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Questions tagged [stochastic-calculus]

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What is the volatility surface of a parameter that depends on a Stochastic process?

Hi this question takes a few steps to setup so please bear with me :) Short version: Does a parameter that depends on the output of a stochastic model inherently have variance? If so why? Long ...
70 views

affine function of random variable

If $X$ is a random variable, then $F(X) = a + bX$ where $a,b$ are constants is an affine function. What is the technical/formal name of a function which is not affine, but for which the following ...
44 views

Ornstein-Uhlenbeck process

Given a multivariate Ornstein-Uhlenbeck process that is a stochastic process, is it correct that each component of this process is a univariate Ornstein-Uhlenbeck process?
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Expectation of $dX_t$ for $X_t$ being an Ito process

Let $X_t$ be an Ito Process: $$dX_t = f(t, X_t)dt + g(t, X_t)dW_t$$ What is $E_t[dX_t]$? How can we compute it and importantly what is the intuitive explanation of $E_t[dX_t]$?
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Continuous-time Kalman filter with no observation/measurement noise

The continuous-time (linear) state space model can be written \begin{align*} \text{d}\mathbf{x}_t &= \mathbf{F} \,\mathbf{x}_t \, \text{d}t + \mathbf{G} \,\text{d} \boldsymbol{\...
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Is an ITO diffusion time slice always Normally distributed?

As the title says, if we take a time slice on any Ito diffusion - are we guaranteed that the data is always Normally distributed? This seems like a useful property for computer generalization and ...
56 views

An intuitive meaning of Stochastic Differential Equation

I'm trying to approach for the first time Ito's calculus and SDE, maybe this is a trivial question. If the following is a generic SDE: $dx = \mu(x)dt + \sigma(x)dB_t$ Can i consider the $dx$ as a ...
26 views

Is this short rate really constant?

Suppose that a financial instrument has a constant short-term rate $r$ and its price $S$ is driven by the equation $$S_t = \mu_t S_t \, {\rm d}t,$$ where $(\mu_t$) is a process adapted to the ...
28 views

Stochastic Integral - Standard Brownian Motion

I was just running through practice exams, we didn't really cover this in class so any guidance (reference etc) would be appreciated. (I'm struggling to use the integral with MathJax so I'm pasting it ...
52 views

Soft Question: What background do I need to understand Feynmann Kac Formulae by Pierre Del Moral?

I am attempting to understand Sequential Monte Carlo(SMC) deeply, but with little theoretical background on probability theory and stochastic processes. Usually, the 'statistics' perspective of markov ...
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Conditional Expectation for Geometric Brownian Motion

Given a geometric Brownian motion: $\frac{dZ}{Z} = \mu dt + \sigma dW$ Is there a closed-form solution to $\mathbb{E}[z_s | (z_s > a)\cap(z_t > b)]$ for $t \geq s$?
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Basic Stochastic Calculus: Integration of wiener proces

How do one compute the following integral: $$\int_{a}^{b}dw_u$$when $W_t$ is a wiener process. My initially guess will be $W_b-W_a$, but I cannot argue for that because I am not sure wether I can use "...
132 views

Question on variance and expectation of Brownian Motion related things

In a mathematical finance text by Ubbo F Wiersema, I came across the following Say $\Delta t$ is very small. $\Delta B(t)$ denotes $\textit{brownian motion increment}$. Then $E[\Delta t\Delta B(t)]=0$...
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How Much And What Kind of Math for Deep and Reinforcement Learning?

I have a lot of books on measure theoretic probability theory, functional analysis, graduate level topology, convex optimization, stochastic calculus, numerical analysis using a functional analysis ...
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Distribution of a slice of a line with normally distributed slope

Give the random variable $X \sim N(\mu,\sigma)$ consider the process $Y_t = tX$ with $t \in I = [t_1,t_2]$, informally a line with a random slope. I would like to find the distribution of the points ...
135 views

How to get a uniformly distributed portfolio allocation vector? [duplicate]

Imagine we have a portfolio allocation vector $(x_1, ..., x_n)$ with $x_1+...+x_n=1$; Also we assume that the vector has only elements $\geq 0$, so we have $|x_1|+...+|x_n|=1$, so no short selling; ...
94 views

Request for Multi-dimensional simulation reference book

Do you have any reference book about multi-dimensional simulation? The random variables are not identically distributed and are correlated. I understand the procedure to generate correlated normal ...
168 views

How to calculate the covariance of a stochastic integral and a Riemann integral?

Right now, I want to figure out the covariance of a stochastic integral and a Riemann integral in the following form: \mathbb{E}\left(\int_{0}^{t}\exp[B(t)-B(s)]ds \cdot \int_{0}^{t}\exp[B(t)-B(s)]...
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conditional distribution of $X_t$ in a jump-diffusion model

I'm working with a variant of the Ornstein Uhlenbeck model that includes Lévy-type jumps, (i.e. a jump-diffusion model with mean reversion). I want the distribution of $X_t$ (given the parameters of ...
62 views

The pdf of last interval of a Poisson process in $[0,T]$

Assume a Poisson point process with rate $\lambda$ in time $[0,T]$. Supoose $X$ is the random variable representing the time between the last arrival and $T$. What is the probability density function ...
207 views

Realize reducible nonstationary kernels as solution to SDEs and its extensions

I am interested in a regression application where my kernel is of the form \begin{equation} k(t,t^{\prime}) = k_s\left(\phi(t),\phi(t^{\prime})\right)= k_s\left(\phi(t)-\phi(t^{\prime})\right), \...
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Book recommendations for probability

I am looking for a book (English only) that I can treat as a reference text (more colloquially as a bible) about probability and is as complete - with respect to an undergraduate/graduate education in ...
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Wanting to learn Dynamic Programming for stochastic optimal control, I need help getting started

I have an optimal stopping and control problem for which the dynamic programming equation is written. I am totally new to this field and type of problem but I have bases in Stochastic Calculus and ...
646 views

Stochastic Differential Equations - A Few General Questions

I just have a few questions about stochastic differential equations. I generally did a lot of pure math but signed up for a course on probability models and stochastic differential equations because I ...
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Distribution of stochastic integral

I would like to find the distributions of the following random variables: $Z_k= \frac{1}{\pi} \int^{2\pi}_{0} cos(kt) dW_t$ $k=1,2,...$ and $(W_t)_{t\geq 0}$ is a Wiener process. What is the ...