# Tagged Questions

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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### Stochastic process on compact spaces

I just heard some strange reasoning that I would like to understand with your help, let me describe the situation (unfortunately, I hesitated to ask the lecturer about it, because I apparently lacked ...
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### Different definitions of local p integrability for local martingales

When talking about cadlag (but not continuous) martingales and local martingales in the context of stochastic integration one can come across different definitions depending on the author. These are: ...
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### $L^p$ integrable local martingale is still $L^p$ integrable when stopped at localizing stopping times.

Assume that $X$ is $L^p$ integrable for $1\leq p\leq \infty$ (i.e., for all $t$, $X_t\in L^p$) and is also a (Cadlag) local martingale. If $T_n$ is a localizing sequence of stopping times for $X$. Is ...
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### What is the min-max argument in mathematics?

In the proof of a theorem the author says that he would prove a special case using the min-max argument. After reading the proof I could not infer what the min-max argument actually does. Could ...
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### Parallel Shifts of Forward Curve and Arbitrage Opportunities

I came accross a phrase in the Paul Glasserman, Monte Carlo Methods In Financial Engineering, page 153 : "a model in which the forward curve makes only parallel shifts admits arbitrage opportunities :...
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### How to arrive the following results?

I am reading the book "stochastic differential equations and diffusion processes" written by Ikeda and Watanabe. In the chapter IV about uniqueness of stochastic differential equation, there is a ...
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### What does Karhunen-Loève expansion have to do with cosine-sine basis expansion?

According to my research, Karhunen-Loève(KL) expansion is a version of Fourier series for stochastic processes and states that under some conditions, a stochastic process $X\left(\omega, t\right)$ can ...
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### Martingale Poisson [closed]

Can somebody help me with working out: $$E[(N_{t}-\lambda t)^2\mid F_{s}]$$ where $N_{t}$ is a Poisson process and $F_{s}$ the $\sigma$-algebra generated by $N_{s}$, $0 \leq s < t$.
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Can somebody help me with determining whether $Z_{n}=\log(2n+S_{n})$ is a martingale, supermartingale or submartingale with $S_{n}=\sum_{i=1}^{n}X_{i}$ and the are i.i.d. random variables with $P(X_i ... 1answer 58 views ### Malliavin derivative under change of measure Let$\widetilde{B}$be a Brownian Motion under the measure$\mathbb{P}$. Let$\theta$be a stochastic process fulfilling the Novikov's condition and$Z_\theta$the relative Radon–Nikodym derivative ... 0answers 31 views ### Why does there exist a right continuous version of the supermartingale$\{P(L >u \vert F_u),u \geq 0)\}$Why does there exist a right continuous version of the supermartingale$\{P(L >u \vert F_u),u \geq 0)\}$where$L$is a measurable random variable Its is clear that not all supermartingales have ... 0answers 61 views ### Why can the solution of a SPDE$\partial_tu(t,x)=\cdots$be viewed as a stochastic process indexed by$t$with values in a space of functions of$x$? Please consider a stochastic partial differential equation of the form $$\partial_tu(t,x)=F(t,x,u(t,x),{\rm D}u(t,x),{\rm D}^2u(t,x))+G(t,x,u(t,x),{\rm D}u(t,x))\partial_tB(t,x)\tag 1$$ where$\...
starting from the folowing SDE (HJM model): $$df(t,T)=\left(\sigma(t,T)'\int_t^T{\sigma(t,u)du}\right)dt+\sigma(t,T)'dW_t$$ And having $r(t)=f(t,t)$, I have two questions : 1) how do we obtain the ...