MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS. Contents

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1 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS ANDREW TULLOCH Contents 1. Lecture 1 - Tuesday 1 March 2 2. Lecture 2 - Thursday 3 March Concepts of convergence 2 3. Lecture 3 - Tuesday 8 March 3 4. Lecture 4 - Thursday 1 March 5 5. Lecture 5 - Tuesday 15 March Properties of paths of Brownian motion 7 6. Lecture 6 - Thursday 17 March 9 7. Lecture 7 - Tuesday 22 March Construction of Brownian motion 1 8. Lecture 8 - Thursday 24 March Lecture 9 - Tuesday 29 March Lecture 1 - Thursday 31 March Lecture 11 - Tuesday 5 March Lecture 12 - Thursday 7 March Lecture 13 - Tuesday 12 March Lecture 14 - Thursday 14 March Lecture 15 - Tuesday 19 March Lecture 16 - Thursday 21 March Lecture 17 - Tuesday 3 May Stochastic integrals for martingales Lecture 18 - Tudsay 1 May Itô s integration for continuous semimartingales Stochastic differential equations Lecture 19 - Thursday 12 May Lecture 2 - Tuesday 17 May Numerical methods for stochastic differential equations Applications to mathematical finance 34 1

2 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS Martingale method Lecture 21 - Thursday 19 May Change of Measure Lecture 22 - Tuesday 24 May Black-Scholes model Lecture 23 - Thursday 26 May Lecture 24 - Tuesday 31 May Lecture 25 - Thursday 2 June 42

3 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 3 1. Lecture 1 - Tuesday 1 March Definition 1.1 (Finite dimensional distribution). The finite dimensional distribution of a stochastic process X is the joint distribution of (X t1, X t2,..., X tn ) Definition 1.2 (Equality in distribution). Two random variables X and Y are equal in distribution if P(X α) = P(Y α) for all α R. We write X = d Y. Definition 1.3 (Strictly stationary). A stochastic process X is strictly stationary if (X t1, X t2,..., X tn ) = (X t1+h, X t2+h,..., X tn+h) for all t i, h. Definition 1.4 (Weakly stationary). A stochastic process X is weakly stationary if E (X t ) = E (X t+h ) and Cov (X t, X s ) = Cov (X t+h, X s+h ) for all t, s, h. Lemma 1.5. If E ( ) Xt 2 <, then strictly stationary implies weakly stationary. Example 1.6. The stochastic process {X t } with X t all IID is strictly stationary. The stochastic process W t with W t N(, t) and X t X s independent of X s (for s < t) is not strictly or weakly stationary. Definition 1.7 (Stationary increments). A stochastic process has stationary increments if for all s, t, h. X t X s d = Xt h X s h 2. Lecture 2 - Thursday 3 March Example 2.1. Let X n, n 1 be IID random variables. Consider the stochastic process {S n } where S n = n X j. Then {S n } has stationary increments Concepts of convergence. There are three major concepts of convergence of random variables. Convergence in distribution Convergence in probability Almost surely convergence Definition 2.2 (Convergence in distribution). X n d X if P(Xn x) P(X x) for all x. Definition 2.3 (Convergence in probability). X n p X if P(( Xn X ϵ)) for all ϵ >. Definition 2.4 (Almost surely convergence). X n a.s. X if except on a null set A, X n X, that is, lim n X n = X. And hence P(lim n X n = X) = 1

