - \mu \to 0$ weakly? I think no: in fact, let $...
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Moment of Ito diffusion computationally

Say, we have an SDE $$ \mathrm d X_t = f(X_t) \mathrm d t + \sigma(X_t) \mathrm d W_t $$ where $W_t$ is a Wiener process. Assuming a strong solution exists globally (so the 1st and 2nd moments should ...
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79 views

How should $\int_0^1 |dX_s|$ be understood for a real valued semimartingale $(X_t)_{t \geq 0}$ of finite variation?

How should $\int_0^1 |dX_s|$ be understood for a real valued semimartingale $(X_t)_{t \geq 0}$ of finite variation? I read this in many sources but I can not find any explanation of this term. I know ...
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22 views

Calculating the covariance involving the Ornstein Uhlenbeck process.

Let $$dX_t=AX_tdt+\sigma dB_t$$ be a $d$-dimensional Ornstein-Uhlenbeck Process with the stationary measure $\pi=N(0,I_d)$. Suppose that $X_0 \sim \pi$. Let $f:\mathbb R^d \to \mathbb R$ be a ...
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41 views

Covariance of stochastic process

I need to calculate the expected value $\mathbb{E}[\eta(s) \cdot \eta( \tilde{s} )]$, where $\eta$ is an Ornstein-Uhlenbeck process defined by the stochastic differential equation \begin{equation*} \...
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30 views

Convolution of random variables - Bernoulli and Binomial

For n ∈ N, p ∈ (0, 1) let X ∼ Binom(n, p) and Y ∼ Ber(p) be two independent random variables. a) Determine the values P(X + Y = k) for k ∈ N. What is the distribution of X + Y ? Does this intuitively ...
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Geometric Poisson process

$\lambda >0$ $ \sigma > 1$ $n_t$ si a Poisson process $S_t = S_0 \exp( N_t \log( \sigma +1) - \lambda \sigma t ) = S_t e^{- \lambda \sigma t } ( \sigma +1 )^{N_t} $ $ M_t= N_t - \lambda t$ is ...
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1answer
57 views

solution to heat equation

We have the follwing PDE: $$ \frac{\partial W}{\partial \tau} = {1 \over 2}\sigma^{2}\, \frac{\partial^{2} W}{\partial^{2}x}\quad \mbox{where}\quad x = \epsilon + \left(r - {1 \over 2}\sigma^{2}\...
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1answer
43 views

$e^{X_t}$ is martingale using Ito's Formula

$B_t$ is the Brownian process. How to prove that $ Y_t=e^{X_t}$ is a martingale using Itô's formula? Here we have, for $f$ deterministic $$X_t = \int_0^t f(s)dB_s-\frac{1}{2}\int_0^t f^2(s)ds $$ and $...
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1answer
67 views

Is there a definition of an extension of a probability space?

I am reading some notes where the extension of a probability space is used but there is no definition given and I can not find one on the internet. Is there a clear definition of what an extension of ...
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Langevin equation and convergence to stationary solutions. Free energy. SDE. FPE.

Let $f\geq 0$ be Lipschtiz. The overdamped Langevin equation \begin{equation}\label{eq overdamped Langevin SDE} dX=-\nabla f(X)dt+\sqrt{2} dW_t \end{equation} with Kolmogorov forward equation \...
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Variance of SDE (Ornstein Uhlenbeck Process)

I am trying to calculate the variance of h, $Var(h_t)$ of the 2D System of SDEs. $\eta$ describes an Ornstein-Uhlenbeck process. The variables $\tau , \tau_{\eta}, \sigma$ are constants. I tried to ...
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Componentwise stationary distribution implies full stationary distribution?

Consider a $d$-dimensional stochastic process $X_t$ having density $\rho(X_t,t)$ at time $t$. Suppose the drift term $b$ of the process depends on the density $\rho = \rho(X_t,t)$: $$ dX_t = b(X_t,\...
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22 views

Correction: Calculating distribution function and determining density function of $Y =2X$

Since we were not provided any solutions to our statistics exercises, I wanted to ask you guys for any corrections or errors I did on this exercise. I will only upload a screenshot from my notes app. ...
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Girsanov: Will the new Brownian motion generate the original filtration?

Setup of the question: Assume you have a Brownian motion $B_t$ on a probability space $(\Omega,\mathcal{F},P)$. Let $\mathcal{F}_t$ be the natural filtration of the Brownian motion, and let $\bar{\...
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1answer
25 views

Is there any closed form solution for the following SDE?

How can I solve the following SDE: \begin{cases} dX_{t}=-\sin X_{t}\cdot\cos^{3}X_{t}dt+\cos^{2}X_{t}dB_{t}\\ X_{0}=x_{0} \end{cases}? This SDE has the form of $$dX_{t}=\frac{1}{2}\sigma'\left(X_{t}\...
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56 views

Calculating the distribution functions from two random variables X and Y

I'm currently trying to catch up on Stochastics for university and I'm really stuck on this one although I feel like its not as difficult as I may think: Let X, Y be random variables. The random ...
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1answer
60 views

Prove that Ito integral is large with large probability

Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion and let $\sigma\colon [0;T] \to \mathbb R$ be deterministic and square-integrable. For some constant $A>0$, I want to bound the ...
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42 views

Solution to PDE

We have the following PDE: $\frac{\partial W}{\partial \tau}=0.5\sigma^2\frac{\partial^2 W}{\partial x^2}$ We are going to look at a special solution of the from: $W(\tau,x)=\tau^af((x-c)/\tau^b)$ ...
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1answer
35 views

PDEs - change of variable question

We have the following PDE $$\frac{\partial U}{\partial \tau}=0.5\sigma^2\frac{\partial^2 U}{\partial \epsilon^2}+(r-0.5\sigma^2)\frac{\partial U}{\partial \epsilon}$$ where U is a function of $\tau$ ...
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1answer
63 views

Is this process optional?

