Rate-optimal tests for jumps in diffusion processes

Suppose one has a sample of high-frequency intraday discrete observations of a continuous time random process, such as foreign exchange rates and stock prices, and wants to test for the presence of jumps in the process. We show that the power of any test of this hypothesis depends on the frequency of observation. In particular, if the process is observed at intervals of length <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$1/n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>n</mi> </mrow> </math> </EquationSource> </InlineEquation> and the instantaneous volatility of the process is given by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$ \sigma _{t}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi mathvariant="italic">σ</mi> <mi>t</mi> </msub> </math> </EquationSource> </InlineEquation>, we show that at best one can detect jumps of height no smaller than <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\sigma _{t}\sqrt{2\log (n)/n}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi mathvariant="italic">σ</mi> <mi>t</mi> </msub> <msqrt> <mrow> <mn>2</mn> <mo>log</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mi>n</mi> </mrow> </msqrt> </mrow> </math> </EquationSource> </InlineEquation>. We present a new test which achieves this rate for diffusion-type processes, and examine its finite-sample properties using simulations. Copyright Springer-Verlag Berlin Heidelberg 2013