Piecewise linear SDE
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From: Richard Hayden on 12 Aug 2010 14:38 Hi, Assume I have the following stochastic differential equation: dX_t = f(X_t) dt + g(t) dB_t where B_t is a Wiener process, and f(.) is *piecewise linear* (that is, on finitely-many regions of the state space of X_t, say A_1,...,A_n, if x \in A_i, we have f(x) = f_i(x), where the f_i are linear functions). I was wondering, is X_t then still Gaussian? Thanks, Richard.
From: Robert Israel on 12 Aug 2010 17:00 > Hi, > > Assume I have the following stochastic differential equation: > > dX_t = f(X_t) dt + g(t) dB_t > > where B_t is a Wiener process, and f(.) is *piecewise linear* (that > is, on finitely-many regions of the state space of X_t, say > A_1,...,A_n, if x \in A_i, we have f(x) = f_i(x), where the f_i are > linear functions). I was wondering, is X_t then still Gaussian? No. Why would it be? Consider e.g. a case where g(t) = 0 for t >= 1, while f(x) = 0 for x <= 0, x for x > 0. So for t >= 1, X_t = { X_1 if X_1 <= 0 { exp(t-1) X_1 if X_1 > 0 If X_1 is Gaussian, X_t for t > 1 won't be. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: Richard Hayden on 12 Aug 2010 17:22
On Aug 12, 10:00 pm, Robert Israel <isr...(a)math.MyUniversitysInitials.ca> wrote: > > Hi, > > > Assume I have the following stochastic differential equation: > > > dX_t = f(X_t) dt + g(t) dB_t > > > where B_t is a Wiener process, and f(.) is *piecewise linear* (that > > is, on finitely-many regions of the state space of X_t, say > > A_1,...,A_n, if x \in A_i, we have f(x) = f_i(x), where the f_i are > > linear functions). I was wondering, is X_t then still Gaussian? > > No. Why would it be? > Consider e.g. a case where g(t) = 0 for t >= 1, while > f(x) = 0 for x <= 0, x for x > 0. So for t >= 1, > X_t = { X_1 if X_1 <= 0 > { exp(t-1) X_1 if X_1 > 0 > If X_1 is Gaussian, X_t for t > 1 won't be. > -- > Robert Israel isr...(a)math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada Hi Robert, Thanks for that - sorry, I see now this is quite obvious. I wonder, then, is such an SDE as that mentioned above any more tractable than for general non-linear f(.)? Would there be any approach other than simulation of traces to obtain information about the distribution of X_t? Thanks. |