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Lecture Notes on Stochastic Calculus (Part II)

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Lecture Notes on Stochastic Calculus (Part II) Lecture Notes on Stochastic Calculus (Part II) Fabrizio Gelsomino, Olivier L´ev^eque,EPFL July 21, 2009 Contents 1 Stochastic integrals 3 1.1 Ito's integral with respect to the standard Brownian motion . 3 1.2 Wiener's integral . 4 1.3 Ito's integral with respect to a martingale . 4 2 Ito-Doeblin's formula(s) 7 2.1 First formulations . 7 2.2 Generalizations . 7 2.3 Continuous semi-martingales . 8 2.4 Integration by parts formula . 9 2.5 Back to Fisk-Stratonoviˇc'sintegral . 10 3 Stochastic differential equations (SDE's) 11 3.1 Reminder on ordinary differential equations (ODE's) . 11 3.2 Time-homogeneous SDE's . 11 3.3 Time-inhomogeneous SDE's . 13 3.4 Weak solutions . 14 4 Change of probability measure 15 4.1 Exponential martingale . 15 4.2 Change of probability measure . 15 4.3 Martingales under P and martingales under PeT ........................ 16 4.4 Girsanov's theorem . 17 4.5 First application to SDE's . 18 4.6 Second application to SDE's . 19 4.7 A particular case: the Black-Scholes model . 20 4.8 Application : pricing of a European call option (Black-Scholes formula) . 21 1 5 Relation between SDE's and PDE's 23 5.1 Forward PDE . 23 5.2 Backward PDE . 24 5.3 Generator of a diffusion . 26 5.4 Markov property . 27 5.5 Application: option pricing and hedging . 28 6 Multidimensional processes 30 6.1 Multidimensional Ito-Doeblin's formula . 30 6.2 Multidimensional SDE's . 31 6.3 Drift vector, diffusion matrix and weak solution . 33 6.4 Existence of a martingale measure . 34 6.5 Relation between SDE's and PDE's in the multidimensional case . 36 7 Local martingales 37 7.1 Preliminary: unbounded stopping times . 37 7.2 Local martingales . 38 7.3 When is a local martingale also a martingale? . 40 7.4 Change of time . 43 7.5 Local time . 44 2 1 Stochastic integrals Let (Ft; t 2 R+) be a filtration and (Bt; t 2 R+) be a standard Brownian motion with respect to (Ft; t 2 R+), that is : - B0 = 0 a:s: - Bt is Ft-measurable 8t 2 R+ (i.e., B is adapted to (Ft; t 2 R+)) - Bt − Bs ??Fs 8t > s ≥ 0 (independent increments) - Bt − Bs ∼ Bt−s − B0 8t > s ≥ 0 (stationary increments) - Bt ∼ N (0; t) 8t 2 R+ - B has continuous trajectories a:s: Reminder. In addition, B has the following properties : -(Bt; t 2 R+) is a continuous ans square-integrable martingale with respect to (Ft; t 2 R+), with 2 quadratic variation hBit = t a.s. (i.e. Bt − t is a martingale). -(Bt; t 2 R+) is Gaussian process with mean E(Bt) = 0 and covariance Cov(Bt;Bs) = t ^ s := min(t; s). -(Bt; t 2 R+) is a Markov process with respect to (Ft; t 2 R+), that is, E(g(Bt)jFs) = E(g(Bt)jBs) a.s., 8t > s ≥ 0 and g : R ! R continuous and bounded. 1.1 Ito's integral with respect to the standard Brownian motion Let (Ht; t 2 R+) be a process with continuous trajectories adapted to (Ft; t 2 R+) and such that Z t 2 E Hs ds < 1; 8t 2 R+: 0 R t It is then possible to define a process ((H · B)t ≡ 0 Hs dBs; t 2 R+) which satisfies the following properties (see lecture notes of the fall semester): Z t 2 2 - E((H · B)t) = 0; E((H · B)t ) = E Hs ds . 0 Z t^s 2 - Cov((H · B)t; (H · B)s) = E Hr dr . 0 - ((H · B)t; t 2 R+) is a continuous square-integrable martingale with respect to (Ft; t 2 R+), with quadratic variation Z t 2 h(H · B)it = Hs ds: 0 - Let n (n) X (n) (n) (n) (H · B)t = H(ti−1) B(ti ) − B(ti−1) ; i=1 (n) (n) (n) where 0 = t0 < t1 < : : : < tn = t is a sequence of partitions of [0; t] such that (n) (n) max jti − ti−1j ! 0: 1≤i≤n n!1 (n) P Then (H · B)t ! (H · B)t as n ! 1, that is, 8" > 0, (n) (H · B)t − (H · B)t > " ! 