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Series of three lectures on \\
{\bf Simple SDE, SPDE, and BSDE models dealing with problems of
climate dynamics and related risk}\\
by Peter Imkeller\\ HU Berlin\\
(joint work with Stefan Ankirchner, Claudia Hein, Michael H\"ogele,
Ying Hu, Matthias M\"uller, Ilya Pavlyukevich, Alexandre Popier,
Goncalo Nu\~nes dos Reis, Torsten Wetzel)
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{\bf \large 1. Meta-stability in some S(P)DE related to simple
climate models}

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\begin{abstract}
Simple models of the earth's energy balance are able to interpret
some qualitative aspects of the dynamics of paleo-climatic data. In
the 1980s this led to the investigation of periodically forced
dynamical systems of the reaction-diffusion type with small Gaussian
noise, and a rough explanation of glacial cycles by Gaussian
meta-stability. A spectral analysis of Greenland ice time series
performed at the end of the 1990s representing average temperatures
during the last ice age suggest an $\alpha-$stable noise component
with an $\alpha\sim 1.75.$ Based on this observation, papers in the
physics literature attempted an interpretation featuring dynamical
systems perturbed by small L\'evy noise. We study exit and
transition between meta-stable states for solutions of stochastic
differential equations and stochastic reaction-diffusion equations
derived from this prototype. Due to the heavy-tail nature of the
$\alpha$-stable component of the noise, the results for L\'evy noise
differ strongly from the well known case of purely Gaussian
perturbations. For SPDE, transitions are governed by the modes with
the largest jumps.
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{\bf \large 2. Martingale optimality, BSDE and cross hedging of
insurance derivatives}
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\begin{abstract}
A financial market model is considered on which agents (e.g.
insurers) are subject to an exogenous financial risk, which they
trade by issuing a risk bond. Typical risk sources are climate or
weather. Buyers of the bond are able to invest in a market asset
correlated with the exogenous risk. We investigate their utility
maximization problem with respect to the correlation, and calculate
bond prices using utility indifference. This hedging concept is
interpreted by means of martingale optimality, and solved with BSDE
tools. Prices are seen to decrease as a result of dynamic hedging.
The increments are interpreted in terms of diversification pressure.
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{\bf \large 3. Meta-stability in SDE related to simple climate
models: model selection; the light tail limit of L\'evy noise}
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Interpreting paleo-climatic time series by simple dynamical systems
with noise leads to statistical model selection problems. For
instance, one needs an efficient testing method for the best fitting
$\alpha$-stable noise component. We develop a statistical testing
method based on the $p$-variation of the solution trajectories of
SDE with L\'evy noise, for example by showing asymptotic normality
or asymptotic $\beta$-stability of their approximations along finite
interval partitions.

It has been suggested that the exit and transition characteristics
of dynamical systems perturbed by small L\'evy noise approach
Gaussian behavior as the heavy tails of their jump laws become
exponentially light of order $\gamma$, i.e. if for $x\to\infty$ they
are given by $\exp(-c x^\gamma)$, and as $\gamma \to 2.$ We show
that this is surprisingly false, by exhibiting an intriguing phase
transition at $\gamma = 1$.
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