Stochastic Differential Equations

Transcription

1 Chapter 5 Stochastic Differential Equations We would like to introduce stochastic ODE s without going first through the machinery of stochastic integrals. 5.1 Itô Integrals and Itô Differential Equations Let us start with a review of the invariance principle. Let {ξ n } n be a sequence of i.i.d. random variables such that Eξ n =, Eξn = 1. Define Xt by X t n = n ξ i, t n = n, n (5.1) and piecewise linear interpolation, then the invariance principle asserts that X d W (5.) in distribution. Alternatively, we can define {Xt n } n=1 by the recursion relation X t n+1 = X t n + t ξ n+1, X =, (5.3) where t = 1. We can think of (5.3) as a forward Euler scheme for solving the differential equation dx t = dw t (5.4) Obviously both (5.3) and (5.4) are a bit unusual. In (5.3), the multiplier in front of ξ n+1 is t instead of the usual t. In (5.4) we write dx t = dw t instead of Ẋ t = Ẇt. This is because Ẇt is not a standard stochastic process, but rather a generalized stochastic process as in generalized functions. In fact one can define Ẇ t as a Gaussian process with mean zero and covariance K(s, t) = δ(t s). 41

2 4 CHAPTER 5. STOCHASTIC DIFFERETIAL EQUATIOS This is a very important process called the Gaussian white noise. But its sample path are not the standard functions, but rather distributions, see [5]. We can now combine (5.4) with standard ordinary differential equation and study dx t = b(x t, t)dt + dw t. (5.5) One can think of this as the distributional limit of the forward Euler scheme X t n+1 = X t n + t b(x tn, t n ) + t ξ n+1 (5.6) where as before {ξ n } n=1 is a sequence of i.i.d. random variables such that Eξ n =, Eξ n = 1. In fact if we denote by X t the process obtained using (5.6) and the piecewise linear interpolation, then Theorem There exists a stochastic process X t such that E X t X t C t (5.7) for t [, 1], where C is independent of t = 1/, and Eg[X[,1] ] Eg[X [,1]] C t (5.8) for any continuous functional g on C[, 1], where C may depend on g but not on t. More generally, we can consider SDE s of the type dx t = b(x t, W [,t], t)dt + σ(x t, W [,t], t)dw t (5.9) where B and sigma are functions of X t and t, and functional of W [,t]. otice that they are nonanticipative functional, i.e. the Wiener process up to time t only enters the right hand-sidde of (5.9). (5.9) is defined it as the limit of the forward Euler scheme X n+1 = X n + t b(x n, W m n, t n ) + tσ(x n, W m n, t n )ξ n+1. (5.1) where, for simplicity we have used the slightly abusive notation X t n = X n. We can also write this as X n+1 = X n + t b(x n, W m n, t n ) + σ(x n, W m n, t n )(W n+1 W n ). (5.11) where W t is the Wiener process and W n = W tn. A special case of (5.9) is stochastic integrals dx t = f(w t, t)dw t, X =, (5.1) where f is continuous, whose solution can be expressed as X t = f(w s, s)dw s. (5.13)

3 5.1. ITÔ ITEGRALS AD ITÔ DIFFERETIAL EQUATIOS 43 The meaning of X t is define as the limit of or in other words X n+1 = X n + f(w n, t n )(W n+1 W n ), (5.14) X n = f(w j, t j )(W j+1 W j ). (5.15) j= We can think of (5.15) as a Riemann sum in which the representative point inside each subinterval is the left-most point. This definition of the stochastic integral is called the Itô integral. Itô integrals have several very special properties. Theorem 5.1. (Itô isometry). Itô integrals satisfy Proof. Let then E f(w s, s)dw s =, ( ) E f(w s, s)dw s = Ef (W s, s)ds. EI n = I n = = f(w j, t j )(W j+1 W j ), j= E(f(W j, t j )(W j+1 W j )) j= Ef(W j, t j )E(W j+1 W j ) =, j= where we use the fact that W j and W j+1 W j are independent. Similarly EI n = = E(f(W i, t i )f(w j, t j )(W i+1 W i )(W j+1 W j )) i,j= Ef (W j, t j )(t j+1 t j ) j= where we use the fact that W i+1 W i and W j+1 W j are independent unless i = j. Taking the limit as t, we obtain the equations in the theorem. Another important property of Itô integral is that X t = f(w s, s)dw s is a martingale, i.e. E Xs (X t ) = X s, where E Xs (X t ) denotes the expectation of X t conditional on X s. However since we will not discuss Martingale theory, we will not pursue this further.

