ON FIRST-PASSAGE PROBLEMS FOR ASYMMETRIC DIFFUSIONS

X(t) a temporally homogeneous, one-dimensional diffusion process over I = (−b, a), (a, b> 0) which is the solution of the SDE: dX(t) = µ(X(t))dt+ σ(X(t))dBt, X(0) = x ∈ I Bt a standard Brownian motion (BM). • µ(x) = µ+(x) if x> 0, µ(x) = µ−(x) if x < 0 • σ(x) = σ+(x) if x> 0, σ(x) = σ−(x) if x < 0 • µ±(x) and σ±(x) regular enough functions. X(t) turns out to be a sign-dependent diffusion. • µ(x) and σ(x) are not assigned at x = 0. We can take µ±(x) and σ±(x) as parametric functions of the state x (e.g. polynomials in the unknown x, with coeffi-cients θ±i, i = 0,..., n and η j, j = 0,...,m, respectively). We suppose that, when X(t) hits 0, it goes right to the position δ> 0 with probability p> 0 and left to the position −δ with probability q = 1−p: the process is reflected at zero rightward or leftward. As a generalization, the infinitesimal coefficients may change when the process X(t) crosses any given barrier, not necessarily the origin. 2 Applications in Mathematical Finance Suppose that X̃(t) is the price of a stock or a commodity, such as gold or oil, or an interest rate, at time t. In practice, when X̃(t) reaches a certain threshold S, it may become much more volatile (or much less volatile). The same thing occurs for the drift term. Thus, volatility and drift may undergo a sharp variation, when X̃(t) crosses the threshold S. If X̃(t) is asymmetric diffusion at S: dX̃(t) = µ̃(X̃(t))dt+ σ̃(X̃(t))dBt, X̃(0) = x • µ̃(x) = µ̃+(x) if x> S, µ̃(x) = µ̃−(x) if x < S • σ̃(x) = σ̃+(x) if x> S, σ̃(x) = σ̃−(x) if x < S Setting X(t) = X̃(t) − S, the process X(t) turns out to be asym-metric diffusion at zero, with infinitesimal parameters µ(x), σ(x), changing when X(t) crosses the origin. dX(t) = µ(X(t))dt+ σ(X(t))dBt where µ(x) = µ̃(x+ S), σ(x) = σ̃(x+ S) Special case: µ ± and σ ± linear functions of x (that is µ±(x) = θ±1 x+ θ 0,