Exact Simulation of Diffusions and Jump Diffusions

Transcription

1 Exact Simulation of Diffusions and Jump Diffusions A work by: Prof. Gareth O. Roberts Dr. Alexandros Beskos Dr. Omiros Papaspiliopoulos Dr. Bruno Casella 28 th May, 2008

2 Content 1 Exact Algorithm Construction EA1 EA2 EA3 2 Jump Exact Algorithm 3 Simulated Example 4 Work in progress

3 Construction Exact Algorithm (EA) Proposed by Beskos, Papaspiliopoulos and Roberts (2006). Exact simulation of a class of Itô s diffusions. Exact in the sense that there is no discretisation error. The EA performs rejection sampling by proposing paths from processes that we can simulate and accepting them according to appropriate probability density ratios. The novelty lies in the fact that the paths proposed are unveiled only at finite (but random) time instances and the decision whether to accept the path or not can be easily taken.

4 Construction Consider X := {X t : 0 t T } a one-dimensional diffusion process solving the SDE: dx t = b(x t )dt + σ(x t )db t, X 0 = x R, t [0, T ] We will restrict our attention to SDEs of the type dx t = α(x t )dt + db t, X 0 = x R, t [0, T ] since the transformation X t η(x t ) can be applied, where η(x) = x z 1 σ(u) du with z being some element of the state space of X.

5 Construction Consider X := {X t : 0 t T } a one-dimensional diffusion process solving the SDE: dx t = b(x t )dt + σ(x t )db t, X 0 = x R, t [0, T ] We will restrict our attention to SDEs of the type dx t = α(x t )dt + db t, X 0 = x R, t [0, T ] since the transformation X t η(x t ) can be applied, where η(x) = x z 1 σ(u) du with z being some element of the state space of X.

6 Construction Q: the probability law of X ; W: the probability law of a Brownian motion starting at x R. The Girsanov Formula implies that { dq T dw = exp α(x t )dx t 1 T } α 2 (X t )dt Under the condition that α is everywhere differentiable and applying Itô s lemma to A(X ), for A(u) := u R, we have u 0 α(y)dy, { dq dw = exp A(X T ) A(x) 1 T ( α 2 (X t ) + α (X t ) ) } dt 2 0

7 Construction Q: the probability law of X ; W: the probability law of a Brownian motion starting at x R. The Girsanov Formula implies that { dq T dw = exp α(x t )dx t 1 T } α 2 (X t )dt Under the condition that α is everywhere differentiable and applying Itô s lemma to A(X ), for A(u) := u R, we have u 0 α(y)dy, { dq dw = exp A(X T ) A(x) 1 T ( α 2 (X t ) + α (X t ) ) } dt 2 0

8 Construction The main idea is to perform rejection sampling. Problem: A(X T ) will typically be unbounded. Solution: To propose candidate paths from a process identical to BM starting at x except for the distribution of its ending point: biased Brownian motion. This biased BM is defined as Ŵ = (W W T h), where h exp{a(u) (u x) 2 /2T }, u R, has to be integrable.

9 Construction Let Z be the probability measure induced by the biased BM. Then It can be shown that dq dz = dq dw dw dz Implying dw dz = f N(X T ; x, T ) exp{ A(X T )} h { dq T ( 1 dz exp 0 2 α2 (X t ) + 1 ) } 2 α (X t ) dt

10 Construction We assume that (α 2 + α )/2 is bounded below implying that dq is bounded. dz It is possible to obtain a non-negative function φ such that Analytically, φ is defined as: { dq T } dz exp φ(x t )dt 1 0 φ(u) = α2 (u) + α (u) k, 2 u R, for k inf u R (α2 +α )(u)/2.

11 Construction Problem: It is not possible to draw complete continuous paths of Z on [0, T ] and calculate the integral involved in the acceptance probability analytically. Solution: Theorem Let X be any continuous mapping from [0, T ] to R, and M(X ) an upper bound for the mapping t φ(x t ), t [0, T ]. If Φ is a homogeneous Poisson process of unit intensity on [0, T ] [0, M(X )] and N is the number of points of Φ found below the graph {(t, φ(x t )); t [0, T ]}, then { T } P(N = 0 X ) = exp φ(x t )dt. 0

12 EA1 - The case when φ is bounded Suppose that M is an upper bound for φ(x t ), t [0, T ]. 1. Produce a realisation {x 1, x 2,..., x τ }, of a Poisson process on [0, T ] [0, M], where x i = (x i,1, x i,2 ), 1 i τ;

13 EA1 - The case when φ is bounded 2. Simulate a skeleton of X Z at the time instances {x 1,1, x 2,1,..., x τ,1 };

14 EA1 - The case when φ is bounded 3. Evaluate N; 4. If N = 0, go to step 5, else go to step 1; 5. Output the currently constructed skeleton S(X ) of X ; X is accepted and X is rejected.