4 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 4 Definition 2.5 (σ-field generated by X). Let X be a random variable defined on a probability space (Ω, F, P). We call σ(x) the σ-field generated by X, and we have σ(x) = { X 1 (B) : B B } where X 1 (B) = {ω, X(ω) B} and B is the Borel set on R. Definition 2.6 (Conditional probability). We have P(A B) = P(A,B) P(B) if P(B). Definition 2.7 (Naive conditional expectation). We have E (X B) = E(XI B) P(B). Definition 2.8 (Conditional density). Let g(x, y) be the joint density function for X and Y. Then we have Y R g(x, y) dx g Y (y). We also have g X Y =y = which defines the conditional density given Y = y. g(x, y) g Y (y) Finally, we define E (X Y = y) = R xg X Y =y(x) dx. Definition 2.9 (Conditional expectation). Let (Ω, F, P) be a probability space. Let A be a sub σ-field of F. Let X be a random variable such that E( X ) <. We define E (X A) to be a random variable Z such that (i) Z is A-measurable, (ii) E(XI A ) = E(ZI A ) for all A A. Proposition 2.1 (Properties of the conditional expectation). Consider Z = E(X Y ) = E(X σ(y )) If T is σ(y )-measurable, then E(XT Y ) = T E(X Y ) a.s. If T is independent of Y, then E(T Y ) = E(T ). E(X) = E(E(X T )) 3. Lecture 3 - Tuesday 8 March Definition 3.1 (Martingale). Let {X t, t } be a right-continuous with left-hand limits. lim t t X t exists Let {F t, t } be a filtration. Then X is called a martingale with respect to F t if (i) X is adapted to F t, i.e. X t is F t -measurable (ii) E( X ) <, t (iii) E(X t F s ) = X s a.s. Example 3.2. Let X n be IID with E(X n ) =. Then {S k, k }, where S k = k i= X i, is a martingale.

5 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 5 Example 3.3. An independent increment process {X t, t } with E(X t ) = and E( X t ) is a martingale with respect to F t = σ{x s, s t} Definition 3.4 (Gaussian process). Let {X t, t } be a stochastic process. If the finite dimensional distributions are multivariate normal, that is, (X t1,..., X tm ) N(µ, Σ) for all t 1,..., t m, then we call X t a Gaussian process Definition 3.5 (Markov process). A continuous time process X is a Markov process if for all t, each A σ(x s, s > t) and B σ(x s, s < t), we have P(A X t, B) = P(A X t ) Definition 3.6 (Diffusion process). Consider the stochastic differential equation dx t = µ(t, x)dt + σ(t, x)db t A diffusion process is path-continuous, strong Markov process such that lim h h 1 E(X t+h X t X t = x) = µ(t, x) lim h h 1 E([X t+h X t hµ(t, X)] 2 X t = x) = σ 2 (t, x) Definition 3.7 (Path-continuous). A process is path-continuous if lim t t X t = X t. Definition 3.8 (Lévy process). Let {X t, t be a stochastic process. We call X a Lévy process (i) X = a.s. (ii) It has stationary and independent increments (iii) X is stochastically continuous, that is, for all s, t, ϵ >, we have Equivalently, X s p Xt if s t. lim P( X s X t ϵ) =. s t Example 3.9 (Poisson process). Let (N(t), t ) be a stochastic process. We call N(t) a Poisson process if the following all hold: (i) N() = (ii) N has independent increments (iii) For all s, t, P(N(t + s) N(s) = n) = e λt (λt) n n! n =, 1, 2,...

6 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 6 The Poisson process is stochastically continuous - that is, P( N(t) N(s) ϵ) = P( N(t s) N() ϵ) = 1 P( N(t s) < α) = 1 P( N(t s) = ) = 1 e λ(t s) as t s The Poisson process is not path-continuous, that is because P(lim t s N(t) N(s) = ) 1 P( t s ϵ N(t) N(s) > δ) P( N(s + 1) N(s) δ) > 4. Lecture 4 - Thursday 1 March Definition 4.1 (Self-similar process). For any t 1, t 2,..., t n, for any c >, there exists an H such that We call H the Hurst index. Example 4.2 (Fractional process). (X ct1, X ct2,..., X ctn ) d = (c H X t1, c H X t2,..., c H X tn ). (1 B) d X t = ϵ t, ϵ t martingale difference BX t = X t 1, < d < 1 Definition 4.3 (Brownian motion). Let {B t, t } be a stochastic process. We call B t a Brownian motion if the following hold: (i) B = a.s. (ii) {B t } has stationary, independent increments. (iii) For any t >, B t N(, t) (iv) The path t B t is continuous almost surely, i.e. P( lim t t B t = B t ) = 1 Definition 4.4 (Alternative formulations of Brownian motion). A process {B t, t } is a Brownian motion if and only if {B t, t } is a Gaussian process E(B t ) =, E(B s B t ) = min(s, t) The process {B t } is path-continuous

