1
vote
0answers
39 views

Ito Integral of a SDE [on hold]

I would like to get help in solving the following Ito stochastic equation: $dY_t=-dW_t \, (Y_t^2+1)$ The process $W_t$ is the standard Brownian motion. Is it possible to get a path solution ...
2
votes
2answers
72 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
69 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
25 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
99 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
37 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
20 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
56 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
25 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
107 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
98 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
0answers
101 views

Linear birth death process extinction probabilities

Given a birth and death process $X$ with $\lambda_n=n\lambda$ and $\mu_n=n\mu$ for $n\ge0$, and letting $P_n(t)=\Pr\{X(t)=n\}$, I need to prove that $P_0'(t)=\lambda P_0(t)^2-(\lambda+\mu)P_0(t)+\mu$. ...
0
votes
1answer
78 views

Solve a special non-linear Backward SDE

It is straigtforward to solve a linear Backward SDE. i.e. $dY_t=Z_tdW_t+ aY_tdt+bZ_tdt$ with $Y_T=\xi$ (where a and b are constants, $\xi$ is bounded Randon Variable.) How can I solve $dY_t=Z_tdW_t+ ...
0
votes
1answer
152 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
55 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
409 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
153 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
44 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
296 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
131 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
92 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)& : ...
2
votes
0answers
116 views

Finding an SDE which satisfies $X(t)$

I am attempting the following problem, and was hoping if you guys could provide any feedback on whether my approach is valid. Thank you in advance for your time! The question is as follows: "Let ...
3
votes
0answers
221 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
247 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
306 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
213 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
212 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 ...
10
votes
2answers
382 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
1k 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
626 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 ...
2
votes
2answers
213 views

How can I obtain from a differential equation a stochastic version?

Suppose $\frac{dx}{dt}=ax+b$ and then assume that $a=c+g$ where $g$ is a Wiener process.
4
votes
1answer
497 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 ...