Does dollar cost averaging actually have any advantage over one time investment? Some banks promote savings plans by stating that investing a fixed amount of money in regular time intervals minimizes the average buying price or the downside risk (dollar cost averaging). Since the regular saving amount is fixed, they claim that, when the stock price is low more shares will be bought, and when the price is high fewer shares will be bought, so in total, you will buy more shares when they are cheap and fewer when they are expensive, hence investing more efficiently.

Although this seems to be a valid argument, intuitively (at least for me) it still feels rather like some sort of fallacy than a sound investment strategy, however I cannot show exactly why. So my question is whether there is an actual mathematical proof that dollar cost averaging has a definite advantage over one time investment (or the opposite)? To be precise, is it really possible to somehow reduce risk and keep the same expected return (or raise expected return and keep the same risk) using this stragegy?
The seminal academic criticism of dollar cost averaging on many specifications of economic conditions is A Note on the Suboptimality of Dollar-Cost Averaging as an Investment Policy (Constantinides (1979)). You might also be interested in these papers:


  Dollar Cost Averaging is an investment system that is widely advocated
  by brokerage firms and mutual funds. In its best known form, an
  investor seeking to put a lump sum into risky assets is counseled to
  invest the money over a period of time in equal installments in order
  to avoid the devastating effect of a market fall immediately after a
  single, lump-sum investment. Using graphical analysis, historical
  stock market returns, and Monte Carlo simulations, this article
  demonstrates that no such benefit accrues to a Dollar Cost Averaging
  Strategy. Two alternative strategies, optimal rebalancing and buy and
  hold achieve better performance in all three analyses.


Nobody gains from dollar cost averaging analytical, numerical and empirical results (Knight and Mandell (1993))


  Some studies find the dollar-cost averaging investment strategy to be
  sub-optimal using a traditional Sharpe ratio performance ranking
  metric. Using both the Sortino ratio and the Upside Potential ratio,
  we empirically test four investment strategies for alternative asset
  investments. We find the relative ranking of dollar-cost averaging
  remains inferior to alternative investment strategies. (JEL G1, G11,
  N2)


An empirical examination of the effectiveness of dollar-cost averaging using downside risk performance measures (Leggio and Lien (2003)) 


  The widespread practice of dollar-cost averaging (DCA) amongst the
  investing public, has puzzled most financial economists, ever since
  Constantinides demonstrated the dynamic inefficiency of this
  strategy under very general conditions. This enduring phenomena has
  forced researchers, such as Statman , to suggest behavioral
  explanations for DCA's popularity, predicated on the prospect theory
  of Kahneman and Tversky .
  
  In this paper we reexamine the payoff structure of DCA via
  continuous-time financial mathematics and then ask the question: Is it
  possible to reconcile the theory and practice of dollar-cost
  averaging?
  
  To answer this question, we take a slightly different approach to the
  issue by using the tools of stochastic calculus and Brownian bridges.
  We demonstrate that engaging in a dollar-cost averaging strategy is
  akin to purchasing a zero strike arithmetic Asian option on the
  underlying security. In other words, people who engage in dollar-cost
  averaging are implicitly purchasing a path-dependent contingent claim.
  We then prove that the expected return from this exotic option — i.e.
  the DCA strategy — conditional on knowing the final value of the
  security will uniformly exceed the return from the underlying security
  for all sufficiently large volatilities.
  
  This leads us to argue that investors may be dollar-cost averaging
  because they have "target prices" for the underlying asset price. The
  strategy of dollar-cost averaging would then exceed the returns from
  lump-sum investing, based on their subjective conditional expectation.
  In fact, the more volatile the underlying security, the greater is the
  benefit to dollar-cost averaging — conditional on knowing the final
  value — which is consistent with common practice.


A CONTINUOUS-TIME REEXAMINATION OF DOLLAR-COST AVERAGING (MILEVSKY and POSNER (2003))


  Dollar Cost Averaging is a strategy for purchasing equity securities
  that is widely recommended by professional investment advisors and
  commentators, but which has been virtually ignored by academic
  theorists and textbook writers. In this paper we explore whether the
  strategy is but another instance of irrational behavior by individual
  investors, or whether it is an investment heuristic that has survival
  value in an environment in which security prices exhibit mean
  reversion behavior that has only belatedly been recognized by academic
  theorists. Our evidence supports the view that the uninformed
  individual investors who follow this strategy in purchasing individual
  stocks to add to an existing portfolio are better off than if they
  followed the ‘rational’ strategies traditionally recommended by
  academics.


Dollar Cost Averaging (Brennan, Li and Torous  (2005))