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.

3,273 questions
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Simplifying a Stochastic Equation

Let $$dG_t=\mu(t)dt+\sigma(t)dW_t$$ where $W_t$ is standard Brownian motion and define $$f(t,u)=E[e^{-r(u-t)}(\mu(u)-rG_u)|F_t]$$ where $(F_t)_{t\in [0,T]}$ is a filtration, $T\ge u\ge t$ and $r$ is a ...
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SDE of the Lévy process

I consider a Lévy process of the form $$X_t = X_0 e^{(\mu-\frac{\sigma^2}{2})t+\sigma B_t-\sum_{i=1}^{N_t} Y_i},$$ where $\mu$ and $\sigma$ are constants, $B_t$ is a Brownian motion, $N_t$ is a ...
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Vasicek model for the evolution of the spot interest rate is parametrised by the reversion rate $\gamma$ and so $\bar{r} = \frac{\eta}{\gamma}$ the average short rate to which the simulated rate ...
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Markov process and non-deterministic random variables

How do I show the following: If $Z_1$ and $Z_2$ are non-deterministic random variables and we define the process $(X_t)_{t\geq 0}$ by $X_t = Z_1 \cos(t)+ Z_2 \sin(t)$. I want to show that this is not ...
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the product of a martingale and process with finite variation. [closed]

If $V_t$ is a process with finite variation and $M_t$ a martingale with $M_0=0$, Do we have $\mathbb{E}[V_t M_t]=0$ ? Thanks for your help