# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...