15 EA1 - The case when φ is bounded The EA returns a Skeleton of the process X defined over the time interval [0, T ] with starting point x. We represent it as: S 0 (x; 0, T ) := {(t 0, x t0 ), (t 1, x t1 ),..., (t M, x tm )} with t 0 = 0 and t M = T. We can simulate further information on the process X using the following representation, immediately derived from elementary Brownian motion constructions: X S 0 (x; 0, T ) M 1 i=0 BB(t i, x ti ; t i+1, x ti+1 ) where BB(s 1, a; s 2, b) is the measure of a Brownian bridge starting in a at time s 1 and ending in b at time s 2.

16 EA2 - The case when either lim sup u φ(u) < or lim sup u φ(u) < If lim sup u φ(u) < or lim sup φ(u) <, it is possible u to identify an upper bound M(m) for φ(x t ), with t [0, T ], after decomposing the proposed path X t at its min or max. If lim sup φ(u) < and m is the minimum of X t in [0, T ] u M(m) = sup{φ(u); u m} If lim sup φ(u) < and m is the maximum of X t in [0, T ] u M(m) = sup{φ(u); u m}

17 EA2 - The case when either lim sup u φ(u) < or lim sup u φ(u) < The EA2 algorithm 1 Initiate a path X t Z on [0, T ] by drawing X T h; 2 Simulate its minimum or maximum m and the moment t m when it is achieved ; 3 Find an upper bound M(m) for φ(x t ); 4 Produce a realization {x 1, x 2,..., x τ }, of a Poisson process on [0, T ] [0, M(m)]; 5 Simulate a skeleton of X (m, t m ) at the time instances {x 1,1, x 2,1,..., x τ,1 } ; 6 Evaluate N; 7 If N = 0, go to step 8, else go to step 1; 8 Output the currently constructed skeleton S(X ) of X. The algorithms to perform steps 2 and 5 are described in Beskos, Papaspiliopoulos and Roberts (2006). ref

18 EA3 - The case when φ is only bounded below So far, we have made the following hypotheses to perform the Exact Algorithm. C0 The Radon-Nikodym derivative of Q w.r.t. W exists and it is given by Girsanov s formula; C1 α C 1 ; C2 α 2 + α is bounded below; C3 The function h(u) = exp{ (u x) 2 /(2T ) + A(u)} is integrable. EA3 requires no further hypotheses on φ. In this case we need to know upper and lower bounds for X t in [0, T ] in order to construct an upper bound for φ(x t ). The bounds of X t are obtained using the idea of a layered Brownian bridge.

19 EA3 - The case when φ is only bounded below Define two increasing sequences of positive real numbers {a i } i 1 and {b i } i 1 with a 0 = b 0 = 0. Given X 0 = x and X T = y, set x = x y and ȳ = x y and define the following events: U i = { sup X t [ȳ + b i 1, ȳ + b i ] 0 t T { L i = inf 0 t T Xt [ x a i, x a i 1 ] } { } { D i = U i Li, i 1. } inf 0 t T Xt > x a i, sup X t < ȳ + b i }, 0 t T Define the random variable I = I (X ) such that {I = i} = D i. {I = i} { x a i < X t < ȳ + b i, t [0, T ]}.

20 EA3 - The case when φ is only bounded below Example: a i = b i and I (X ) = 4.

21 EA3 - The case when φ is only bounded below The EA3 algorithm 1 Initiate a path X t Z on [0, T ] by drawing X T h; 2 Simulate I (X ) ; 3 Find an upper bound M(I ) for φ(x t ); 4 Produce a realization {x 1, x 2,..., x τ }, of a Poisson process on [0, T ] [0, M(I )]; 5 Simulate a skeleton of X I at the time instances {x 1,1, x 2,1,..., x τ,1 } ; 6 Evaluate N; 7 If N = 0, go to step 8, else go to step 1; 8 Output the currently constructed skeleton S(X ) of X. The algorithms to perform steps 2 and 5 are described in Beskos, Papaspiliopoulos and Roberts (2007). ref

22 EA3 - The case when φ is only bounded below If we want to simulate a diffusion conditioned on the ending point X T = y, the paths are proposed from a BB(0, x; T, y). The following paper proposes some methods for inference in diffusions based on the Exact Algorithm: Beskos, A., Papaspiliopoulos, O., Roberts, G, and Fearnhead, P. (2006), Exact and computationally efficient likelihood-based inference for discretely observed diffusion processes (with discussion), J. R. Stat. Soc. B 68(3),

23 Jump Exact Algorithm (JEA) Proposed by Casella and Roberts (2008). Generalises the Exact Algorithm for the simulation of jump diffusions.