4 44 CHAPTER 5. STOCHASTIC DIFFERETIAL EQUATIOS Theorem (Itô formula). Let f be a smooth function, assume that X t is the solution of the SDE (5.9), then g t = f(x t, t) satisfies the SDE df(x t, t) = f(x t, t)dt + f (X t, t)dx t + 1 f (X t, t)σ (X t, t)dt = f(x t, t)dt + ( f (X t, t)b(x t, t) + 1 f (X t, t)σ (X t, t) ) dt + f (X t, t)σ(x t, t)dw t, where f(x, t) = f/ t, f (x, t) = f/ x. (5.16) Itô formula is the analog of chain rule in ordinary differential calculus. However ordinary chain rule would give df(x t ) = f(x t, t)dt + f (X t, t)dx t. Here because of the non-smooth nature of X t, we have the additional term that depends on f. The proof of Itô formula can be outlined as follows. For simplicity we assume that f does not depend explicitly on t. We Taylor expand f(x n+1 ) f(x n ) using (5.1) and keeping terms up to O( t) using W j+1 W j = O( t): f(x n+1 ) f(x n ) = f (X n )(X n+1 X n ) + 1 f (X n )(X n+1 X n ) + = f (X n )(X n+1 X n ) + 1 f (X n )( tb(x n, t n ) = f (X n )(X n+1 X n ) + σ(x n, t n )(W n+1 W n )) + O(( t) 3/ ) + 1 f (X n )σ (X n, t n )(W n+1 W n ) + O(( t) 3/ ). The Itô formula follows in the limit as n since 1 f (X n )σ (X n, t n )(W n+1 W n ) 1 f (X t )σ (X t, t)dt as implied by the following lemma which can be written formally as (dw t ) = dt. Lemma Let W j = W tj with t j = j/. Then as n,, n/ t. (W j W j 1 ) t a.s. j=1 Proof. The limit of this sum can be estimated from 1 ξj = n 1 n j=1 ξj. In this last expression n/ t by assumption; the remainder is the rescaled sum of the i.i.d random variables ξj with mean equal to one which therefore converges to 1 as n by SLL. j=1

5 5.1. ITÔ ITEGRALS AD ITÔ DIFFERETIAL EQUATIOS 45 Example We compute W s dw s. Using the definition of Itô integral, this integral is the limit of W j (W j+1 W j ) = 1 (W j W j+1 + W j + W j+1 )(W j+1 W j ) j Therefore = 1 j (Wj+1 Wj ) 1 (W j+1 W j ). j j W s dw s = 1 W t 1 t. (5.17) This result can also be obtained via a straight forward application of Itô formula which gives for f(x) = 1 x 1 dw t = W t dw t + 1 dt (5.18) Integrating (5.18) we obtain (5.17). Compared with the standard relation f(s)df(s) = 1 (f (t) f ()) for smooth functions, we see the presence of the extra term 1 t. It is forced to be there by the relation E W sdw s =. We can express (5.17) as W s dw s = 1! t H (W t / t), where H (x) = 4x is the second order Hermite polynomial. In the same fashion, we have Using Itô formula, we have ( s W u dw u )dw s = 1 (W s s)dw s. Ws dw s = 1 3 W t 3 W s ds. Hence, using we obtain sdw s = tw t ( s W u dw u )dw s = 1 3! W s ds, ( ) 3/ t H 3 (W t / t),

6 46 CHAPTER 5. STOCHASTIC DIFFERETIAL EQUATIOS where H 3 (x) = 8x 3 1x is the third order Hermite polynomial. As shown below, we have for n dw t1 1 n 1 dw t... dw tn = 1 n! where H n (x) is the n-th order Hermite polynomial ( ) n d H n (x) = ( 1) n e x e x. dx Example We solve Itô s formula with f(x) = log x gives Integrating we get ( ) n/ t H n (W t / t), d t = α t dt + β t dw t (5.19) d log t = 1 (α t dt + β t dw t ) 1 t t β t dt. t = exp ( αt 1 β t + βw t ). ote that if W t were smooth, we would have obtained t = exp(αt + βw t ). For = 1 and α =, t reduces to Using Rodrigues formula, this can be expressed as t = t = exp ( 1 β t + βw t ) e az a = n= H n (z)a n n= H n (W t / t) n! n! ( ) n/ t β n. But iteration of the equation for t shows that it can also be expressed as We deduce that as asserted before. t = 1 + βw t + β n dw t1 n= n 1 dw t1 dw tn = 1 n! n 1 dw tn. ( ) n/ t H n (W t / t)

