Model set up Sample Clauses

Model set up. We consider a market consisting of two correlated assets S and V with discounted prices given by t dSt = (µ1 − r)Stdt + σ1StdW 1 t t dVt = (µ2 − r)Vtdt + σ2Vt(ρdW 1 + √1 − ρ2dW 2), (1) for t0 ≤ t ≤ T ≤ ∞, where W = (W 1, W 2) is a standard two–dimensional Brownian motion. We suppose further that the client can trade dynamically by holding Ht units of the asset St and investing the remaining of his wealth in a bank account with normalized value Bt = er(t−t0 ) for a constant interest rate r. It follows that the discounted value of the corresponding self– financing portfolio satisfies dXπ = πt(µ1 − r)dt + πtσ1dW 1, t0 ≤ t ≤ T, (2) t t where πt = HtSt. At a given time t0, the client borrows an amount L from a bank leaving the asset with value Vt0 as a collateral. We assume that the bank collects the dividends paid by the underlying asset V at a rate δ for the duration of the loan. In addition, the bank charges the client a fee c and stipulates an interest rate α to be charged on the loan amount L, so that the client can redeem the asset with value er(t−t0 )Vt at time t0 ≤ t ≤ T by paying an amount eα(t−t0 )L. At the maturity time T , we assume that the client needs to decide between repaying the loan or forfeiting the underlying asset indefinitely. In other words, at the beginning of the loan the client gives the bank an asset worth Vt0 and receives a net amount (L − c) plus the option to buy back an asset with market price er(t−t0 )Vt for an amount eα(t−t0 )L. Denoting the cost of this option for the bank by Ct0 , the loan parameters are related by c = L + Ct0 − Vt0 (3)

Model set up. The simplest approximation of the structural model developed in the previous chapter is a single target fault (Porthtowan fault zone) embedded into an impermeable matrix. Despite its simplicity, this type of model appears to be a realistic representation of other EGS reservoirs (Chapter 2.2). Following the numerical modelling approach of Xxxxxx et al. (2010), the conceptual model of the Porthtowan fault zone, represented by a single subvertical rectangular structure, was implemented into a hydro-mechanical numerical model which was set up using the finite element software COMSOL Multiphysics®. With this model, hydraulic pressure diffusion, seismic failure and mechanical stress redistribution can be simulated. The fault structure is intersected by an injection well at a depth of 2500 m and by a production well at a depth of 4500 m. The fault is divided into quadratic patches of 20 m side length that, assuming seismic behaviour, slip whenever the ratio of shear to effective normal stress exceeds the frictional strength of the patch. Each patch may slip independently, but since the patches are mechanically coupled to their neighbours via a so- called “block-spring model”, stress redistribution occurs (see Chapter 3.1.1, Figure 4). This may cause the neighbouring patches to experience overcritical stress conditions, thus triggering further co-seismic slips. With each slip event seismic energy is released, whereas the total amount of energy and, correspondingly, the seismic moment are proportional to the size of the slipped area, i.e. to the sum of slipped patches. During slip, a local stress drop occurs which restores the subcritical state of the patches.