This coursework is marked out of 100 and counts for 20% of the overall course grade. The assignment is due by 1600 GMT on Wednesday week 10. If you use any materials (including code) from another source in your submission then you must add an appropriate citation – you do not need to cite course materials. Your submission should consist of 1. A single PDF file containing your solutions all ”pen and paper” questions. 2. A completed Jupyter notebook containing your solutions to questions related to code Your solutions to ”pen and paper” questions should be clearly legible and be in complete sentences. You do not need to typeset your solutions – although you may choose to. The code in the jypyter notebook that you submit should be clear – the purpose of different parts of your code should be easy to identify from a brief review of the code. All output should be easy to understand, and plots should be appropriately formatted and clearly labelled. Questions Question 1 1.1 Assume that X(t) satisfies the stochastic differential equation dX(t) = aX(t)dt + � bX(t)dW(t), X(0) = 1 for 0 ≤ t ≤ T where a, b > 0. Use Ito’s formula to find E (X(t)), E (X2(t)), E (X3(t)) exactly. [15 marks] 1.2 Assume that X(t) satisfies the stochastic differential equation dX(t) = �1 3X1/3(t) + 3X2/3(t) � dt + X2/3(t)dW(t), X(0) = 1. Use Ito’s formula with u(t, X(t)) = (X(t))1/3 to prove that the exact solution is X(t) = � t + 1 + 1 3W(t) �3 . [10 marks]. Question 2 Let W(t) be a standard one dimensional Brownian motion and let Q(t) be the Brownian Bridge on [0, 1], i.e. Q(t) = W(t) − tW(1), t ∈ [0, 1]. 2.1 Show that Q(t) is a Gaussian process with EQ(t) = 0, E(Q(t)Q(s)) = min(t, s) − ts. 1 [7 marks] 2.2 Now consider the stochastic process Y (t) = � t 0 Q(ℓ)dℓ which is related to the area under the Brownian bridge curve. Show that E (Y (t)Y (s)) = ts2 2 − s3 6 − (ts)2 4 , s ≤ t [10 marks] 2.3 Show that an equivalent definition to Brownian motion is given by W(t) = (t + 1)Q � t t + 1 � (Hint: you can use without proof the fact that if X(t) is a Gaussian process then Y (t) = X(f(t)) is Gaussian if f(t) is a strictly increasing function ) [8 marks] 2.4 Show that the solution to the eigenvalue problem � 1 0 R(t, s)φk(s)ds = λkφk(t), R(t, s) = min(t, s) − ts, φk(0) = φk(1) = 0 (1a) � 1 0 φk(s)φℓ(s)ds = δkℓ (1b) is given by λk = 1 π2k2 , φk(t) = √ 2 sin kπt Hence explain why the following holds Q(t) = ∞ � k=1 √ 2sin kπt kπ ζk, (2) where ζk are i.i.d N(0, 1) random variables. [10 marks] Question 3 Consider the following deterministic ODE describing the harmonic oscillator problem dX = Y dt, (3a) dY = −Xdt. (3b) 3.1 Show that d dt(X2(t) + Y 2(t)) = 0 [5 marks] 3.2 Now consider (3) perturbed by noise dX = Y dt (4a) dY = −Xdt + γdW (4b) where W(t), is a one dimensional Brownian motion. Show using Ito’s formula that the following ODE holds for the average energy of the system E(t) = E(X2(t) + Y 2(t)) dE(t) dt = γ2 (5) [8 marks] 3.3 From now on consider X(0) = 1, Y (0) = 0, and consider the discretization of (4) by the Euler-Maryuama method Xn+1 = Xn + Yn∆t (6a) Yn+1 = Yn − Xn∆t + γ∆Wn (6b) 2 The code in the notebook calculates an approximation of E(n∆t) by E(X2 n + Y 2 n ) for γ = 1 and plots its evolution as a function of tn = n∆t. Modify the plotting in such a way that includes the true evolution of the E(t) as predicted by (5). What do you observe? [5 marks] 3.4 By writing a recurrence relationship between E(X2 n+1 + Y 2 n+1) and E(X2 n + Y 2 n ) deduce that E(X2 n + Y 2 n ) ≥ e( 1 2 ∆t)tn (Hint: You can use without proof that log (1 + x2) ≥ 1 2x2) [10 marks] 3.5 Now consider the following numerical method Xn+1 = Xn + Yn∆t (7a) Yn+1 = Yn − Xn+1∆t + γ∆Wn (7b) Modify the code for the Euler-Maruyama method so as to implement this method and compare numerically in a similar way the evolution of the numerical average energy with the true average energy. What do you observe? Comment on which of the two methods you think is more suitable for numerical integration of (4) when the final integration time is large. [12 marks]