7 5.. UMERICAL SOLUTIOS OF SDE S 47 Example We solve dx t = γx t dt + σdw t (5.) This is the Ornstein-Uhlenbeck process. Using Itô formula we have Therefore we get d(e γt X t ) = γe γt X t dt + e γt dx t = σe γt dw t X t = e γt X + σ e γ(t s) dw s. This is Duhammel principle applied to (5.). Let Q t = e γ(t s) dw s. Q t is a Gaussian process with mean and variance EQ t = EQ t = E(e γ(t s) ) ds = e γt e γs ds = 1 γ (1 e γt ) As t, the variance goes to 1/γ. Thus as t ( ) d X t, σ γ (5.1) (5.) is a special case of the so-called Langevin equations. More generally, Langevin equation with potential function V (x) is given by dx t = V (X t )dt + σdw t (5.) 5. umerical Solutions of SDE s We will discuss accuracy and stability issues Accuracy Let {Xt t } be the numerical solution of a SDE with time step size t, and {X t } be the exact solution. There are two different notions of measuring the error X t Xt t. We say that {Xt t } converges to {X t } with strong order α if max t T E X t t X t C( t) α (5.3)

8 48 CHAPTER 5. STOCHASTIC DIFFERETIAL EQUATIOS holds with a constant C independent of t. We say that {Xt t } converges to {X t } with weak order β if max t T Eg(X t t ) Eg(X t ) C( t) β (5.4) holds with a constant C independent of t but which may depend on the smooth function g. It can be shown that α β, since ( Eg(X t ) Eg(Xt t ) E X t Xt t where M = max g. 1 ME X t X t t, g (X t + θ(x t t ) X t ) )dθ Exercise Show that α = 1, β = 1 for the forward Euler scheme discussed earlier. In order to find numerical schemes that improve the order of accuracy, we proceed as follows. Consider the SDE From Itô formula, we have dx t = b(x t )dt + σ(x t )dw t (5.5) df(x t ) = f (X t )dx t + 1 f (X t )σ (X t )dt = (L 1 f)(x t )dt + (L f)(x t )dw t where Integrating, we get (L 1 f)(x) = f (x)b(x) + 1 f (x)σ (x) (L f)(x) = f (x)σ(x) f(x t+ t f(x t ) = + t (L 1 f)(x s )ds + t t + t (L f)(x s )dw s. (5.6) Apply (5.6) with f(x) = b(x), and f(x) = σ(x) respectively, and use the result in X t+ t X t = we get + t b(x s )ds + t t X t+ t = X t + tb(x t ) + σ(x t )(W t+ t W t ) t s t t + t s (L 1 b)(x τ )dτds + (L 1 σ)(x τ )dτdw s + + t s + t t t + t s t t t t σ(x s )dw s, (L b)(x τ )dw τ ds (L σ)(x τ )dw τ dw s. (5.7)

9 5.3. PATH ITEGRAL REPRESETATIO 49 If we just keep the first two terms at the right hand side, we get Euler scheme. Out of the four double integrals, the last one is the biggest. Approximate the last integral by + t s t t (L σ)(x t ) (L σ)(x τ )dw τ dw s + t s t t dw τ dw s = 1 (L σ)(x t ) ( (W t+ t W t ) t ). (5.8) Since E(W t t) = EW t 4 tew t + ( t) = 3( t) ( t) + ( t) = ( t), the term in (5.8) is of order t; we also have E(W t t) =. So the total error from the final term in (5.7) can be estimated as E(total error) 1 t O( t ) = O( t). (5.9) This suggests that the error for the forward Euler scheme scales as O( t). This analysis also suggests an obvious extension of the forward Euler scheme that promises to be more accurate. X t+ t = X t + b(x t ) t + σ(x t )(W t+ t W t ) + 1 (σσ )(X t )((W t+ t W t ) t). This is the well-known Milstein scheme. Despite the fact that Milstein s scheme is formally more accurate, it also uses the derivative of σ. This is unpleasant for many applications. 5.. Stability 5.3 Path Integral Representation We have seen in Chapter 4 that the Wiener measure over [, T] can be formally expressed as ( ) The solution of the SDE dµ W = Z 1 exp 1 T ḣ (t)dt dx t = b(x t, t) + σ(x t, t)dw t. Dh( )