I have a rightcontinuous adapted $\bar{\mathbb{R}}$-valued stochastic process $(X_t)_{t\geq 0}$ with $X_s=\infty$ implies $X_t=\infty$ for $t>s \geq 0$. Since $\bar{\mathbb{R}}$ is homeomorph to $[-...
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1answer
50 views

Application of Doob's maximal inequality

Consider a sequence of predictable processes $H_n$ converging to a predictable process $H$, each of which is bounded. Consider $$E \left[ \sup_{t \in [0,T]} \left| \int_0^t (H_n(s)-H(s)) dM(s) \right|...
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Change of measure in multivariate Itô diffusion processes

Let $X_t$ and $Y_t$ be $d$-dimensional Itô diffusion processes that solve following SDEs, $\mathrm{d}X_t = \alpha X_t \mathrm{d}t + \Sigma \mathrm{d}B_t\,$ where $\,B_t$ is a standard brownian motion,...
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1answer
33 views

Definition of continuous semi-martingale

According to William's Diffusion Markov process X is a continuous semi-martingale if \begin{align*} X=X_0+M+A \end{align*} where $M$ is a continuous local martingale null at 0, and $A$ is a ...
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19 views

transition probability density of Follmer process

If Follmer diffusion process $\{X_t\}_{t \in [0,1]}$ satisfies the following SDE $$dX_t= dB_t+\nabla \log P_{1-t}f(X_t)dt, \quad X_0=0$$ where $f=\frac{d\mu}{d\gamma}, \gamma(dx)=(2\pi)^{-\frac{n}{2}}...
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1answer
31 views

Assumption in Itô-Tanaka formula

The Itô-Tanaka formula usually require that $f$ is the difference of two convex functions. But I do see this condition is used in the proof. I think the proof only needs the fact that $f$ is convex ...
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1answer
25 views

Expected value of vector with density $\frac{3}{2}x_1^2x_2 + \frac{1}{2}(x_2 - \frac{1}{2}) - \frac{3}{4} x_1^2 $

Let $X=(X_1, X_2)$ be a two-dimensional continuous random vector with the following density: $$f(x_1,x_2) = \begin{cases}\frac{3}{2}x_1^2x_2 + \frac{1}{2}(x_2 - \frac{1}{2}) - \frac{3}{4} x_1^2 & \...
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1answer
91 views

Apply Girsanov theorem to a Ornstein-Uhlenbeck process

From the original paper by Schwartz (free access here, page 926): Assume that the commodity spot price follows the stochastic process$^5$ $$\tag1 dS = \kappa(\mu-\log S)Sdt+\sigma SdW, $$ where $...
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90 views

The meaning of $d\langle Z(t,x),Z(t,y)\rangle =c(x,y)\,dt$ where $Z$ is a random field

This question is influenced by this paper. Consider a random field $Z(t,x)$ with a correlation structure given by $$d\langle Z(t,x),Z(t,y)\rangle =c(x,y)\,dt.$$ For example, a choice for the ...
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44 views

Generalization of lambda-prescription for stochastic integral and change of variable

Using the change of variable i have found the Ito's integration formula in terms of the ordinary integrals: $$ \int^t_{t_0} h'(x(\tau)) g(x(\tau),\tau)dB(\tau)=h(x(t))-h(x(t_0))- \int^t_{t_0} \left[h'(...
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1answer
50 views

How to prove $W_t-tW_1$ is Markov?

For a Brownian motion $W_t$, how do we prove the bridge process $W_t-tW_1$ is Markov? Essentially, we need to prove for $s< t$, \begin{align} \mathbb{P}(W_t-tW_1\in x\mid \mathcal{F}_s)=\mathbb{P}(...
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1answer
92 views

Why does Ito Chain Rule make no sense?

$$dY\Big(X(t)\Big) = \bigg(\frac{d}{dt}Y + \frac{d}{dX}Y \cdot \mu(t) + \frac{d^{2}}{dX^{2}}Y \cdot \frac{1}{2}\sigma^{2}(t)\bigg) \cdot dt + \Big(\frac{d}{dX}Y \cdot \sigma(t)\Big) \cdot dW$$ ...
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59 views

Infinitesimal generator in context of SDEs

Consider the following SDE: $$X(t)= s + \int_{t_0}^t b(X(s)) dt + \int_{t_0}^t a(X(s)) dW(t) $$ where $X(t_0)=s$ and $a:\mathbb{R}^d \rightarrow \mathbb{R}^{d \times N} $,$b:\mathbb{R}^d \rightarrow \...
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82 views

Proving the set of stochastic integrable functions is dense in $\mathscr{L}_2$

I am not understanding a proof of a lemma in the following text Chung,Williams,2014, "Introduction to Stochastic Integration". Settting Consider $\mathscr{L}^2(\mathbb{R}^+\times\Omega,P,\...

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