7 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 7 Proof. ( ) For all t 1,..., t m, we have (B tm B tm 1,..., B t2 B t1 ) N(, Σ) as B t has stationary, independent increments. Hence, (B t1, B t2,,..., B tn ) is normally distributed. Thus B t is a Gaussian process. We can also show that E(B t ) = and E(B s B t ) = min(t, s). ( ) TO PROVE Corollary. Let B t be a Brownian motion. The so are the following: {B t+t B t, t } { B t, t } {cb t/c 2, t, c } {tb 1/t, t } Proof. Here, we prove that {X t } = {tb 1/t } is a Brownian motion. Consider n α i X ti. Then by a telescoping argument, we know that the process is Gaussian (can be written as a sum of X t1, X t2 X t1, etc). We can also easily show that E(X t ) = and E(X t X s ) = min(s, t) as required. We now show that lim t X. We must show t = a.s. Fix ϵ P { X t 1 n } = 1 n=1 m=1 <t< 1 m However, as X t has the same distribution as B t (as they are both Gaussian with same mean and covariance), we have that this is equivalent to P { B t 1 n } n=1 m=1 <t< 1 m which is clearly one. 5. Lecture 5 - Tuesday 15 March Theorem 5.1 (Properties of the Brownian motion). We have P(B t x B t = x,..., B tn = x n ) = P(B t x B s = x s ) = Φ( x x s t s ) Theorem 5.2. The joint density of (B t1,..., B tn is given by n g(x 1,..., x n ) = f(x tj x tj 1, t j t 1 ) where f(x, t) = 1 2πt e x2 2t

8 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 8 Theorem 5.3 (Density and distribution of Brownian bridge). Let t 1 < t < t 2. Then ( g(b t B t1 = a, B t2 = b) N a + (b a)(t t 1), (t ) 2 t)(t t 1 ). t 2 t 1 t 2 t 1 The density of B t B t1 = a, B t2 = b is given as g t1,t,t 2 (a, b, t) g t1,t 2 (a, b) Theorem 5.4 ( Joint distribution of B t and B s ). We have P(B s x, B t y) = P(B s x, B t B s y B s ) = = x y z x y 5.1. Properties of paths of Brownian motion. 1 e x 2πs 1 e x 2πs 2 1 2s 2 1 2s 1 e z 2 2 2(t s) dz1 dz 2 2π(t s) Definition 5.5 (Variation). Let g be a real function. Then the variation of g over an interval [a, b] is defined as V ([a.b]) = lim g(t j ) g(t j 1 ) where is the size of the partition a = t,..., t n = b of [a, b]. Definition 5.6 (Quadratic variation). The quadratic variation of a function g over an interval [a, b] is defined by [g, g] = lim g(t j ) g(t j 1 ) 2 Theorem 5.7 (Non-differentiability of Brownian motion). Paths of Brownian motion are continuous almost everywhere, by definition. Consider now, the differentiability of B t. We claim that a Brownian motion is non-differentiable almost surely, that is, We claim that B t B s lim = a.s. t s t s (i) Brownian motion is not differentiable almost surely for any t. (ii) Brownian motion has infinite variation on any interval [a, b], that is, V B ([a, b]) = lim B(t j ) B(t j 1 ) = a.s.

9 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 9 (iii) Brownian motion has quadratic variation t on [, t], that is [B, B]([, t]) = lim B(t j ) B(t j 1 ) 2 = t a.s. Proof. We know that (B ct1,..., B ctn ) d = (c H B t1,..., c H B tn ). This is true as B ct N(, ct) = c 1/2 N(, t) as required. Now suppose X t is H-self-similar with stationary increments for some < H < 1 with X =. Then, for any fixed t, we have X t X t lim = a.s. t t t t Consider X t X t P(lim sup M) t t t t which by stationary increments, is equal to X t P(lim sup M) = lim t t t P( B tn B t n t n t k n M) and as the RHS goes to zero, we have > lim n P( B t n B t t n t M) = lim n P( N(, 1) M (t n t ) 1/2 ) P( N(, 1) M (t n t ) 1/2 ) 1 as required. Now, Assume V B ([, t]) < almost surely. Consider Q n = n B t j B tj 1 2. Then we have Let. Then Q n max j n B t j B tj 1 lim Q n lim max B t j B tj 1 j n B tj B tj 1 B tj B tj 1 lim max j n B t j B tj 1 V B ([, t]) V B ([, t]) = because B s is uniformly continuous on [, t]. This is a contradiction to V B ([, t]) <.