0 P n!1 3 Remark. In general,(H ·B) is not a Gaussian process; it does not have neither independent increments, R t 2 nor stationary increments. Moreover, h(H · B)it = 0 Hs ds is not deterministic. R t Remark. Processes such as 0 H(t; s) dBs are not martingales in general: at each time t, the integrand H changes. Nevertheless, the above isometry properties remains valid: Z t Z t 2! Z t 2 E H(t; s) dBs = 0; E H(t; s) dBs = E H(t; s) ds 0 0 0 and Z t Z s Z t^s Cov H(t; s) dBr; H(s; r) dBr = E H(t; r) H(s; r) dr : 0 0 0 1.2 Wiener's integral R t 2 Let f : R+ ! R be a deterministic continuous function (so 0 f(s) ds < 1, 8t 2 R+). Then the process ((f · B)t; t 2 R+), in addition of all the above properties (f is a particular case of H), satisfies also: -(f · B) is a Gaussian process, with mean and covariance: Z t^s 2 E((f · B)t) = 0; Cov((f · B)t; (f · B)s) = f(r) dr: 0 -(f · B) has independent increments. R t 2 - h(f · B)it = 0 f(s) ds is deterministic. R t Remark. In general, (f · B) does not have stationary increments and processes such as 0 f(t; s) dBs do not have independent increments. 1.3 Ito's integral with respect to a martingale Let (Ft; t 2 R+) be a filtration and (Mt; t 2 R+) be a continuous square-integrable martingale with respect to (Ft; t 2 R+). Reminder. The quadratic variation of M is the unique process (hMit; t 2 R+) which is increasing, 2 continuous and adapted to (Ft; t 2 R+), such that hMi0 = 0 a.s. and (Mt − hMit; t 2 R+) is a martingale with respect to (Ft; t 2 R+). Lemma 1.1. For all t > s ≥ 0, 2 E (Mt − Ms) j Fs = E(hMit − hMis j Fs): Proof. 2 2 2 2 2 2 E (Mt − Ms) j Fs = E(Mt − 2MtMs + Ms j Fs ) = E(Mt j Fs) − 2E(Mt j Fs)Ms + Ms 2 2 2 2 2 = E(Mt − hMit + hMit j Fs) − 2Ms + Ms = Ms − hMis + E(hMit j Fs) − Ms = E(hMit − hMis j Fs): R t Remarks. - Since t 7! hMit is increasing, it is a process with bounded variation, so 0 Hs dhMis is a well-defined Riemann-Stieltjes integral, as long as H has continuous trajectories. 2 - In general, hMit is not deterministic, but when M has independent increments, then hMit = E(Mt ) − 2 E(M0 ) (and is therefore deterministic). 4 Let (Ht; t 2 R+) be a continuous process adapted to (Ft; t 2 R+) such that Z t 2 E Hs dhMis < 1; 8t 2 R+: 0 R t It is then possible to define a process ((H · M)t ≡ 0 Hs dMs; t 2 R+) which satisfies the following properties: Z t 2 2 - E((H · M)t) = 0; E((H · B)t ) = E Hs dhMis . 0 Z t^s 2 - Cov((H · M)t; (H · M)s) = E Hr dhMis . 0 - ((H · M)t; t 2 R+) is a continuous square-integrable martingale with respect to (Ft; t 2 R+), with quadratic variation Z t 2 h(H · M)it = Hs dhMis: 0 - Let n (n) X (n) (n) (n) (H · M)t = H(ti−1) M(ti ) − M(ti−1) ; i=1 (n) (n) (n) where 0 = t0 < t1 < : : : < tn = t is a sequence of partitions of [0; t] such that (n) (n) n!1 max jti − ti−1j ! 0: 1≤i≤n (n) P Then (H · M t −! (H · M)t as n ! 1, that is, 8" > 0, (n) (H · M)t − (H · M)t > " ! 0: P n!1 Let us give here a short explanation regarding the construction of the integral in this case and the isometry property. For a simple predictable process of the form n X Hs(!) = Xi(!) 1]ti−1;ti](s); s 2 [0; t]; i=1 where 0 = t0 < t1 < : : : < tn = t is a partition of [0; t] and Xi is Fti−1 -measurable and bounded, the stochastic integral H · M is defined as n X (H · M)t = Xi (M(ti) − M(ti−1)): i=1 Let us then compute n 2 X E (H · M)t = E(Xi Xj (M(ti) − M(ti−1)) (M(tj) − M(tj−1))) i;j=1 n X 2 2 = E(E(Xi (M(ti) − M(ti−1) j Fti−1 )) i=1 X +2 E(E(Xi Xj (M(ti) − M(ti−1)) (M(tj) − M(tj−1)) j Ftj−1 )): i<j Since Xi is Ftj−1 -measurable and Xi, Xj and M(ti) − M(ti−1) are Ftj−1 -measurable for i < j, we have n 2 X 2 2 E (H · M)t = E(Xi E((M(ti) − M(ti−1)) j Fti−1 )) i=1 X +2 E(Xi Xj (M(ti) − M(ti−1)) E(M(tj) − M(tj−1) j Ftj−1 )): i<j 5 Since M is a martingale, E(M(tj) − M(tj−1) j Ftj−1 ) = 0, so the second term on the right-hand side drops.
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