Subject: (Stochastic) Difference vs Differential Equations
From: lones@lones.mit.edu (Lones A Smith)
Date: 2 Jan 1997 18:23:51 GMT
Given is the difference equation y_{n+1}-y_n=-g(y_n), for g(0)=0 with g'>0.
If x'=-g(x), with x(0)>= y_0>0, we have x(n) > y_n.
Conversely, by a result to to Anthony Quas, if g'<1 exists,
and (1-g'(z))z'=-g(z) and y_0>= z(0)-g(z(0)), then y_n >= z((n)-g(z(n)).
Thus, we can bound a difference equation aboive and below by a differential
equation, and can provide precise limiting analysis of the discrete
dynamics using continuous dynamics.
Question:
Does this extend for STOCHASTIC difference vs differential equations?
For simplicity, given a nonnegative stochastic difference eqn
y_{n+1} = g(y_n) with chance p(y_n) and =h(y_n) with chance 1-p(y_n),
and assume g(0)=h(0)=0, with g',h'>0. Do there exist natural stochastic
differential equations (x(t)) that probabilistically tracks the discrete
solution near the 0 stationary point? (namely, x(0)>y_0>0 then with
positive chance x(n)>y_n for all n, and another one z(t) that bounds
y_n below with positive probability).
This is such a general result, it would be quite useful to have for
one and all. Thanks for cites or proofs or inisghts.
Lones
.-. .-. .-. .-. .-. .-.
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Lones Smith, Economics Department, M.I.T., E52-252C, Cambridge MA 02139
(617)-253-0914 (work) 253-6915 (fax) lones@lones.mit.edu
