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If we have an a process $X_t$ with values in $\mathbb{R}^{n \times n}$ which solves a linear Stratonovich SDE $$ dX_t = A_t X_t dt + B_t X_t \circ dW_t $$ then the inverse of $X_t$ exists and solves $$ dZ_t = - Z_t A_t dt - Z_t B_t \circ dW_t $$ It is easy to see that $X_tZ_t =$ Id by using the product rule. My question is this:

If $X$ instead solves the affine SDE $$ dX_t = (A_t X_t + a_t) dt + (B_t X_t + b_t) \circ dW_t $$ does it have an inverse?

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    @ mathman : I guess you have to (at least formally), calculate $d(1/X_t)$ using multidimensional Itô's lemma and check under what conditions the result holds true. Best regards2012-06-15
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    @TheBridge : What is $1/X_t$ if $X_t$ is a matrix?2012-06-15
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    even in 1d , with b fixed and non zero this process will *always* hit zero. When $X_t$ gets small it looks like an ordinary brownian motion.2012-06-15
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    @mike : In 1-d this is a geometric Brownian motion, so does not hit zero.2012-06-15
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    not when you add a constant to the volatility. that's $b_t$ (little b) that should be non-zero.2012-06-15
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    @mike : Sorry, I see what you are saying now. Do you have a reference for proving that say $X_t = x + \int_0^t a_s ds + \int_0^t b_s dW_s$ hits zero a.s.?2012-06-18
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    almost any place that discusses the scale function, Breiman's probability book, Freedman's Brownian Motion & diffusion, Karlin & Taylor A second course in ...2012-06-19

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