10 5 CHAPTER 5. STOCHASTIC DIFFERETIAL EQUATIOS can be viewed as a map between the Wiener path W [,T] and X [,T] : W [,T] Φ X[,T]. Consequently, Φ induces another measures on C[, T], which is nothing but the distribution of X [,T]. We now ask the question how the measure dµ W change under the mapping Φ? Let us first consider the case when σ = I. In this case, we claim the measure dµ X can be formally expressed as ( ) dµ X = Z 1 exp 1 T (ḣ(t) b(h(t), t)) dt We proceed at the discrete level. The recurrence relation Dh( ) X n+1 = X n + b(x n, t n ) t + W n+1 W n, X = x, in principle allows to express X n as a function of W m n and x, i.e. X n = Φ n (W m n, x). Therefore the expectation of any function F(X n ) can be expressed as EF(X n ) = Z 1 F(Φ n (y m n, x))exp ( I tn (y n ) ) Dy n R where Dy n = dy 1 dy, Z is a normalization factor, and I tn (y n ) is the kernel defined in the last chapter I tn (y n ) = 1 In the expression above, we wish to integrate over (y i y i 1 ) t i t i 1, y = t = h n = Φ n (y m n, x), instead of y n. Since the recurrence relation for X n implies that W n+1 W n = X n+1 X n b(x n, t n ) t, it follows that the kernel I tn (y n ) can be expressed in terms of h n simply as Ī tn (h n ) = 1 (h i h i 1 b(h i 1, t i 1 ) t) t i t i 1, y = t = On the other hand, the Jocobian of the transformation for y n to h n is one because the matrix h n y m

11 5.3. PATH ITEGRAL REPRESETATIO 51 is a lower triangular matrix (i.e. it has zero element above the diagonal) with ones on the diagonal and hence has determinant one. Therefore we have EF(X n ) = Z 1 F(h n )exp ( Īt n (h n ) ) Dh n R In the limit as t, this expression formally gives EF[X [,T] ] = F[h [,T] ]dµ X [h [,T] ], with the measure dµ X given before. Quite remmarkably, we can give a rigorous meaning to these formal manipulations. To see how, notice that the kernel Īt n (h n ) can be expanded as (using t i t i 1 = t) Ī tn (h n ) = 1 (h i h i 1 ) b(h i 1, t i 1 )(h i h i 1 ) + 1 t b (h i 1, t i 1 ) t = I tn (h n ) b(h i 1, t i 1 )(h i h i 1 ) + 1 b (h i 1, t i 1 ) t. Therefore we can also express the expectation of EF(X n ) as the ratio EF(X n ) = A/B. Here A = Z 1 F(h n )e Q(hn ) exp ( I tn (h n ) ) Dh n R ( = E F(Wn x x )) )eq(w n, and where W x n = x + W n and B = Z 1 e Q(hn ) exp ( I tn (h n ) ) Dh n R = Ee Q(W x n ) Q(h n ) = b(h i 1, t i 1 )(h i h i 1 ) 1 b (h i 1, t i 1 ) t Taking the limit as t, we deduce that ) E (e Q[W x [,T] ] F[W[,T] x ] EF(X [,T] ) =, Ee Q[W x [,T] ]

12 5 CHAPTER 5. STOCHASTIC DIFFERETIAL EQUATIOS where W x t = x + W t is the shifted Wiener process and Q[W x [,T] ] = T b(w x t, t)dw t 1 T b (W x t, t)dt. Quite remarkably, this expression simplify even more because one can show that Ee Q[W [,T]] = 1 and hence ) EF(X [,T] ) = E (e Q[W x [,T] ] F[W[,T] x ] This formula is known as the Girsanov formula. To see that Ee Q[W x [,T] ] = 1, let so that Therefore B t = b(w x s, s)dw s 1 b (W x s, s)ds db t = b(w x t, t)dw t 1 b (W x t, t)dt, B =, and we obtain de Bt = e Bt db t + 1 ebt b (W x t, t)dt = ebt b(w x t, t)dw t e Bt = 1 + The first Itô isometry then implies that e Bs b(w x s, s)dw s Ee Bt = 1 t and, in particular, Ee B T = EeQ[W [,T]] = 1. The Girsanov formula asserts that dµ X and dµ W are absolutely continuous with respect to each other with Radon-ikodym derivative given by ( dµ 1 X ( = exp b(w x dµ s, s)dx s 1 b (Ws x, s)ds )). W A special case of this representation is the Cameron-Martin formula, for the transformation X t = W t + φ(t) (5.3) where φ is a smooth function. This can be obtained from SDE with b(x, t) = φ(t). In this case, we get dµ X dµ W = exp ( 1 ( φ(s)dws 1 φ (s)ds) ). A slight generalization is the Girsanov formula. Consider two SDE s: { dx t = b(x t, t)dt + σ(x t, t)dw t, X = x dy t = (b(y t, t) + c(y t, t))dt + σ(y t, t)dw t, Y = x

13 5.3. PATH ITEGRAL REPRESETATIO 53 (ote that the initial conditions are the same.) Then the distributions of X [,T] and Y [,T] are absolutely continuous with respect to each other. Moreover the Radon-ikodym derivative is given by ( dµ T ) T Y [X.] = exp φ(x t, t)dw t 1 dµ φ(x t, t) dt, X where φ is the solution of σ(x, t)φ(x, t) = c(x, t).