10 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 1 Proof. We now show that E((Q n t) 2 ) as n. In fact, we have E(Q n ) = E(B tj B tj 1 ) 2 We now show L 2 convergence. We have E(Q n t) 2 = E((Q n E(Q n )) 2 ) = E( = = i=1 (t j t j 1 ) = t Y j where Y j = B tj B tj 1 2 E( B tj B tj 1 2 ) E(Y 2 j )) t j t j 1 2 E N(, 1) 4 C t as. Thus we have convergence in L Lecture 6 - Thursday 17 March Theorem 6.1 (Martingales related to Brownian motion). Let B t, t be a Brownian motion. Then the following are martingales with respect to F t = σ(b s, s t). (1) B t, t. (2) B 2 t t, t. (3) For any u, e ubt u2 t 2 Proof. (1) is simple. a.s. (2). We know that E( B 2 t ) is finite for any t. We can also easily show E(B 2 t t F s ) = B 2 s s Theorem 6.2. Let X t be a martingale satisfying X 2 t t is also a martingale. Then X t is a Brownian motion. Definition 6.3 (Hitting time). Let T α = inf {t,bt=α} (1) If α =, T =. (2) If α >, then P(T α t) = 2P(B t α) = 2 2πt We clearly have T α > t sup s t < α a e x2 2t dx

11 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS Lecture 7 - Tuesday 22 March Theorem 7.1 (Arcsine law). Let B t be a Brownian motion. Then P(B t =, for at least once, t [a, b]) = 2 π arcos a b Example 7.2. Processes derived from Brownian motion (1) Brownian bridge X t = B t tb 1 t [, 1] Consider X F (x), with X 1,... X n data. Our empirical distribution F n (x) = 1 1 {Xj X n We can then prove that (2) Diffusion process i=1 n F n (x) F (x) X t t (, 1) X t = µt + σb t This is a Gaussian process with E(X t ) = µt, Cov(X t, X s ) = σ 2 min(s, t). (3) Geometric Brownian motion This is not a Gaussian process. (4) Higher dimensional Brownian motion X t = X e µt+σbt B t = (B 1 t,..., B n t ) where the B i are independent Brownian motions, then 7.1. Construction of Brownian motion. Define a stochastic process ˆB n t = S [nt] E(S [nt] ) n With Then we can prove that ˆB B t n t n if t = i n = otherwise B t n B t on [,1] Definition 7.3 (Stochastic integral). We now turn to defining expressions of the form A X t dy t

12 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 12 with X t, Y t stochastic processes. Definition 7.4 ( f(b s) ds). We have f(t) dt exists if f is bonded and continuous, except on a set of Lebesgue measure zero. Thus, we can set f(t) dt = lim f(b yj )(t j t j 1 ) if f(x) is bounded. We now seek to find B s ds. Consider Q n = n B y j (t j t j 1 ). As the sum of normal variables, we know that Q n N(µ n, σn). 2 Since B s ds is normally distributed with mean, we have E(X 2 ) = E = = = 1/3 B s dx E(B s B t ) ds dt min(s, t) ds dt B t dt 8. Lecture 8 - Thursday 24 March Recall f(b s ) ds = lim B s ds N(, 1 3 ) = lim Consider I = X s dy s. We have the following: f(b yj )(t j+1 t j ) i=1 B yj (t j+1 t j ) Theorem 8.1. I exists if (1) The functions f, g are not discontinuous at the same point x. (2) f is continuous and g has bounded variation or, (2) f has finite p-variation and g has finite q-variation, where 1/p + 1/q = 1

