{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

HO3-sol-08

# HO3-sol-08 - Math 236 Stochastic Dierential Equations...

This preview shows pages 1–2. Sign up to view the full content.

Math 236, Stochastic Differential Equations Solutions to problem set 3 February 17, 2009 Problem 1 Let X t be the one-dimensional diffusion process satisfying the Ito SDE dX t = b ( X t ) dt + σ ( X t ) dB t , X 0 = x where b ( x ) and σ ( x ) satisfy the Ito conditions (linear growth bound and Lipschitz condition). Let x ( α, β ) and let τ = τ x = inf { t > 0 : X t [ α, β ] c } be the exit time from the interval [ α, β ] starting from x in its interior, and let L be the generator of the process L = 1 2 σ 2 ( x ) 2 ∂x 2 + b ( x ) ∂x Show using Ito’s formula and the optional stopping theorem that if there is a non-negative and bounded function u ( x ) for x [ α, β ] such that L u ( x ) ≤ − 1, then E x { τ } < and so τ x < with probability one. Solution First we note that t τ are stopping times that are bounded since t τ t . From Ito’s formula we have du ( X s ) = L u ( X s ) + σ ( X s ) u ( X s ) dB s and integrating u ( X t τ ) = u ( x ) + integraldisplay t τ 0 L u ( X s ) ds + integraldisplay t τ 0 σ ( X s ) u x ( X s ) dB s Assuming that u x is bounded (not explicitly stated in the statement), taking expectations and using the optional stopping theorem we have E x { u ( X t τ ) } = u ( x ) + E x { integraldisplay t τ 0 L u ( X s ) ds } Using the hypotheses on u we have therefore that E x { t τ } ≤ u ( x ) Since t τ τ

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}