0
votes
0answers
25 views

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 ...
1
vote
0answers
17 views

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 ...
3
votes
1answer
89 views

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 ...
0
votes
1answer
50 views

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 ...
2
votes
2answers
84 views

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 ...
1
vote
0answers
93 views

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 ...
-1
votes
1answer
27 views

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 ...
1
vote
1answer
126 views

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 ...
3
votes
0answers
40 views

Merton's Problem Stochastic Differential Equation

Solve the following numerical case of Merton's optimal portfolio selection problem: find an optimal policy function $(s, y) \mapsto u(s, y)$ such that for the Ito diusion determined by $dX_t =X_t(u(t, ...
2
votes
0answers
25 views

Stochastic input to LTI systems

Papoulis stochastics processes book question 9-6 says: Show that if $R_\nu(t_1,t_2)=q(t_1)\delta(t1-t2)$ and $\mathbf{w''}=\mathbf{v}(t)U(t)$ and $\mathbf{w}(0)=\mathbf{w'}(0)=0$ then ...
0
votes
2answers
66 views

Stochastic differential

Im really new in the stochastic procceses please help me. How can I solve this stochastic differential equation? $$dX = A(t)Xdt$$ $$X(0) = X_0$$ If $A$:[0,$\infty$]$\to$ $R$ is continous and $X$ is ...
1
vote
0answers
27 views

Probability of winding number in 2D Brownian motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...
2
votes
1answer
114 views

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 = ...
0
votes
0answers
16 views

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 ...
0
votes
1answer
114 views

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 ...
0
votes
1answer
163 views

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 ...
0
votes
2answers
57 views

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 ...
1
vote
1answer
425 views

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 ...
0
votes
1answer
183 views

Kolmogorov forward equations of a simple birth process.

I have got a simple birth process such that $ q(n,n+1) = -q(n,n)= \lambda n$ I want to solve the KFE. $d/dt.p_t(n,n) = -\lambda n p_t(n,n)$ $p_0(n,n) = 1$ $d/dt p_t(n,n+k) = -\lambda (n+k) ...
0
votes
1answer
47 views

Numerically solve SDE

I am not really into solving stochastic differential equations, but I was trying to numerically solve an OED given by: $\frac{dy}{dt} = f(t,y,p) + N(0,\sigma^2)$ where normal noise with 0 mean and ...
5
votes
1answer
304 views

Computing the limit of the expectation of a function of a stochastic process (phew!)

I state my problem in a few lines then describe what I have already done. I have a quite simple stochastic differential equation (SDE): $dx=-2x \, dt+\sqrt{1-x^2} \, dW$ with $W$ a brownian. I ...
1
vote
0answers
138 views

damped harmonic oscillator driven by a stochastic momentum (not force)

Could you give references for solutions or solutions to the following problem: Given: damped harmonic oscillator driven by stochastic force of very short duration (= stochastic momentum). Find: ...
0
votes
0answers
99 views

How to transform a stochastic jump diffusion equation to a Levy stochastic differential equation?

If I have this type of stochastic differential equation : $$ dX(t) = A(X(t),t)\ dt +B(X(t),t)\ dW(t) + C(X(t),t)\ dP(t) $$ With $$ \begin{align} dW(t)& : \text{A wiener process}\\ dP(t)& : ...
3
votes
0answers
231 views

Harmonic oscillator with stochastic forcing

It's well known that the solution of the differential equation: $$\ddot x(t)+\omega^2x(t)=\sin(\psi t)$$ has the form: $$x(t)=C_1 \sin(\omega t)+C_2 \cos(\omega t)-\frac{\sin(\psi ...
1
vote
1answer
284 views

Solutions to a stochastic differential equation

$$dX_t = -\frac{1}{2}e^{-2X_t}\ \ dt+e^{-X_t}dB_t, X_0=x_0$$ Hint: solve this equation using the substitution $X_t=u(B_t)$, show that the solution blows up at a finite random time.
2
votes
2answers
329 views

Differential equation with gaussian noise

The equation has the following form: $$x'' + w^2 x=n$$ $w=1$, $x(0)=1$, $n$ is Gaussian noise with mean $0$ and standard deviation of $1$. Without the Gaussian noise, i can easily solve the ...
4
votes
2answers
218 views

Stochastic predator-prey

My system is a simple $P$ vs $I$ foxes- vs rabbits model given by: $$ \begin{cases} \frac{\mathrm{d}I}{\mathrm{d}t}=& \alpha_I+\lambda_IP- \gamma_II -\delta_IPI;\\ ...
1
vote
0answers
226 views

Show that this is the unique solution of that Stochastic Differential Equation

Reading through a paper, I stumbled across the stochastic differential equation $ dS_t = \sigma S_{t-} dX_t $. The claim there was that $ S_t = S_0 \exp(\alpha N_t - \beta t) $ should be its unique ...
9
votes
2answers
428 views

SDE with no strong solutions and Euler-Maruyama

Tanaka's SDE [see wikipedia article] $\text{d}X_t = \operatorname{sgn}( X_t ) \; \text{d}B_t$ with $X_0 = 0$, where $\operatorname{sgn}(x) = 1$ if $ x\ge 0$ and $\operatorname{sgn}(x) = -1$ if ...
2
votes
2answers
2k views

Analytic Solution to a Generalized Ornstein-Uhlenbeck Process?

An Ornstein Uhlenbeck process $x_t$ satisfies the following stochastic differential equation: ${\large dx_t = \theta (\mu - x_t ) dt + \sigma dW_t }$ where $\theta,\mu, \sigma > 0$, and $W_t$ ...
15
votes
1answer
642 views

Stochastic interpretation of Einstein Equations

Einsteins theory of gravitation, general relativity, is a purely geometric theory. In a recent question I wanted to know what the relation of Brownian Motion to the Helmholtz equation is and got a ...
4
votes
1answer
530 views

Solution of a Hamilton-Jacobi-Bellman (HJB) equation

I am trying to solve a ODE that arises from a Hamilton-Jacobi-Bellman (HJB) equation. The equation is ...