Stochastic differantial equations and an application to portfolio analysis

Stochastic Differential Equations and An Application to Portfolio Analysis SUMMARY Key Words: Stochastic Integral, Ito Formula, Stochastic Differential Equation, Wiener Process In this study, it is introduced an application of stochastic differential equations in portfolio analysis. Firstly, it is given the definition of stochastic integral with respect to wiener (brownion motion) process and the definition of stochastic differential with respect to stochastic function S(t) as follows dS(t)=a(t)dt+a(t)dW(t) (1) where a(t) and o(t) are the coefficient of equation (1). It follows Ito Formula, important in stochastic analysis, given by df(S(t),tMft(S(t),t)+^^ where fix,t) are twice continuously differentiable in xe(0,oo) and once continuously differentiable in te[0,T]. In the last chapter, it is studied on that how one can determine an investment model including stock and bond. The stock whose price S(t) per share at time t is governed by the stochastic diferential equation dS(t)=n S(t)dt + o S(t)dW(t), t>0 where \&0, oX) are constants, and n may be interpreted as the mean rate of return, and o2 may be interpreted as the variance of the rate of return. The bond is a riskless asset whose price 0(t) per share at time t is given by P(t) = 0(O)ert, where r>0 is the interest rate and (3(0) is a positive constant. Finally, it is indicated that how one might guess the rational value and identify an associated self-financing strategy. viii