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MAST31801 Mathematical Finance II, Spring 2019. Problem Sheet 8 (4-8.4.2019)

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MAST31801 Mathematical Finance II, Spring 2019. Problem Sheet 8 (4-8.4.2019) UH|Master program Mathematics and Statistics|MAST31801 Mathematical Finance II, Spring 2019. Problem sheet 8 (4-8.4.2019) 1. Show that a local martingale (Xt : t ≥ 0) in a filtration F, which is bounded from below is a supermartingale. Hint: use localization together with Fatou lemma. 2. A non-negative local martingale Xt ≥ 0 is a martingale if and only if E(Xt) = E(X0) 8t. 3. Recall Ito formula, for f(x; s) 2 C2;1 and a continuous semimartingale Xt with quadratic covariation hXit 1 Z t @2f Z t @f Z t @f f(Xt; t) − f(X0; 0) − 2 (Xs; s)dhXis − (Xs; s)ds = (Xs; s)dXs 2 0 @x 0 @s 0 @x where on the right side we have the Ito integral. We can also understand the Ito integral as limit in probability of Riemann sums X @f n n n n X(tk−1); tk−1 X(tk ^ t) − X(tk−1 ^ t) n n @x tk 2Π among a sequence of partitions Πn with ∆(Πn) ! 0. Recalling that the Brownian motion Wt has quadratic variation hW it = t, write the following Ito integrals explicitely by using Ito formula: R t n (a) 0 Ws dWs, n 2 N. R t (b) 0 exp σWs σdWs R t 2 (c) 0 exp σWs − σ s=2 σdWs R t (d) 0 sin(σWs)dWs R t (e) 0 cos(σWs)dWs 4. These Ito integrals as stochastic processes are semimartingales. Com- pute the quadratic variation of the Ito-integrals above. Hint: recall that the finite variation part of a semimartingale has zero quadratic variation, and the quadratic variation is bilinear. For con- tinuous semimartingales only the local martingale part contributes to the quadratic variation, and Z · @f Z t@f 2 hf(X)it = (Xs)dXs = (Xs) dhXis 0 @x t 0 @x 1 5. Let Xt;Yt be continuous semimartingales with quadratic variations hX; Xit, hY; Y it and cross variation hX; Y it Show that Z t 1 1 Zt := exp(Xt − hX; Xit) z0 + exp(−Xs + hX; Xis) d Ys − hX; Y is 2 0 2 satisfies the pathwise linear differential equation dZt = ZtdXt + dYt; with initial value Z0 = z0, meaning that Zt satisfies Z t Zt = z0 + ZsdXs + Yt − Y0 0 where the integral is an Ito integral. Hint : When Xt are Yt continuous with finite variations on finite intervals Var[0;t](X) < 1 Var[0;t](Y ) < 1, you can show that the solution of the linear equation dZt = ZtdXt + dYt is given by Zt = z0 exp(Xt) , when Yt ≡ constant , otherwise Z t Zt = exp(Xt) z0 + exp(−Xs)dYs : 0 When Yt is constant and X has quadratic variation hX; Xit > 0, it follows from Ito formula that 1 Z = z expX − hX; Xi t 0 t 2 t solves the linear equation dZt = ZtdXt. Use the stochastic integration by parts formula d(XtYt) = XtdYt + YtdXt + dhX; Y it : 6. Show that the Ornstein Uhlenbeck process Z t −αt −α t αs Xt = e x + e e dWs 0 solves the Langevin equation satisfies the linear Langevin equation dXt = −αXtdt + dWt;X0 = x (0.1) where Wt is Brownian motion, and the differential is understood in the sense of Ito. (0.1). Use the martingale decomposition together with the property that martingales have constant expectation to compute 2 the expectations E(Xt) and E(Xt ). 2 7. Show that the Ornstein Uhlenbeck process is Gaussian (meaning that all its finite dimensional distributions are Gaussian). Hints. (Xt1 ;:::;Xtn ) is a linear functional of the Brownian motion. 8. Let Xt be a continuous semimartingale with quadratic variation hXit > 0, and let ∗ Xt := max Xs; t ≥ 0 0≤s≤t be its running maximum ∗ Show that the map t 7! Xt is continuous. 1;2 Let F (x; y) 2 C , with continuous partial derivatives Fx(x; y);Fxx(x; y);Fy(x; y). Use the Ito formula to represent the Ito integral Z t ∗ Fx(Xs;Xs )dXs 0 As a special case compute the Ito integral Z t ∗ Xs dXs 0 9. Let (W (t): t ≥ 0) ?? (B(t): t ≥ 0) be two mutually independent Brownian motions adapted to the same filtration F. And let Dn = n −n ti = i2 ; i 2 N be the sequence of dyadic partitions of R+. Show that 2 X n n n n L lim W (ti ^ t) − W (ti−1 ^ t) B(ti ^ t) − B(ti−1 ^ t) −! 0 n!1 i2N where we consider convergence in L2(P ) space. This means that W and B have cross variation hW; Bit = 0 By using Borel-Cantelli lemma show that we have also P -almost sure convergence. 10. Let (Wt) and (Bt) two independent Brownian motions, and consider the market model with two stocks St and Xt, satisfying dSt = St(µ dt + σ dWt) ;S0 > 0 dXt = Xt(b dt + a dBt);X0 > 0 3 with µ, σ; a; b; 2 R, S0;X0 > 0. Since St;Xt > 0 8t P -almost surely, we can choose each of these two assets as numeraire. Let Set = St=Xt = f(St;Xt) Use Ito-formula (or stochastic integration by parts ) to compute the Ito differential dSet. 11. For which values of σ; µ, a; b; Set is a martingale with respect to P ? R t 12. Let Wt be a Brownian motion and consider the Ito integral 0 WsdWs = 2 (Wt − t)=2. Check by a direct calculation the Ito isometry for Z t 2 Z t 2 2 2 E (Wt − t)=2 = E WsdWs = E(Ws )ds 0 0 4 13. Use the martingale decomposition to compute E(Wt ). Hint: use Ito 4 formula write the Ito differential dWt , and recall that martingales have constant expectation. 4.
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