The 3,3-dimethyl-2-(2-methyl-1-propenyl)cyclopropanecarboxylic acid compounds of formula (1):
wherein R is hydrogen, substituted or unsubstituted alkyl, substituted or unsubstituted aryl, or substituted or unsubstituted aralkyl, are very important compounds as the intermediate for the synthesis of pyrethroid-type household agents for epidemic prevention, pesticides, or the like.  There have also been developed a number of analogs in which the 2-methyl-1-propenyl groups attached to the cyclopropane rings are replaced with various alkenyl groups, using the 3,3-dimethyl-2-(2-methyl-1-propenyl)cyclopropanecarboxylic acid compound of the above formula (1) as the key compound, as well as a number of household agents for epidemic prevention, pesticides, and the like using these analogs. 
As the process for producing the analogs in which the 2-methyl-1-propenyl groups attached to the cyclopropane rings are replaced with various alkenyl groups, there has been known, for example, a process in which the 3,3-dimethyl-2-formylcyclopropanecarboxylic acid derivative of formula (2):
wherein R is as defined above, are reacted with Wittig reagents (see, e.g., J. Labelled Compounds and Radiopharmaceuticals, 13, 561(1977)).  The 3,3-dimethyl-2-formylcyclopropanecarboxylic acid derivatives of the above formula (2) become important compounds in the synthesis of the above analogs. 
As the processes for the production of the 3,3-dimethyl-2-formylcyclopropanecarboxylic acid derivative of formula (2), there have been known, for example, a process in which the 3,3-dimethyl-2-(2-methyl-1-propenyl)-cyclopropanecarboxylic acid compound of the above formula (1) are oxidized in the presence of an osmium tetroxide catalyst (see, e.g., J. Labelled Compounds and Radiopharmaceuticals, 13, 561(1977)) and a process in which the 3,3-dimethyl-2-(2-methyl-1-propenyl)cyclopropanecarboxylic acid compounds of the above formula (1) are oxidized with ozone (see, e.g., JP-B 46-24695).  However, since the former process uses highly toxic osmium tetroxide and the latter process has a tendency to need large-scale equipment, both cannot be said to be production processes suitable on an industrial scale. 
Journal of the Chemical Society, Perkin Transactions I, 1980 pages 1711-1717 discloses a process for preparing cis-2,2-formyl-3,3-diemethylcyclopropanecarboxylate by oxidizing a chrysanthemate with sodium metaperiodate in the presence of the catalyst, osmium tetraoxide.  However, this process also uses highly toxic osmium tetroxide and cannot be said to be a process suitable on an industrial scale.  EP-A 0 444 708 discloses a process for preparing a ketone or aldehyde compound by oxidizing an olefin compound having a β-lactam structure.  JP-A 5-229981 discloses a process for preparing an aromatic acetaldehyde by oxidizing an allyl substituted aromatic compound with sodium periodate in the presence of a ruthenium catalyst and a phase transfer catalyst.  JP-A 55-087739 discloses a process for preparing an aromatic aldehyde by oxidizing an α, β unsaturated aromatic compound in the presence of an oxidizing agent and a ruthenium catalyst.  However, the oxidization disclosed in these processes are not compared to the above oxidization of the compound of the above formula (1).