Stochastic Processes and Financial Mathematics
(part two)
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19.5 Exercises on Chapter 19 \(\msconly \)
On random graphs and debt contagion
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19.1 Consider the following graph \(G\). Write down the distribution of \(D_G\).
If we sample a uniformly random edge, and position ourselves at the head of this edge, what is the distribution of the out-degree \(O\) of the (random) node that we end up at?
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19.2 Consider the following graph, as a banking network in the Gai-Kapadia model (as described in Section 19.2).
Let the contagion probabilities be \(\eta _j=\frac 1j\). Suppose that the bank marked \(X\) fails. What is the probability that the bank marked \(Y\) also fails?
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19.3 Consider the following graph, as a banking network in the Gai-Kapadia model.
Let the contagion probabilities be \(\eta _j=\frac 1j\). Suppose that the bank marked \(X\) fails. What is the probability that the bank marked \(Y\) fails?
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19.4 Consider the following graph (known as a binary tree) as a banking network in the Gai-Kapadia model.
This graph is a tree, with infinitely many nodes, in which every node except for \(v_0\) has one in-edge and two out-edges.
Let the contagion probabilities be \(\eta _j=\alpha \), where \(\alpha \in (0,1)\) is constant. Suppose that the bank marked \(V_0\) fails. Explain how the cascade of defaults that results can be represented
as a Galton-Watson process.
We say that a ‘catastrophic default’ occurs if an infinite number of banks fail. Under what condition on \(\alpha \) does this event have positive probability?
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19.5 Consider the following graph, as a banking network in the Gai-Kapadia model.
Let the contagion probabilities be \(\eta _j=\frac {1}{1+j}\). Suppose that the bank marked \(A\) fails.