13 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 13 For any p, we define p-variation by lim f(t j ) f(t j 1 ) p Theorem 8.2. If J = f(t) dg(t) exists for any continuous f, then g must have finite variation. Theorem 8.3. B s has bounded p variation for any p 2 and unbounded q-variation for any q < 2. Proof. We can write (for p 2), p = 2 + (p 2). Then we have t = lim max B tj B tj 1 p 2 B tj B tj 1 2 and hence t exists. Corollary. Thus db t is well defined if f has finite variation (as setting q = 1, p 2 gives 1/p + 1/q > 1). Consider B s db s - is this an R-S integral? Consider 1n = B tj (B tj+1 B tj ), 2n = B tj+1 (B tj+1 B tj ) We have that and Thus we know 2n 1n = 1n + 2n = (B tj+1 B tj ) 2 1 Bt 2 j+1 Bt 2 j ) = B1. 2 2n 1 2 (B ) 1n 1 2 (B2 1 1) Definition 8.4 (Itô integral). The Itô integral is defined by evaluating f(b tj ), the left-hand endpoint at each partition interval [t j, t j+1 ) 9. Lecture 9 - Tuesday 29 March Definition 9.1 (Itô integral). Consider f(s) db s. Where f(s) is a real function, B s a Brownian motion. We define the integral in two steps.

14 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 14 (1) If f(s) is a step function, define I(f) = f(s) db s = = m j+1 t j f(s) db s m c j (B tj+1 B tj ) (2) If f(s) L 2 ([, 1]), then let f n be a sequence in L 2 ([, 1]) such that f n f in L 2 ([, 1]). Then define I(f) to be the limitation in such situations such that E(I(f n ) I(f)) 2 in L 2 ([, 1]) or Let I(f) = lim n I(f n ) in probability. Remark. If f(x), g(x) are given step functions then αi(f) + βi(g) = I(αf + βg). Remark. If f(x) is a step function, then I(f) N(, f 2 (s) ds) Proof. I(f) = m c j (B tj+1 B tj ) N(, σ 2 ) where σ 2 = m c2 j (t j+1 t j ). Theorem 9.2. I(f) is well defined (independent of the choice of f n.) Proof. Let f n, g n f in L 2 ([, 1]). We then need to only compute n,m = E(I(f n ) I(g m )) 2 as m, n. In fact, n,m = E(I(f n g m )) 2 = (f n g m ) 2 dx 2 (f n f) 2 dx + 2 (g m f) 2 dx as n, m.

15 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 15 Remark. If f is continuous bounded variation, then f(s) db s = (R.S.) ) 1 f(s) db s Proof. Case 1). = lim δ f(t j )B(t j+1 B tj ) αi(f) + βi(g) = lim n [αi(f n) + βi(g n ) = lim n [I(αf n + βg n )] = I(αf + βg) and thus αf n + βg n αf + βg L 2 ([, 1]). Case 2). If I(f) = lim δ I(f n ) in probability. Then if σ 2 n f 2 (x) dx. In fact, as f 2 n dx = = I(f n ) N(, σ 2 n) N(, N(, (f n f + f) 2 dx f 2 dx + f 2 dx f 2 n(s) ds) f 2 (s) ds) (f n f) 2 dx + as other terms tend to zero by L 2 convergence and Hölder s inequality. Remark. (R.S.) f(s) db s exists if f is of bounded variation. Remark. If f is continuous then f 2 dx < and f n (t) = f(t j )I [tj,t j+1 ) f(t) in L 2 ([, 1]) Thus, I(f) = lim I(f n ) δ m = lim f(t j )(B tj+1 B tj ) δ (f n f)f dx