24 Consider X := {X t : 0 t T } a one-dimensional jump diffusion process solving the SDE: dx t = b(x t )dt + σ(x t )db t + dj t, X 0 = x J t is a jump process where the jump times follow a Poisson process of rate λ(t, X t ) and the jump sizes are given by a function g(z t, X t ), where z t has a distribution f Zt.

25 Between any two jumps, the process X behaves as a homogeneous diffusion process with drift b and coefficient σ. Jump times and size may depend on time and on the state of the process. The jump times are generated by a Poisson process on [0, T ] with intensity function λ(t, X t ). A random variable Z i f Z (z; t i ) is associated to each jump time t i. Z i and the state of the process X t determine the amplitude g(z ti, X t ) of the jump in t i. Once more, we apply the transformation X t η(x t ) to work with dx t = α(x t )dt + db t + dj t, X 0 = x

26 Between any two jumps, the process X behaves as a homogeneous diffusion process with drift b and coefficient σ. Jump times and size may depend on time and on the state of the process. The jump times are generated by a Poisson process on [0, T ] with intensity function λ(t, X t ). A random variable Z i f Z (z; t i ) is associated to each jump time t i. Z i and the state of the process X t determine the amplitude g(z ti, X t ) of the jump in t i. Once more, we apply the transformation X t η(x t ) to work with dx t = α(x t )dt + db t + dj t, X 0 = x

27 Constructing the JEA We make the hypothesis that λ(t, X t ) is bounded. To break down the state-dependency of the jump times we use the following result. Poisson Thinning To simulate a Poisson process of rate λ(t, X t ) λ on [0, T ] do: 1 Simulate a Poisson process of rate λ on [0, T ]. 2 Keep each point with probability R(t i, X t ) = λ(t i, X t i λ i i = 1,..., n, where n is the number of points simulated on step 1. ), for

28 1. Simulate a Poisson process of rate λ on [0, T ];

29 2. Apply the Exact Algorithm to [0, T ];

30 3. Perform Poisson thinning until one point t 1 is kept or no point is kept;

31 4. Simulate the jump at t 1 ;

32 5. Apply the Exact Algorithm to [t 1, T ];

33 6. Perform Poisson thinning until one point t 2 is kept or no point is kept;

34 7. Simulate the jump at t 2 ;

35 8. Apply the Exact Algorithm to [t 2, T ];

36 9. Perform Poisson thinning until one point t 3 is kept or no point is kept;

37 10. If no point is kept...

38 10. Output skeleton S(X ) of X.

39 dx t = tanh(x t )dt + db t + dj t, X 0 = 0 X 2 t λ(x t ) = 1 + Xt 2 < 1, g(z ti, X t ) = Z Z N(0, 9) and T = 20 The JEA is compared to the Euler approximation scheme by comparing the distribution of the ending point.

40 One realization of JEA

41 Comparison with Euler approximation CPU time JEA 15.94s Euler t = s t = s t = s t = s t = s t = m 6s

42 Work in progress Extend the Jump Exact Algorithm to simulate jump diffusions with unbounded jump rate. Extend the Jump Exact Algorithm to simulate jump diffusions conditioned on the ending point. Develop methods for inference in jump diffusion processes based on the JEA. Unknown parameters in the drift and in the diffusion coefficient; Unknown parameters in the jump rate; Unknown parameters in the distribution of the jump sizes.

43 Thank you! Contact:

44 References Beskos, A; Papaspiliopoulos, O. and Roberts, G. O. (2006), Retrospective exact simulation of diffusion sample paths with applications. Bernoulli, 12(6), Return Beskos, A., Papaspiliopoulos, O., Roberts, G, and Fearnhead, P. (2006), Exact and computationally efficient likelihood-based inference for discretely observed diffusion processes (with discussion), J. R. Stat. Soc. B 68(3), Beskos, A; Papaspiliopoulos, O. and Roberts, G. O. (2007), A factorization of diffusion measure and finite sample path constructions. To appear in Methodology and Computing in Applied Probability. Return Casella, B. and Roberts, G. O. (2008), Exact simulation of jump-diffusion processes with Monte Carlo Applications. In preparation.