MATH10053 Applied Stochastic Differential Equations
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Applied Stochastic Differential Equations
MATH10053
Friday 11th December 2020
1300-1500 † *
† All students: you have an additional 1 hour to assemble and submit your PDF. Final submission deadline: 16:00.
* Students with a Schedule of Adjustment: You are entitled to a further fixed additional 1 hour for this remote examination.
Final submission deadline: 17:00
Attempt all questions
Important instructions
1. Start each question on a new sheet of paper.
2. Number your sheets of paper to help you scan them in order.
3. Only write on one side of each piece of paper.
4. If you have rough work to do, simply include it within your overall answer – put brackets at the start and end of it if you want to highlight that it is rough work.
MATH10053 December 2020 MAT-1-ASDE
Applied Stochastic Differential Equations
1. Let W (t) be a one dimensional Brownian motion and consider X(t) = e_k1 αt W (ek2 αt ) where k1 , k2 , α > 0.
(a) Calculate µ(t) = E(X(t)) as well as the covariance function
C(t, s) = E [(X(t) _ µ(t))(X(s) _ µ(s))] .
[8 marks]
(b) For which values of k1 , k2 is X(t) a weakly stationary process? [5 marks]
(c) Is it also strictly stationary? (Hint: you can use without proof the fact that if Z(t) is a Gaussian process then Y (t) = Z(f(t)) is Gaussian if f(t) is a strictly increasing function). [3 marks]
(d) Now let P = {0 = t0 < t1 < . . . < tn = T} denote a partition of the interval [0, T].
0 < k < n _ 1, 0 < l < n _ 1 the following holds
E [(W (tk+1) _ W (tk ))(W (te+1) _ W (te ))] = αδke , δke = 
[4 marks]
(e) Use the results from (d) and the definition of the Ito stochastic integral to calculate
0T W (s)dW (s). (Hint: The following identity might be useful for your calculation a(b _ a) = 
(b2 _ a2 ) _ 
(a _ b)2 ). [4 marks]
(f) Hence deduce that
Ee 0(尸) W (s)dW (s) = e 
[6 marks]
2. Let f : R |→ R and W (t) be a one dimensional Brownian motion. Now consider the
Ito SDE
dX = 
fo (X)f(X)dt + f(X)dW. (1)
(a) Write down its Euler-Maryama and Milstein discretizations. What are the strong
and weak orders of convergence of each of these discretisations? [7 marks]
(b) Convert this SDE into its Stratonovich form.
(c) Use the definition of the Stratonovich integral to explain why Xn+1 = Xn + f ╱ 
、 ∆Wn
[6 marks]
(2)
is a suitable numerical scheme for solving the corresponding Stratonovich SDE.
[6 marks]
(d) Starting by (2) and by using the Taylor expansion
f ╱ 
、 = f (Xn ) + 
fo (Xn )(Xn+1 _ Xn ) + o((Xn+1 _ Xn )2 ),
obtain an explicit numerical scheme (you should ignore any term that is higher than o(∆W2 )). Compare it with the schemes derived in part (a). Hence deduce its strong and weak order [8 marks]
(e) Now consider the case where f (X) = X ln(X), and X(0) = e and solve equation (1). [8 marks]
3. An Ornstein-Unlenbeck process in the plane is governed by the SDEs
dX = _Xdt + dW1
dY = _Ydt + dW2
(3)
(4)
where W1 and W2 are independent one dimensional Brownian motions and X(0) = 1, Y (0) = 1.
(a) By using Ito formula or otherwise derive an ODE for E(X(t)), E(X2 (t)), E(Y (t)), E(Y2 (t)).
[8 marks]
(b) Solve these ODEs to deduce what is the behaviour of these expressions when t → 与. What does this imply for the distribution of (X(t), Y (t)) for large times? [8 marks]
(c) Now let (R, θ) be polar coordinates, with X = R cos θ, Y = R sin θ .
(i) Show that R satisfies
dR = ╱ _R + 
、dt + dW,
where W is a Brownian motion such that dW = cos θdW1 + sin θdW2 .
[8 marks]
(ii) Write down the corresponding forward Kolmogorov (Fokker-Planck) equation.
[4 marks]
(iii) Show that the stationary distribution of this equation is given by pR (r) = 2re_r2 .
You can use without proof that limr →( pR (r) = limr →( pR(o)(r) = 0. [7 marks]
2021-11-28