16 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 16 We have to prove if f is continuous. (f n f) 2 dx We have 1. Lecture 1 - Thursday 31 March (Itô ) f(s) db s N(, f(t) 2 dt if f is continuous, and of finite variation. In this case, we can write (R.S.) f(s) db s = (Itô ) f(s) db s = lim f(t j )(B tj+1 B tj ) Example 1.1. We have (1 t) db t = (Itô ) and by integrating by parts, we have Now consider (1 t) db t = (1 t)b t 1 (1 t) db t N(, (Itô ) B t d(1 t) = + where we now allow X s to be a stochastic process. X s db s Write F t = σ(b s, s t). Let π be the collection of X s such that (1 t) 2 dt) = N(, 1 3 ) B t dt N(, 1 3 ) (1) X s is adapted to F s, that is, for any s, X s is F s measurable. (2) X2 s ds < almost surely (R.S) Let π = {X s X s π, E(X2 s ) < }. Then π π. Let X s = e B2 s. Then E(Xs 2 1 ) = E(e B2 s ) = 1 4s s < 1 4 s 1 4 Definition 1.2 (Itô integral for stochastic integrands). We proceed in two steps. (1) Let X s = n ζ i1 [tj,t j+1) where ζ j is F tj measurable. Then I(X) = ζ j (B tj+1 B tj ). (2) If X π, there exists a sequence X n π such that X n are step process with 1 X n s X s 2 ds

17 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 17 as n in probability or in L 2 ([, 1]) if X s π. Proof. We show only for X s continuous, the general case can be found in Hu-Hsing Kuo (p.65). As X s is continuous, then it is in π. Then choose Xs n = X + j = 1 n X tj 1 [tj,t j+1) Then X s pi and for any s (, 1). We can also show that E( X n s X s 2 ) E( X n s X s 2 ) < and so by the dominated convergence theorem, we have lim n E( X n s X s 2 ) ds =. Finally if X s π, the Itô integral X s db s is defined as in probability or in L 2 if X π. I(X) = lim δ I(X n ) 11. Lecture 11 - Tuesday 5 March Let I(X) = X s db s in the Itô sense. We then require (1) X s is F s = σ{b t, t s}-measurable (2) X2 s ds < almost surely. Then I(X) = lim n I(X n ) where X n is a sequence of step functions converging to X in L 2, that is, (X n s X s ) n ds We then show that this definition is independent of the sequence of step functions. For any Y s step process, we have (Y n s Y m s ) 2 dx 2 Theorem 11.1 (Properties of the Itô integral). (1) For any α, β R, X, Y π, (Y m s X s ) 2 ds + 2 I(αX + βy ) = αi(x) + βi(y ) (X n s X s ) 2 dx

18 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 18 (2) For any X s π, we have If X π, Y s π, then (3) If X s continuous then E(I(X)) =, E(I 2 (X)) = E(I(X)I(Y )) = I(X) = lim δ E(X s Y s ) ds E(X 2 s ) ds X tj (B tj+1 B tj ) in probability. (4) If X is continuous and of finite variation and (R.S.) = (Itô ) B s dx s < then X s db s = X 1 B 1 X B B s dx s Proposition We now show why we require (1) X s is F s = σ{b t, t s}-measurable (2) X2 s ds < almost surely. Proof. Motivation for (1) X s = ζ j 1 (tj,t j+1) I(X) = ζ j (B tj+1 B tj ) E(I(X)) = = = E(ζ j (B tj+1 B tj )) E(E(ζ j (B tj+1 B tj ) F tj )) E(ζ j E(B tj+1 B tj F tj ))

19 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 19 Motivation for (2) E(I 2 (X)) = E( ζ j (B tj+1 B tj )) 2 = 2 i<j E(E(ζ i ζ j (B tj+1 B tj )(B ti+1 B ti )) F ti ) + = = E(ζj 2 )(t j+1 t j ) E(X 2 t ) dt E(ζj 2 (B tj+1 B tj ) 2 ) We now show that there X n s π. such that E(Xn s X s ) 2 ds. We have E(I 2 (X)) = E(I(X)) I(X n ) + E(I(X n s )) 2 = E(I 2 (X n )) + 2E(I(X n )(I(X) I(X n ))) + E(I(X) I(X n )) 2 By Cauchy-Swartz, the middle term tens do zero, and by definition, the third term tends to zero. Thus we have lim n E(I2 (X n )) = E(I 2 (X)) Let Xs n be step processes. We now show By definition, we have E(X n s X s ) 2 I(X) = lim n I(Xn ) = lim and we proved the required result in lectures. E(X n s X s ) 2 ds δ X tj (B tj+1 B tj ). We now show that the (R.S.) integral exists if X is continuous and of finite variation, and B s dx s <, the our integration by parts formula holds. We have (R.S.) which is our Itô integral by definition. X s db s = lim δ = lim δ X yj (B tj+1 B tj ) X tj (B tj+1 B tj )

