Lecture Notes on Stochastic Calculus (Part II)Fabrizio Gelsomino, Olivier L ev eque, EPFLJuly 21, 2009Contents1Stochastic integrals31.1Itos integral with respect to the standard Brownian motion . . . . . . . . . . . . . . . . .31.2Wieners integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41.3Itos integral with respect to a martingale . . . . . . . . . . . . . . . . . . . . . . . . . . .42Ito-Doeblins formula(s)72.1First formulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72.2Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72.3Continuous semi-martingales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82.4Integration by parts formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92.5Back to Fisk-Stratonovi cs integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103Stochastic differential equations (SDEs)113.1Reminder on ordinary differential equations (ODEs) . . . . . . . . . . . . . . . . . . . . .113.2Time-homogeneous SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113.3Time-inhomogeneous SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133.4Weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144Change of probability measure154.1Exponential martingale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154.2Change of probability measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154.3Martingales under P and martingales underePT. . . . . . . . . . . . . . . . . . . . . . . .164.4Girsanovs theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .174.5First application to SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .184.6Second application to SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .194.7A particular case: the Black-Scholes model. . . . . . . . . . . . . . . . . . . . . . . . . .204.8Application : pricing of a European call option (Black-Scholes formula). . . . . . . . . .2115Relation between SDEs and PDEs235.1Forward PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .235.2Backward PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .245.3Generator of a diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265.4Markov property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .275.5Application: option pricing and hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . .286Multidimensional processes306.1Multidimensional Ito-Doeblins formula. . . . . . . . . . . . . . . . . . . . . . . . . . . .306.2Multidimensional SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .316.3Drift vector, diffusion matrix and weak solution . . . . . . . . . . . . . . . . . . . . . . . .336.4Existence of a martingale measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .346.5Relation between SDEs and PDEs in the multidimensional case . . . . . . . . . . . . . .367Local martingales377.1Preliminary: unbounded stopping times . . . . . . . . . . . . . . . . . . . . . . . . . . . .377.2Local martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .387.3When is a local martingale also a martingale? . . . . . . . . . . . . . . . . . . . . . . . . .407.4Change of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .437.5Local time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4421Stochastic integralsLet (Ft, t R+) be a filtration and (Bt, t R+) be a standard Brownian motion with respect to (Ft, t R+), that is :- B0= 0 a.s.- Btis Ft-measurable t R+(i.e., B is adapted to (Ft,t R+)- Bt Bs Fst s 0 (independent increments)- Bt Bs Bts B0t s 0 (stationary increments)- Bt N(0,t) t R+- B has continuous trajectories a.s.Reminder. In addition, B has the following properties :- (Bt, t R+) is a continuous ans square-integrable martingale with respect to (Ft, t R+), with quadratic variation hBit= t a.s. (i.e. B2t t is a martingale).- (Bt, t R+) is Gaussian process with mean E(Bt) = 0 and covariance Cov(Bt,Bs) = ts := min(t,s).- (Bt, t R+) is a Markov process with respect to (Ft, t R+), that is, E(g(Bt)|Fs) = E(g(Bt)|Bs) a.s., t s 0 and g : R R continuous and bounded.1.1Itos integral with respect to the standard Brownian motionLet (Ht, t R+) be a process with continuous trajectories adapted to (Ft, t R+) and such thatE?Zt0H2sds?0,P?(H(n) B)t (H B)t? ? n03Remark. In general, (H B) is not a Gaussian process; it does not have neither independent increments, nor stat