-1
votes
0answers
31 views

Distribution of the supremum of a transformed Brownian Motion?

I have a stochastic process given by $z_{t}=w_{t}/\alpha\left(t\right)$ , where $w_{t}$ follows a Wiener process (a standard $\left(0,1\right)$ Brownian Motion) starting from $w_{0}=0$ , ...
1
vote
0answers
51 views

Stochastic differential equation for a Fokker-Planck-type equation with a non-derivative term

I have something similar to a Fokker-Planck equation of the form $\frac{\partial}{\partial t}f( x,t) = A(x,t)f(x,t)- \frac{\partial}{\partial x}[B(x,t) f(x,t)] +\frac{1}{2}\frac{\partial ^2}{\partial ...
1
vote
0answers
23 views

Stochastic differential equation of a falling body

It's well known the motion of a falling body in a constant gravity model, for high speed is given by: $$m\ddot{x}(t)=g-\beta\dot{x}(t)^2$$ where $\beta$ is he drag coefficient. In a turbolent flow we ...
3
votes
1answer
98 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 ...
2
votes
1answer
149 views

Black Derman & Toy Model

The BDT model is given by $$d(\ln\,r)=\left(\theta(t)-\frac {d(\ln\sigma(t)}{dt}\ln r\right)\,dt+\sigma(t) \, dW.$$ How can one rewrite the BDT model as $dr=A\,dt+B\, dW$, using It$\hat o$?
0
votes
0answers
16 views

Is there a solution for this stochastic differential equation or analogous ordinary differential equation?

I'm trying to analyze the following Ito stochastic differential equation: $$dX_t = \|X_t\|dW_t$$ where $X_t, dX_t, W_t, dW_t \in \mathbb{R}^n$. Here, $dW_t$ is the standard Wiener process and ...
1
vote
1answer
150 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 ...
0
votes
0answers
21 views

DE: $pU^2 = (rx-1)U' + Ur + \frac{1}{2}\sigma^2U''$?

I have been trying to solve the following ordinary differential equation that results from a problem in stochastic control theory. $U$ is a function of $x$. $pU^2 = (rx-1)U' + Ur + ...
0
votes
2answers
77 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
29 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
129 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
1answer
137 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 ...
5
votes
1answer
310 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
48 views

Stochastic differential equations and experimental data

If we have a set of experimental data: $$X=\{x_1,x_2,\ldots,x_N\}$$ is it possible to write down an equation of the kind: $$dx(t)=b(x(t))\,dt+\sigma(x(t))\,dB(t)$$ describing the process from which ...
0
votes
0answers
101 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)& : ...
4
votes
1answer
102 views

Bessel differential equation with random parameter

I know that the following differential equation: $$x^2\frac{d^2y(x)}{dx^2}+x\frac{dy(x)}{dx}+(x^2-\alpha^2)y(x)$$ has the solution: $$y(x)=C_1\cdot J_\alpha(x)+C_2\cdot Y_\alpha(x)$$ In my case, the ...