20 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 2 Definition 11.3 (Itô process). Suppose Y t = X s db s, t is well defined, for X s π. Then Y t is an Itô processes. To show a process Y t is an Itô process, we need to show that X s ds)m a.s. and X s 2 ds < a.s. Theorem We have that Y t is continuous (except on a null set), is of infinite variation, as j+1 Y tj+1 Y tj = X s db s t j C min X s B tj+1 B tj s n B tj+1 B tj 12. Lecture 12 - Thursday 7 March From before, consider the Itô process Y t = X s db s. Lemma E( s X udb u F s ) =. Proof. Let X u = n ζ j1 [tj,t j+1). Then E( s X u db u F s ) = E(ζ j (B tj+1 B tj ) F s ) = = = E(E(ζ j (B tj+1 B tj ) F s ) F tj ) E(E(ζ j (B tj+1 B tj ) F tj ) F s ) where the third equality follows from the fact that (F t ) is an increasing sequence of σ-fields and the final follows from the fact that B tj+1 B tj is independent of F tj. Definition 12.2 (Local martingale). A process Y t is a local martingale if there exists a sequence of stopping times τ n, n 1 such that Y min(t,τ),ft is a martingale. Proposition We have the following. (1) If X s π, that is, E(X2 s ) ds <, then (Y t, F t ) is a martingale. (2) If X s π, that is X2 s ds < a.s., then (Y t, F t ) is a local martingale. (3) For f(x) satisfying f 2 (z) dz <., we have Y t = f(s) db s

21 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 21 is a Gaussian process. Proof. We have Y t is trivially F t -measurable. We have E( Y t ) < a.s. as E(Yt 2 ) = E(X2 s ) ds < by assumption. We have from the previous lemma. E(Y t F s ) = E( s X u db u + = E(Y s F s ) + E( = Y s Now assuming X s π, the exists an X n s as n. Set Y n t = Xn u db u. Then s X u db u = Y t Y s s s X u db u F s ) X u db u F s ) a step process such that E(X n u X u ) 2 du = Y t Y n t + Y n t Y n s + Y N s Y s Then for each n, we have have Z = E( s X u db u ) = E(Y t Yt n F s ) + E(Ys n Y s F s ) We only need to prove E( Z 2 ) which implies E(Z 2 ) = and thus Z = almost surely. We by definition of Y n t. E(Z 2 ) 2E(Y t Y n t ) 2 + 2E(Y s Y n s ) Lecture 13 - Tuesday 12 March Theorem Let f be continuous. Then lim f(θ j )(B tj+1 B tj ) 2 = δ f(b s ) ds

22 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 22 Proof. Let Q n = f(θ j ) f(b tj ) B tj+1 B tj 2 Note that n f(b t j )(B tj+1 B tj ) 2 f(b s) ds in probability. It only needs to show that Q n in probability. We have Q n max 1 j n f(θ j) f(b tj ) B tj+1 B tj 2 t = in probability from the quadratic variance of the brownian motion Theorem Let f be bounded on [, 1]. Then lim f(x tj )(B tj+1 B tj ) 2 = δ f(x s ds) Proof. We only prove for cases where E(X2 s ) ds is finite. In this case, there exists X n s step process such that as n. E(X n s X s ) 2 ds In fact, we can let Xs n = n X t j 1 [tj,t j+1]. Let Y n t = Then Yt n j+1 Yt n j = j+1 t j Xs n db s. Therefore, (Y tj+1 Y tj ) 2 = X n s db s (Y tj+1 Yt n j + Z 1j + Z 2j ) where Z 1j = Y tj+1 Y n t j+1 and Z 2j = Y tj = Y n t j. Continuing, we have = = (Y tj+1 Yt n j ) 2 + error Xt 2 j (B tj+1 B tj ) 2 + error if the error term goes to zero. We have X 2 s ds R n 2 Z 2 1j + 2 Z 2 2j + 2 Z 1j Y n t j+1 Y n t j + 2 Z 2j Y n t j+1 Y n t j + 2 Z 1j Z 2j + 2( Z 2 1j) 1/2 A 1/2 n + 2( Z 2 2j) 1/2 A 1/2 n + 2( Z 2 1j) 1/2 ( Z 2 2j) 1/2

