Questions tagged [stochastic-calculus]
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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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Solution of SDE with exp in integral
Consider the linear stochastic differential equation $$ dX_t = \left[AX_t+K \right]dt + g(t)dW_t, ~~~X_0 =x_0 $$ where $A$ is a $n\times n$ matrix, $K$ is a vector and $W_t$ is a Wiener process. The ... linear-algebra stochastic-calculus stochastic-integrals stochastic-differential-equations- 39
Solution of two-dimensional SDE in R
How can the solution of a two dimensional stochastic differential equation can be plotted in R? I can use existing functions, I want to plot the exact solution given by applying Itos lemma, not an ... stochastic-processes numerical-methods stochastic-calculus stochastic-differential-equations- 39
Equivalence of different versions of Markov property
Let $X_t,t\ge 0$ be continuous, $n$-dimensional process and filtration $\mathcal{F}_t=\sigma(X_s,0\le s \le t)$. We want to show the equivalence of the two definition in the following: $$ E[f(X_t)|X_{... stochastic-processes stochastic-calculus stochastic-integrals stochastic-analysis- 319
Brownian motion and martingale
The following theorem (picture below) is from Peres-Morters book: Brownian motion. Consider an open $O \subset \mathbb{R}^d$ and a harmonic function function $f: \overline{O} \to \mathbb{R}$ such that ... probability-theory stochastic-processes stochastic-calculus brownian-motion stochastic-analysis- 103
Determine the solution of the particular Ornstein-Uhlenbeck SDE
Consider the following stochastic differential equation: \begin{equation} dS_t=\mu S_t dt+\sigma dW_t. \end{equation} I tried to solve it by referring to the general Ornstein-Uhlenbeck model, i.e. to \... stochastic-processes stochastic-calculus stochastic-analysis stochastic-differential-equations- 160
the proof of existence of solution of stochastic equation
I am going through the proof of the existence of strong solution of SDE on Brownian motion and Stochastic Calculus by Karatzas and Shreve. However, there is a single line I really cannot see how to ... stochastic-processes stochastic-calculus- 591
Interpreting $\frac{ n^k }{ k! }$ [closed]
So I'm trying to figure out what kind of sense the following formula has: $$\frac{ n^k }{ k! }$$ Does this mean: I choose $k$ Elements from a container of $n$ Elements, where $k$ Elements can be ... combinatorics stochastic-calculus- 19
Does there exist a version of the Burkholder-Davis-Gundy inequality for multidimensional continuous local martingales?
Let $(\Omega,\mathcal{F},\mathbb{F}=(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a stochastic basis and $M$ a continuous local martingale on it with start in $0$. Then the Burkholder-Davis-Gundy ... stochastic-processes stochastic-calculus stochastic-analysis- 437
Markov alternative definition
On wiki, there are two definitions on Markov property, which are Let $(\Omega ,{\mathcal {F}},P)$ be a probability space with a filtration $({\mathcal {F}}_{s},\ s\in I)$, for some (totally ordered) ... stochastic-processes stochastic-calculus stochastic-integrals stochastic-analysis- 319
Decompose the Laplace transform of autocorrelation function of a stachastic process using Wiener-Kolmogorov whitening procedure
Here I read that it is possible to use the Wiener-Kolmogorov whitening procedure to decompose the Laplace transform of autocorrelation function into the product of white noise and a system function: $$... reference-request stochastic-processes stochastic-calculus stochastic-differential-equations- 7,350
Question on Markov property
I am self-studying "Introduction to Stochastic Integration" by Hui-Hsiung Kuo and have some doubts on the lemma. In the book it first defines that the Markov property on page 198 as follow ... stochastic-calculus stochastic-integrals stochastic-analysis stochastic-differential-equations- 319
How to calculate the autocorrelation of Ornstein Uhlenbeck process
Let us consider the following SDE: $$ d x(t)=-\alpha \cdot(x(t)-\mu) \cdot d t+\sqrt{2 \cdot \alpha} \cdot \sigma \cdot d W(t) $$ Its solution is a stationary stochastic process called the Ornstein-... stochastic-processes stochastic-calculus- 7,350
Prove that realized variance converges in probability to integrated variance for Ito process
Let $P_{t}$ be a price process of an asset. Assume the log-price $p_{t}=\ln P_{t}$ follows a generalized Ito process $$ d p_{t}=\mu_{t} d t+\sigma_{t} d w_{t}, \quad t \in R, \quad (1) $$ where $\mu_{... stochastic-processes stochastic-calculus finance stochastic-differential-equations- 7,350
How can I prove that $\Bbb{P}(X>0)\geq \frac{\Bbb{E}(X)^2}{\Bbb{E}(X^2)}$
I have the following problem: Let $X$ be a nonnegative random variable such that $\Bbb{E}(X^2)<\infty$ Then show that $$\Bbb{P}(X>0)\geq \frac{\Bbb{E}(X)^2}{\Bbb{E}(X^2)}$$ I would like to get ... probability probability-theory expected-value stochastic-calculus cauchy-schwarz-inequality- 1,509
Can I use the Chebychev inequality to prove this statement?
Let $X$ be a random variable with $\Bbb{E}(X^2)<\infty$ and $a>0$ I need to show that $$\Bbb{P}(X>a)\leq \frac{\Bbb{E}(X^2)}{a^2}$$ My idea was the following. Let me first remark that $$\{X&... probability probability-theory expected-value stochastic-calculus- 1,509
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