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Questions tagged [stochastic-calculus]

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Wiener process definition as Gaussian summation

In this lectures Wiener process is defined by summing white Gaussian random variables and then limit them when sample time go to zero. $$ {\bf{w}}(t) = \int_0^t {{\bf{\tilde q}}(\tau )} d\tau = \...
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Practicing Understanding Stochastic Differential Equations using R

Is there a book or set of notes that I can use to practice differential equations using R-Studio or Python. I don't want a solver, I want a way of visualizing them and understanding their properties. ...
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43 views

Variance of a linear (and/or nonlinear) combination of an integral function

Let \begin{align} I(t) = K \int_{-T/2}^{T/2} e^{i 2\pi f_X \left(t + X(t) \right)} \cos\,(2\pi f_Y (t + Y(t)) - \phi))\,dt, \end{align} where $X, Y$ are independent random variables which are the ...
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42 views

Understanding policy gradient theorem - What does it mean to take gradients of reward wrt policy parameters?

I am looking for a little clarity on what the policy gradient theorem means. My confusion lies in the fact that the reward $R$ in reinforcement learning is non-differentiable in the policy parameters. ...
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28 views

What are the prerequisites for stochastic calculus?

I am considering learning stochastic calculus myself, but do not have math background. Could you please suggest a list of books which will help to understand stochastic calculus?
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How to prove that $X_t=\int^t_0f(u)dW_u$ and $X_t-X_s$ are independent?

Let $X_t=\int^t_0f(u)dW_u$ for a deterministic function $f$ and $W_t$ is a brownian motion. How can I compute $E[\exp(\lambda_1 X_s + \lambda_2(X_t-X_s))]$ and prove that $X_s$ and $X_t-X_s$ are ...
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42 views

Stochastic Differential equation: CAPM

Let $R = (R_1, \dots , R_M)'$ denote a vector of excess returns of $M$ assets observed at $n$ time points, $0 < t_1 < t_2 < \cdots < t_n < T$, within a time span $T > 0$. We wish ...
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Change in function value between correlated variables

Suppose i have dependent random variables $X_1,X_2,\ldots,X_n$ and a function $z = f(X_1,X_2,\ldots,X_n)$ and suppose i want to perturb one of the features $X_1$ by $\Delta X_1$ (i'm happy to use a ...
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184 views

How to solve / fit a geometric brownian motion process in Python?

For example, the below code simulates Geometric Brownian Motion (GBM) process, which satisfies the following stochastic differential equation: The code is a condensed version of the code in this ...
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15 views

Derivative of Time-Transformed Stochastic Process

Given a continuous time stochastic process X(t), we can define the functional transformation, $$f(X)(t)=(X(t))^2−2X(t)$$ and evaluate the Hadamard derivative. Given a transformation on the real ...
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stochastic ordering of counting processes

Let $N_1(t)$ be a delayed renewal process and $N_2(t)$ be an ordinary renewal process such that $N_1(t)\geq_{st}N_2(t)$. Consider a renewal process $Z(t)$ with the same inter-arrivall distribution as $...
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When can a Gaussian Process solve an SDE?

Considering an SDE of the form: $$dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t$$ ... (where $W_t$ is a Weiner process) is there a set of necessary and sufficient conditions on the structure of the ...
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21 views

First moments of GBM-like process with non-normal shocks

First consider a standard GBM process of the form, $$\frac{dS_t}{S_t} = \mu dt+ \sigma dW_t$$ but instead of the normal $W_t \sim N(0,1)$ , instead we have that $W_t \sim EMG^-(0,1,\lambda)$. Where ...
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2answers
391 views

Dealing with different definitions of the Ornstein-Uhlenbeck process

I've run up against a wall in reconciling two different definitions of the Ornstein-Uhlenbeck process, and would appreciate some help. On the one hand, as discussed here, we can define an Ornstein-...
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42 views

Correlation and Covariance of lognormal random variables

Let $X_i$ be lognormally distributed stochastic variable and $j$ < $i$ then is this the correct formula to calculate their correlation (please specifically note the index of t): $$ Correl(i,j) = ...
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Is this inhomogeneous Poisson process?

I asked this question in math stack exchange but since it is more related to probability and statistics, I thought I can ask here. Consider the population growth model where $P′(t)=rP(t)$, where $P(...
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1answer
234 views

Correlation between Ornstein-Uhlenbeck processes

Consider the Ornstein-Uhlenbeck process, $U(t)$, whose evolution follows: $$ \mathrm{d}U(t) = -\theta U(t) \mathrm{d}t + \sigma \mathrm{d}W(t), $$ where $\theta \in (0,2)$ is the mean-reversion rate, $...
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45 views

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 ...
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52 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 ...
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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 ...
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25 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 ...
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51 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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62 views

What is the mean and variance of a general stochastic integral?

$$X_T = x_t+\int_t^T\mu(s,X_s)ds+\int_t^T\sigma(s,X_s)dW_s$$ where $W$ is a Wiener process. What is the variance and mean of this process? It is well known $$E\left[\int_t^T\sigma(s,X_s)dW_s\right]=0....
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31 views

Expectation and variance of a stochastic time process conditioned on its past

$$dV_t=-k(V_t-1)dt+ \epsilon\sqrt{V_t}dW_t$$ $W_t$ is wiener process and the rest is just some parameters. For $T_{i+1}>T_{i}$ how do I find the expectation and variance of $V_{T_{i+1}}$ ...
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1answer
32 views

How to transform Gaussian variables so certain criterias are satisfied?

Given two standard normal distributed variables $Z_1$ and $Z_2$ then $N$ is defined as $$N=\alpha Z_1 + \beta Z_2$$ How do I derive $\alpha$ and $\beta$ such that $N$ is standard normally distributed ...
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1answer
28 views

Why does the theoretical value of the difference between these 2 stochastic integrals differ from the observed value in r?

Consider the stochastic integral $$ 2 \int_0^1 W_t \hspace{2mm} dW_t $$ Using r, this may be evaluated using one of the following summations $$ S_1 = 2 \sum_{j=0}^{n-1} \left[ W_\frac{j}{n} \left( W_\...
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173 views

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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1answer
43 views

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 "...
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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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1answer
575 views

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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1answer
30 views

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 ...
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1answer
111 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; ...
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1answer
92 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 ...
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0answers
164 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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1answer
133 views

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 ...
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0answers
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 ...
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2answers
205 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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7answers
2k views

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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2answers
169 views

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 ...
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3answers
595 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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1answer
196 views

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 ...
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497 views

Textbooks on stochastic calculus and stochastic differential equations

I am looking for key reference books in stochastic calculus, Stochastic Differential Equations (SDEs) as well as Stochastic Partial Differential Equations (SPDEs), from the most theoretical to the ...
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1answer
64 views

Financial Random Walks

Does anyone know of any good and accessible papers on the random walk modelling of financial data from a statistics perspective? Most of the papers I've found have been written by economists or ...