# Tagged Questions

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

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### Extended CIR and discretization

Did someone know how to discretize this process efficiently : $dX(t) = \kappa [\theta(t)-X(t)]dt + \sigma \sqrt{X(t)}dW(t)$ I am looking for something more sophisticated than the trivial Euler ...
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### What is wrong in this GBM simulation?

I am trying to generate a few samples of GBM using the following very simple MATLAB code: ...
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### Transformation to reduce standard deviation without changing median

Consider some negative skew and high kurtosis return time-series $X_t$. I do not know the functional form of the pdf of $X_t$ and have about 150,000 data points. Suppose that I was to create an ...
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### CIR model - nth moment generation $E^*[r_T^n]$

I am analyzing the nth moment generation process for $r_t$ with dynamics defined by CIR model $r_t$ has following dynamics $$dr_t=a(b-r_t)dt+\sigma \sqrt{r_t} dW_t^* \quad \quad (1)$$ for some ...
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### The relation between exchange rate SDE and respective interest rates

The exchange rate between a domestic currency money market and a foreign currency money market can be expressed as $$dQ(t) = (r_d - r_f)Q(t)dt + \sigma Q(t)d\tilde{W}(t)$$ where $r_d$ is the ...
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### Expected value of log-GARCH process

Is there a way to analitycally compute expectation of log-GARCH process? The GARCH(1,1) process: $dU_t = \theta(\omega - U_t) dt + \xi U_t d W_t$ The log-GARCH(1,1) process: $e^{U_t}$ The ...
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### Conditional expectation of a non stochastic process

In an example I was working through it was shown that $W_{t}^{2} - t$ was a martingale with respect to the Brownian motion filtration $\mathcal{F}_{s}^{W}$ with $t>s$. Everything was fine except a ...
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### What are $d_1$ and $d_2$ for Laplace?

What are the formulae for d1 & d2 using a Laplace distribution?
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### Why is the value of an adaptive stochastic process known at time t?

I am having a hard time to understand the concept of an adapted stochastic process. Using an analogy to finance, I have been told we can think of adaptiveness of a stock price process as having an ...
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### Arbitrage and dominant strategies

If there is no arbitrage there is no dominant trading strategy, but there may be arbitrage opportunities even if there are no dominant trading strategies. Could you explain this statement and bring ...
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### Law of a geometric brownian motion first hitting time (formula dont match Monte Carlo Simulation)

I posted this question before on MSE I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all ...
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### Time series (stochastic process) estimating parameters using characteristic function

I have a time series of assets ${A_1, A_2, ..., A_n}$, which is described by a sophisticated distribution having the following characteristic function: $\phi(u; t;\theta)$, where $\theta$ is a vector ...
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### Foward-start option pricing

Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and is generated by $1 d$- ...
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### Speed of mean reversion of an interest rate model

I would like to have a bit more of intuition about the concept of "speed of mean reversion" for an interest rate model, e.g. Vasicek or CIR. In particular, is a negative speed of mean reversion ...
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### Distribution of minimum of hazard functions

Suppose I have two random variables, $X_1$ and $X_2$, that are independent (but not identically distributed) and assume both have hazard functions $\lambda_1(s)$ and $\lambda_2(s)$, for $s > 0$. ...
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### Meaning of w in SDE

I'm missing meaning of $w$ in typical SDE like $dX_t(w) = f_t(X_t(w)) + \sigma(X_t(w))dW_t$, in context of $w \in F_{xxx}$. Does it mean that both $w$ is one of events that could happen before ...
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### From discrete time series models to continuous

Is it possible to convert an SARIMA model to a continuous model? If so, what is the methodology to do that?
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### $\mathop{\mathbb{E^{}}}\left\lbrace 1_{S_T > K} \; S_T \right\rbrace$ ? Exp. of an indicator funct and a diffusion with non-proportional vol

How to compute $\mathop{\mathbb{E^{}}}\left\lbrace 1_{S_T > K} \; S_T \right\rbrace$ ? where $dS_t = S_t r dt + \sigma dW_t$ and $1_{S_T > K}$ is the indicator function being one when ...
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### How to use Euler discretization for this interest rate model?

How can I perform Euler discretization on this model where $\delta t=1$ and $\delta x_t = x_t-x_{t-1}$
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### Feynman Kac Formula for path-dependent options

Consier geometric Brownian motion: $dS_t/S_t=\mu dt+\sigma dW_t$ Feynman Kac theorem tells us that the conditional expectation $v(t,x)=E[ e^{-rT}\Psi(S_T) | S_t=x]$ can be computed by solving the ...
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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 ...
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