In this talk, I will introduce a Brownian motion to the Lotka-Volterra system within its economical interpretation, the Goodwin model:

\[

\begin{array}{ccl}

dx_t & = & x_t (\Phi(y_tt) − \alpha + v^2(y_t ))dt + x_t v(y_t )dW_t \\

dy_t & = & y_t (\kappa(x_t ) − \gamma + v^2(y_t))dt + y_t v(y_t)dW_t\\

\end{array} \]

We motivate this by heterogeneity of workers productivity growth rate in the capitalist economy. We expect it to provide a richer set of possible trajectories than the deterministic counterpart, the latter allowing a rather simple form for business cycles and growth rate evolution. The addition of stochasticity, in the treatment of this system implies different methods also. This model is thus an opportunity to manipulate stochastic Lyapunov techniques. First, we prove the existence of a solution to the system, even with $\phi$ or $\kappa$ exploding for finite values. Then we provide estimates using a Lyapunov function for the deterministic system. Finally, we study the trajectories of the system.

In the deterministic system, we obtain closed orbits around a non-hyperbolic equilibrium point. Here, we obtain successive recurrent domains around a non stable point. This provides an opportunity to develop additional stochastic stability results. We prove under a specific assumption that trajectories indeed loop around the centre in finite time almost surely. However, other regimes of variations are possible.

This is part of a Research Project with the Fields Institute for Research in Mathematical Sciences.

Guaranteed minimum maturity benefits (GMMB) are similar to financial options on the fund value of a variable annuity. They are usually financed through a fee set as a fixed percentage of the fund, which is paid continuously throughout the term of the contract. From the policyholder's point of view, when the option is deep out-of-the-money, this fixed fee represents an important incentive to lapse. We first present an integral form for the value of a contract with a surrender option, and we analyse the optimal surrender region. We discuss the link between the continuous fee and the value of the surrender option, and present ideas for more advanced fee structures that aim to decrease the value of the surrender option, thus reducing the incentive to lapse. In particular, we introduce a new type of fee structure for variable annuities, where the fee rate depends on the moneyness of the guarantee. We present some numerical examples and sensitivity analysis.

The implementation of the convolution method for the numerical solution of backward stochastic differential equations (BSDEs) presented in Hyndman and Oyono Ngou (2013) uses a uniform space grid. Locally, this approach produces a truncation error, a space discretization error and an additional extrapolation error. Even if the extrapolation error is convergent in time, the resulting absolute error may be high at the boundaries of the uniform space grid. In order to solve this problem, we propose a tree-like grid for the space discretization which suppresses the extrapolation error leading to a globally convergent numerical solution for the BSDE. On this alternative grid the conditional expectations involved in the BSDE time discretization are computed using Fourier analysis and the fast Fourier transform (FFT) algorithm as in the initial implementation. The method is then extended to reflected BSDEs and numerical results demonstrating convergence are presented.

We will review the Watts' friendship cascade model from Watts (2006) and the extension of the Watts' friendship cascade model with random edge weights as specified by Hurd and Gleeson (2012). We will discuss the implecations of cascades in these models when viewed as part of a financial network. Unlike most cascade models, both of these models can be analyzed analytically when the LTIA condition is satisfied. This iterative method as described in Hurd and Gleeson (2012) requires the initial state vector to satisfy the LTIA condition but does not produce a state vector satisfying the LTIA condition as a solution, forcing the initial condition to be remembered throughout the entire process. We propose a Markovian approach towards analysing the extended Watts model by reducing the information contained in the state vector after each iteration. We will demonstrate the ability for this method to analyze cascades models with multiple cascade mechanisms, such as stress, exogenous shocks, and defaults, on a stylized financial network, leading to a more accurate measure of systemic risk.

This is joint work with T. Hurd, and H. Cheng.