Tagged Questions

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Can the transition probabilities of an inhomogeneous Markov chain be written as an exponential?

If $Z_t$ is a homogeneous continuous-time Markov chain with finite state space $E=\{1,\ldots,p\}$, transition matrices $(P(t))$ and intensity matrix $Q$, it holds that $$P(t) = \exp(tQ),$$ see for ...
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characterising attractors for master equations

I have a master equation for $(x,y,z)$ with the constraint $x+y+z=N$. $x$ can be regarded as the number of animal of a certain species in the whole system. In other words, I have a differential ...
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Problem on Solving Stochastic Differential Equation

Let $(Xt)$ be a solution to the equation $dX_t = aX_t dt + \sqrt{(1+X_t^2)} dW_t$ where $W_t$ is a Brownian motion process at time t Let $Y = F(X_t)$ for a certain function $F$. Find $F$ for which ...
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Birth and Death Process Questions

Consider a birth and death process with the birth rate $\lambda_m = \lambda (m\ge 0)$ and death rate $\mu_m = m \mu (m \ge 1)$. A. How would I derive the stationary distribution? Only information I ...
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Understanding basic stochastic differential equations

This is from a physics course in economics, the literature provides a bare minimum of mathematical explanations. I am trying to understand how to work with stochastic differential equations given in ...
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Differential equation with random variable

How can I derive analytically or compute numerically the solution to following differential equation $$dy/dt = y\cdot X\cdot (y\cdot X - g(y,X))\cdot X$$ where X is a random variable (e.g. from a ...
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find the stochastic differential eqution with ito

I was trying to do some ito problems but I don't grasp how to apply the formula (which is the process). If somebody could give me a hand it would be great! Thanks so much in advance. I have the ...
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Background for studying and understanding Stochastic differential equations

Assume I have back ground of the following knowledge based on the textbook as : ODE : ODE by Tenenbaum Baby probability : Ross 's baby probability Baby real anlysis : Bartle's introduction to real ...
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Square root of a stochastic process

i need help with the following problem. how can i derive d√v using Ito's lemma for the following process: d√v=(α−β√v)dt+δdX The parameters α, β, δ are constant. Using Itô's lemma show that dv = ...
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Stochastic differential equation log displace by a deterimnistic function [duplicate]

Do you know how to solve the stochastic differential equation: $$dS_t=(\alpha S_t+ f_t)dW_t$$ with $W_t$ a Brownian Motion $F_t$-measurable, $\alpha$ a constant and $f_t$ a known determistic ...
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Second derivative of Brownian motion?

My question is, we give a meaning to the following expression: $$dX(t) = \mu(t,X(t))dt + \sigma(t,X(t))dW(t), \ \ X(0)=x.$$ where $W$ is a Wiener process. This equation can be thought as ...
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Proving a Probability Generating Function satisfies a partial differential Equation

We have N animals grazing in a field. The animals graze independently, and periods of grazing and resting alternate for the animals. If an animal is resting at time t, the probability it begins ...
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Specifying differential equation that describes a particular set of dynamics.

There are $S$ individuals who are susceptible to infection, and $I$ who are infectious. $S + I = N$, where $N$ is the total size of the population. Each infectious transmit the disease to a ...
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Linear birth death process, probability of extinction by time t

I have a linear birth death process with birth rates $\lambda n$ and death rates $\mu n$ . Let r(t) be the probability of extinction by time t. If there is 1 individual alive at time 0 explain why ...
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