Quantitative Finance Stack Exchange
Quantitative Finance Stack Exchange

Questions tagged [stochastic-processes]

stochastic processes is a collection of random variables representing the evolution of some system of random values over time.

Given a probability space $(\Omega\,\mathcal{F},P)$ and a measurable space $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n)) ,$ an $\mathbb{R}^n$-valued stochastic process is a collection of $\mathbb{R}^n$-valued random variables on $\Omega$ , indexed by a totally ordered set $\{ t|t\ge0 \}$

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Are BSDE's used in practice?

In the academic applied probability/math finance community, Backwards Stochastic Differential Equations (BSDE's) are extremely popular, and they provide a single framework for several different problems, notably hedging and utility maximization,…
quasi
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Why does jump process has to be Cadlag and not the other way around

In all books and references that I have been exposed to, the jump processes have been defined to be Cadlag(right continuous with left limits). But no one has explained why this is the preferable case, why can't it be Caglad? I suspect it has…
Kenneth Chen
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Dynamic Programming: Dynamic Card Game

I'm reading an interview book called A Practical Guide to Quantitative Finance Interview and I have some doubts about the solution provided by the book, so I really appreciate your advice if my doubt is correct or not. Question description (from…
M00000001
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Gyöngy Theorem Proof

Can you please point me to a publicly available text that discusses the proof for the Gyöngy Theorem? Gyöngy, I. (1986), “Mimicking the One-Dimensional Marginal Distributions of Processes Having an Ito Differential,” Probability Theory and…
Discretizer
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Central Limit Theorem and Lévy processes

Lévy processes are self-decomposable and independent on any non-overlapping interval, so how come the distribution of the process at time T,$\phi(T)$, which is the sum of N i.i.d with law $\phi(T/N)$ is not normally distributed ? I cannot find what…
Amir Yousefi
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Covariation of these processes

Let $N_t \sim \text{Poisson}(\lambda t)$ and $M_t \sim \text{Poisson}(\theta \lambda t)$. We know that if $N$ and $M$ were independent, $dNdM = 0$ using polarization identity. We also know that $(dN)^2 = dN$; but now that these two processes are…
user57062
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Is there uniform stochastic process?

Shall I construct a stochastic process $X(t)$ such that $X(s+t)-X(s)\sim U(-t,t)$ ? Or is there already any similar formula?
Zhiyuan Wang
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demonstrate that a Square-root process is Non-central Chi-squared distributed

how can i prove that the value at some future time $t'$, $x_{t'}$, of the Square-root process at current time $t$, $x_t$, is Chi-squared distributed? $dx_t = k(\theta - x_t)dt + \beta \sqrt{x_t}dz_t$ explicitely: $x_{t'} = x_t e^{-k(t'-t)} + \theta…
davidpaich
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Stationary distribution for square root process

Consider the process, $$ dX_t=(-aX_t+b(1-X_t))dt + \sqrt{X_t(1-X_t)}dW_t $$ How do I show that the stationary distribution for the transition density is a beta distribution? I tried expanding the corresponding Kolmogorov Forward Equation but it…
Danny
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How to calculate the covariance involving Stochastic process

I was looking at some old post : Variance of time integral of squared Brownian motion I failed to grasp 2 derivations - $\text{Cov}\left(\int_{0}^{t}W^3_sdW_s\,,\,\int_{0}^{t}W^2_sds\right)$. I know this can eventually be written as $\mathbb{E}…
Bogaso
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Integration over function of Wiener process

I wish to calculate below 3 expectations of a typical Wiener process - $E \left[ \int\limits_{0}^{T} tdW_t \right]$ $E \left[ \left( \int\limits_{0}^{T} tdW_t \right)^2 \right]$ $E \left[W_T \int\limits_{0}^{T} tdW_t \right]$ How should approach…
Bogaso
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How to create a Stochastic Process through pre specified points?

I want to create a random (quasi random) process which goes through pre determined points and constraints. E.g. I have a daily price series but want to generate intra-day prices with the same OHLC properties. Also I am exploring the possibility to…
Suminda Sirinath S. Dharmasena
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Underlying Sample Space in Continuous Market Model

E.g., a model for $N$ stocks might have each follow a GBM $dS_i = \mu_i S_i dt + \sigma_i S_i dW_i$, where each $W_i$ is independent of the others. Letting $(\Omega, \mathcal{F}, P)$ be the underlying probability space, what should I be thinking of…
bcf
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$X_{T-}$ is $\mathcal{F}_{T-}$ measurable

By definition, $\mathcal{F}_{T-}=\mathcal{F}_0 \vee \sigma(A\cap \{ t
An old man in the sea.
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Martingale Definition notation

I am reading Stochastic Calculus by Shreve and am a bit confused by the notation when he first introduces a Martingale with the definition: $E_n(X_{n+1})=X_n $ What I don't understand is why the $X_n$ is capitalized. I thought that when we refer to…
robot112
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