Space-time analyticity of weak solutions to linear parabolic systems with variable coefficients

作者：Takac Peter^*

来源：Journal of Functional Analysis, 2012, 263(1): 50-88.

DOI：10.1016/j.jfa.2012.04.008

• Space-time analyticity
• Parabolic PDE
• Holomorphic continuation
• Hardy space
• Market completeness
• European option

Analytic smoothing properties of a general, strongly parabolic linear Cauchy problem of second order in R-N x (0, T) with analytic coefficients (in space and time variables) are investigated. They are expressed in terms of holomorphic continuation of global (weak) L-2-type solutions to the system. Given 0 %26lt; T%26apos; %26lt; T %26lt;= infinity, it is proved that any L-2-type solution u : R-N x (0, T) -%26gt; R-M possesses a bounded holomorphic continuation u(x + iy, sigma - i tau) into a complex domain in C-N x C defined by (x, sigma) is an element of R-N x (T%26apos;, T), vertical bar y vertical bar %26lt; A%26apos; and vertical bar tau vertical bar %26lt; B%26apos;, where A%26apos;, B%26apos; %26gt; 0 are constants depending upon T%26apos;. The proof uses the extension of a solution to an L-2-type solution in a domain in C-N x C. such that this extension satisfies the Cauchy-Riemann equations. The holomorphic extension is thus obtained in a Hardy space H-2. Applications include market completion by European options in Finance.

• 出版日期2012-7-1