1
vote
0answers
19 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
0answers
42 views

Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where ...
1
vote
1answer
48 views

References for numerical stochastic differential equations

I am currently working on a topic in physics which requires me to solve stochastic differential equations (specifically stoch. Schrödinger equation). I am a physicist and have not had any ...
0
votes
0answers
15 views

interpolation linear for a sample path

I am looking for a couple of references: interpolation linear for a sample path of Brownian Motion
0
votes
0answers
34 views

Numerical methods for stochastic differential equations [duplicate]

What type of SDEs can be solved by numerical methods? E.g. can the following differential equation $$ dy/dt = y\cdot X\cdot (y\cdot X - g(y,X))\cdot X $$ where X is a random variable and ...
1
vote
0answers
99 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 ...
3
votes
1answer
59 views

Error of apprximation

Have anyone read book "Paul Glasserman Monte Carlo MIFE", it's good, but i'm stuck in chapter 6 page 341 let $$ dX_t=a(X_t)dt+b(X_t)\,dW_t $$ they said that $$ \int_{t}^{t+\Delta t}a(X_{u}) \, ...
2
votes
1answer
91 views

Brownian bridge distribution: $\sup_{a \leq u \leq b} \frac{|W^0(u)|}{\sqrt{u(1-u)}}, 0 < a < b <1 $

If $W^0$ is a tied-down Wiener process (Brownian bridge) on the range $(0, 1)$, what is the distribution of \begin{equation} \sup_{a \leq u \leq b} \frac{|W^0(u)|}{\sqrt{u(1-u)}} \end{equation} ...
1
vote
0answers
49 views

How to conserve probability using a numerical integration scheme?

I have an iterative operator that conserves probability given by $P_{n+1}(z_j) = \int_a^b P_B(x+z_j)P_n(x) dx$, where $P_n$ is the PDF at time step $n$ and $P_B$ is a PDF that is fixed with compact ...
3
votes
0answers
96 views

Inadmissibility of Simpson's rule

Let $B_t$, $t\ge0$ be a standard Brownian motion and suppose $0<x_1<x_2<\cdots<x_n<1$. Then the conditional expectation $$ \mathbb E\left(\int_0^1 B_t\,dt \,\middle\vert\, B_0, ...
2
votes
0answers
63 views

Initial Conditions for Finite Difference of PDE

I am having trouble with figuring out what my initial conditions should be for a simple finite difference algorithm I wrote in Matlab. Specifically, I'm trying to show that the regular 1-Dimensional ...
7
votes
1answer
296 views

PDE - Feynman-Kac vs. finite difference methods

I've heard that in greater than three dimensions, it's more efficient to solve a second-order parabolic PDE using a Monte-Carlo method based on the Feynman-Kac formula that it is to use finite ...
10
votes
2answers
583 views

Relation of Brownian Motion to Helmholtz Equation

one can obtain solutions to the Laplace equation $$\Delta\psi(x) = 0$$ or even for the Poisson equation $\Delta\psi(x)=\varphi(x)$ in a Dirichlet boundary value problem using a random-walk approach, ...