Finding a stochastic differential equation as limit of a discrete stochastic process

2011-12-24T19:35:33Z

I stumbled upon a problem that seems simple but I cannot tackle it. Let $X_n$ be a discrete process defined by the following algorithm.

Choose $X_0\in[0,1]$, set $\kappa>0$ small enough and
$X_{n+1}=X_n+\kappa(I_n-X_n)$
with $I_n=1$ with probability $X_n$ and $I_n=0$ with probability $1-X_n$.

In other words the $X_n$ decreases with probability $1-X_n$ by $\kappa X_n$ and increases with probability $X_n$ by $\kappa(I_n-X_n)$ so $E[X_{n+1}]=X_n$.

The point is that $\kappa$ can be arbitrarily small so we can take its limit to $0$ while decreasing linearly the time step. This naturally should give an SDE (in this case I would expect it to be non-linear). So my question is how can one find this SDE or the PDE that gives the probability density.

I should add that for short times it looks like a random walk (which is expected I guess) with the variance being proportional to $\kappa^2 t^2 X_0(1-X_0)$, with $t$ small. However since $X_n\in[0,1]$, $1$ is an upper bound for the variance.

Answer by Jason Swanson for Finding a stochastic differential equation as limit of a discrete stochastic process

2012-07-19T17:46:13Z

Here is a sketch of how you might approach this. Let $$ \xi_j = \frac{\kappa(I_{j-1} - X_{j-1})} {\sqrt{X_{j-1}(1 - X_{j-1})}}. $$ The two "hard" results that must be proven are: (1) For each $t$, $$ \lim_{\kappa\to0} E\big[\max\{|\xi_j|: 1 \le j \le \lfloor\kappa^{-2}t\rfloor\}\big] = 0, $$ and (2) for each $t$, $$ \sum_{j=1}^{\lfloor\kappa^{-2}t\rfloor} \xi_j^2 \to t, $$ in probability as $\kappa\to0$. The rest of the proof would then be the following "soft" argument based on general theory.

First, let $W^\kappa(t)=\sum_{j=1}^{\lfloor\kappa^{-2}t\rfloor} \xi_j$. Using the two results above, one can use the martingale central limit theorem (Theorem 7.1.4 in Ethier & Kurtz) to prove that $W^\kappa\Rightarrow W$, where $W$ is a standard Brownian motion.

Next, we take the difference equation which defines the sequence ${X_n}$ and rewrite it as an integral equation. More specifically, if we define $X^\kappa(t)=X_{\lfloor\kappa^{-2}t\rfloor}$, then we may write $$ X^\kappa(t) = X_0 + \int_0^t \sqrt{X^\kappa(s-)(1 - X^\kappa(s-))}\,dW^\kappa(s). $$ There is nothing deep here, just a change of notation, really.

Finally, we use Theorem 5.4 in Kurtz & Protter to prove that $(X_0,X^\kappa,W^\kappa) \Rightarrow(X_0,X,W)$, where $X$ is the unique strong solution to $dX=\sqrt{X(1-X)}\,dW$, $X(0)=X_0$.

A watered-down version of Theorem 5.4 in Kurtz & Protter is available as Theorem 2.3 in these lecture notes. This version is sufficient for your purposes, and it may be easier to digest. Also, to use this theorem, you must show that, for every version of $(X_0,W)$, the limiting SDE has a unique strong solution for all time. This follows, for example, from Proposition 5.2.13, Theorem 5.5.4, and Corollary 5.3.23 in Karatzas & Shreve.

Finding a stochastic differential equation as limit of a discrete stochastic process - MathOverflow

Finding a stochastic differential equation as limit of a discrete stochastic process

Answer by Jason Swanson for Finding a stochastic differential equation as limit of a discrete stochastic process