A. Kyprianou: errata

Errata for publications and preprints.

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There is no such thing as a perfectly written paper. Mistakes are made (and sometimes propogated from other papers). There is no point in being shy about it so here are the mistakes that I have found in my papers which might cause some confusion.

• Slow variation and uniqueness of solutions to the functional equation in the branching random walk. J. Appl. Probab. (1998) 35 795--802.
  The following unfortunate errors went undetected in the original manuscript ! They do not affect the proof or validity of the argumet if corrected as below.
  • p798, 7th line from bottom: remove words 'and in mean'.
  • p798, 6th line from bottom: 'transient' should read 'oscillating'.
  • p799, 6th line from top: remove words 'in mean'.
  
  The following point was brought to my attention by Dr. David Grey for which I am grateful. On pp798, in the construction of $g(\beta)$, it is not clear that $g$ will be finite for all $\beta$. The left hand side of equation (5) is finite for all $\beta\geq 1$ becasue $L$ is monotone decreasing. It is also valid when the sum on the right hand side is taken over individuals in the $n$th generation since $\Psi$ also satisfies the functional equation where the product is over the $n$th generation. As we know the terms $y_u$ are almost surely tending to zero as $n$ tends to infinity, then in the '$n$th generation' version of (5) on the RHS we can identify that g must be finite for arguments less than one.

• Some remarks on first passage of Lévy process, the American put and pasting principles. (with Larbi Alili) Annals of Applied Probability (2005) 15, 2062--2080.
  Some typos which will affect the reading of Remark 10.
  • p2077 10th line from the top. 'strictly smaller' should be 'greater'
  • p2077 17th line from top. 'less than' should be 'greater than'
  • p2077 21th line from top. 'greater than' should be 'less than'.
  • p2077 Throughout Remark 10 when talking of the variable 'y' one actually should read 'e^y'.
• Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels. (with B. Surya) Finance and Stochastics. (2007), 11, 131-152. In the proof of Theorem 5.1 it is claimed that $\Theta^{(q)}(x)$ is a potential density and therefore decreasing in $q$. This claim is correct, but it is not quite the potential density of the decending ladder height process. In the expression on p144 of the Finance and Stochastics article, or p15 of the PDF link just above which displays the role of $\Theta^{(q)}(x)$ as a density, on the left hand side of that identity the exponent $qt$ should be replaced by $q L^{-1}_t$.
• Spine changes of measure. This is an error which has propogated through many papers concerning this topic (not just my own papers). It is commonly known that a change of measure using an additive martingale for a given branching process induces a spine decomposition. It is often asserted that if $W_t$ is the additive martingale then $1/W_t$ is a martingale under the change of measure. This is in general not true. It is true if there is no extinction with probability one. Otherwise the best one can say is that $1/W_t$ is a supermartingale. Lucikly this does not affect any of the arguments in any of the papers that I know of which use spine decompositions.

  The above observation is thanks to Matt Roberts (a PhD student at Bath University) in late 2006.

• Some calculations for Israeli options. Finance and Stochastics. (2004), 8, 73 - 86. The claim in Remark 5 regarding the constant k* should be considered in the light of the final Remark of The Shepp-Shiryaev stochastic game driven by a spectrally negative Lévy process. (with Erik Baurdoux)