Abstract
The present paper attains a Harnack inequality for the option pricing (or Kolmogorov) equation with gradient estimate arguments. We then perform a no-arbitrage analysis of a financial market.
Highlights
The study of Harnack inequality has been an active field in the past three decades
Harnack inequality states that the values of the nonnegative solution to a harmonic function are comparable
Carciola et al [3] proved the Harnack inequality of option pricing equations based on the fundamental solution of the Kolmogorov operator equations and studied the no-arbitrage bounds for a financial market by using this Harnack inequality
Summary
The study of Harnack inequality has been an active field in the past three decades. Harnack inequality states that the values of the nonnegative solution to a harmonic function are comparable. Many articles have studied the no-arbitrage condition mainly by functional analysis, the Martingale approach, and convex optimization. These articles yield plentiful rich results [6,7,8]. Carciola et al [3] proved the Harnack inequality of option pricing (or Kolmogorov operator) equations based on the fundamental solution of the Kolmogorov operator equations and studied the no-arbitrage bounds for a financial market by using this Harnack inequality. We consider a similar no-arbitrage problem to some Kolmogorov-like operator equations based on a Harnack inequality with gradient estimate arguments instead of the argument of a fundamental solution for a financial market.
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