# Questions tagged [stochastic-processes]

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

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### Analytic Hull White model with correlated stochastic processes

I am trying to price a path dependent option which uses two underlyings (a stock index and an interest rate index). I am using Hull White model for interest rate modelling and local vol for stock ...
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### Aggregate Portfolio Simulation vs. Underlying Assets

Background: I am currently implementing a correlated Monte Carlo simulation model using Cholesky decomposition to create the sampling distribution. Question: What is the difference between creating ...
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### multivariate geometric brownian motion equivalent martingale measure

Suppose $W$ is a $\mathbb{P}$-Brownian motion and the process $S$ follows a geometric $\mathbb{P}$-Brownian motion model with respect to $W$. $S$ is given by \begin{equation} dS(t) = S(t)\big((\mu - ...
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### Confusion about the formula for gain process in a financial market

In this wikipedia page, we consider the following financial market The formulas for the stocks are given here And the gain process of a portfolio $\pi$ is defined such that From what I understand, ...
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### Bond-pricing under the Vasicek short rate model

I'm currently studying the Vasicek model of the short interest rate $$dr_t=a(\mu-r_t)dt+\sigma dW_t$$ I know how to solve this stochastic differential equation (SDE) and how to find expectation and ...
1 vote
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### Did I derive the Kelly criterion correctly?

$$\frac{dX_t}{X_t}=\alpha\frac{dS_t}{S_t}+(1-\alpha)\frac{dS^0_t}{S^0_t}$$ where $\alpha$ is proportion of the investment in the risky asset $S_t$ and $S^0_t$ is the risk-free asset. $S_t$ follows a ...
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### integral of adapted process with respect to semimartingale is a martingale

Fix $T > 0$ a finite time horizon. Let $H$ be an adapted (or progressively measurable, if needed) continuous process and S be a continuous semi martingale, both on $[0,T]$. Under what conditions is ...
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### Deriving an Analytical Expression for Standard Deviation of Log Returns

I am looking to find an expression for the standard deviation log returns of a stock price process. I have a stock price which follows the following dynamics: $dY(t) = Y(t)(r(t)dt + η(t)dW(t))$ Here,...
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### Bessel Correction and Geometric Brownian Motion

Does it make sense to use bessel's correction for standard deviation and variance when fitting the drift and volatility parameters of geometric brownian motion to historical return data for a security....