# 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.

4,914 questions
Filter by
Sorted by
Tagged with
10 views

### Find the distribution of sum

Find the distribution of $W_1 + W_2 + W_3$, where $W_t$ - standard Wiener process
1 vote
33 views

• 1,771
1 vote
16 views

### $exp$ of Half-Normal Distribution

I know that the Half-Normal Distribution has moments of all orders - that is, if $X\sim\mathcal{N}(\mu,\sigma)$, then, $$E[|X|^p]<\infty$$ However, do we also have $$E[e^{|X|}]<\infty$$ ? ...
• 352
31 views

### Find the mean and the variance of $X(1)$ for stochastic differential equation: $dX(t)=-1.5X(t)dt+0.85dW(t)$ with $X(0)=0.7$

Suppose that $X(t)$ satisfies $\hspace{5cm}$ $dX(t)=-1.5X(t)dt+0.85dW(t)$ with $X(0)=0.7.$ Find the mean and the variance of $X(1).$ I know that $E[X(1)]$ will result in mean and $E[(X(1))^{2}]$ in ...
• 359
28 views

### How to prove or calculate $E[\int_{t_{i-1}}^{t_i} e^{-μ({t_i}-s)}\sigma B_s |x_{t_{i-1}}]=0$?

$B_s$ is brownian motion. Because $\int_{t_{i-1}}^{t_i} e^{-μ({t_i} -s)}\sigma dB_s$ has a Brownian component, it is normally distributed with the mean zero according to Taylor & Karin 1998 's ...
24 views

### Solution to a linear Backward SDE

In my last question, Jose and I discussed about the solution to a linear backward SDE. I followed his steps and made it clear. Besides, I read a paper from Professor Peng talk about the linear ...
• 141
12 views

### Radon–Nikodym derivative of two multivariate Gaussians

Let two Gaussian distributions $P_1$, $P_2$ with mean $0$ and covariance matrix $\boldsymbol{\Sigma}_1,\boldsymbol{\Sigma}_2\in\mathbb{R}^d$ be given. I want to calculate the Radon–Nikodym derivative ...
• 824
18 views

### Bound Hypergeometric distribution by Gauss distribution

I want to lower bound the pdf of a Hypergeometric distribution $H(N,K,K)$, which has equal number of success states and number of draws with the pdf of a normal distribution. With the central limit ...
• 824
27 views

### $E[|T\cap S|^2]$ for random sets $S$, $T$ with fixed number of member $|S|=|T|=d$

Let two sets $T,S\subset\{1,\dots, p\}$ be given. Both sets have $d<p$ elements, i.e. $|S|=|T|=d$ Find $E[|T\cap S|^2]$ for random sets $S$, $T$, i.e. if $\mathcal{S}$ is the set over all those ...
• 824
23 views

### european call with binomial model in python

I have this exercise to resolve. Calculate the prices of a European and an American call option in an N = 3 period binomial model with $S_0 =$ 1, $u = \frac{1}{2} =$ 2 and $r = - \frac{1}{4}$. ...
30 views

### Constructing the solution to a linear backward SDE

For a linear backward stochastic differential equation (BSDE): for any given $\xi \in L^2(\mathscr{F}_T)$, $$-dY_s = (a_s Y_s + b_s Z_s +c_s)ds-Z_sdB_s$$ Where $a_t,b_t,c_t \in L^2_\mathscr{F}(0,T;R)$,...
• 141
8 views

• 197
1 vote
63 views

39 views

### Expectation of product of brownian motion and stochastic integral

Let $f:[0,\infty)\to\mathbb{R}$ be a deterministic continuous function and $B$ a Brownian motion with $B_0=0$. I need to prove that $$\mathbb{E}\left(B_t\int_0^tf(s)dB_s\right)=\int_0^tf(s)ds.$$ I ...
45 views

### Finding a weak solution to an SDE

Consider the SDE $$dX_t=\text{sign}(X_t)dB_t$$ with $X_0=0$ and where $$\text{sign}(x)=\begin{cases}-1&\text{if }x\leq0\\1&\text{if }x>0\end{cases}.$$ I am asked to find a weak solution to ...
67 views

• 81
### How do I prove that the distribution of $x_t$ in $x_t=x_{t-1}e^{-μt}+ θ(1-e^{-μΔt}) + \int_{t-1}^t e^{-μ(t-s)}\sigma dB_s$ is a normal distribution?
$B_s$ is brownian motion. Because $\int_{t-1}^t e^{-μ(t-s)}\sigma dB_s$ has a Brownian component, it is normally distributed according to Taylor & Karin 1998 's Introduction to Stochastic ...