23 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 23 and as Z ij 2 in probability, we have our result. Theorem 13.3 (Itô s first formula). If f(x) is a twice-differentiable function then for any t, f(b t ) f(b s ) = f (B u ) db u + 1 s 2 s f (B u ) du Proof. Let s = t 1,..., t n = t. We have [ f(b t ) f(b s ) = f(btj+1) f(b tj ) ] Applying Taylor s expansion to f(b tj+1 ) f(b tj ) we get f(b tj+1 ) f(b tj = f (B tj )(B tj+1 B tj ) f (θ j )(B tj+1 B tj ) 2 and so f(b t ) f(b s ) = lim = δ s f (B tj )(B tj+1 B tj ) + lim δ f (B u ) db u s f (B u ) du 1 2 f (θ j )(B tj+1 B tj ) 2 Example Lecture 14 - Thursday 14 March Definition 14.1 (Covariation). The covariation of two stochastic processes X t, Y t is defined as [X, Y ](t) = 1 ([X + Y, X + Y ](t) [X Y, X Y ](t)) 4 where [, ] is the quadratic variation previously defined. Definition 14.2 (Stochastic differential equation). Let X t be an Itô process. Then We write X t = X a + µ(s) ds + b a a σ(s) db s dx t = µ(t) dt + σ(t) db s By convention, we write dx t dy t = d[x, Y ](t). In particular, (dy t ) 2 = d[y, Y ](t). Theorem Let Y t be path continuous, and let X t have finite variation. Then [X, Y ](t) =.

24 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 24 Proof. We have [X, Y ](t) = 1 ([X + Y, X + Y ] [X Y, X Y ]) 4 = lim (X tj+1 X tj )(Y tj+1 Y tj ) δ lim δ max Y tj+1 Y tj X tj+1 X tj as Y t is path continuous and X t has finite variation. Corollary. From this theorem, we then have db t dt =, (dt) 2 =, (db t ) 2 = dt Corollary. For an Itô process X t given above, we then have d[x, X](t) = dx t dx t = (µ(t)dt + σ(t) db t ) 2 = σ 2 (t) dt Corollary. If f(x) has a twice continuous derivative, then [f(b t ), B t ](t) = Proof. From Itô s formula, we have f (B s ) ds df(b t ) = f (B t ) db t f (B t ) dt Then from above, we have d[f(b t ), B t ](t) = df(b t ) db t = f (B t )dt Theorem 14.4 (Itô s lemma). By Taylor s theorem, we have df(t, X t ) = f(t, X t) f(t, x) dt + t x dx t f(t, x) x=xt 2 x 2 (dx t ) 2 x=xt f(t, x) 2 x t dx t dt x=xt = f(t, X t) f(t, x) dt + t x (µ(t) dt + σ(t) db t ) f(t, x) x=xt 2 x 2 σ 2 dt x=xt ( f = t + f x µ(t) ) f 2 x 2 σ2 (t) dt + f x σ(t) db t

25 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 25 Example Let f(b t ) = e Bt. Then df(b t ) = e Bt db t ebt dt Theorem Assume T f 2 (t) dt <. Let X t = f(s) db s 1 t 2 f 2 (s) ds. Then Y t = e Xt is a martingale with respect to F t = σ(b s, s t). Proof. We have dy t = e X t dx t ex t (dx t ) 2 = e Xt [f(t) db t 1 2 f 2 (s) dt] ext f 2 dt = e X t f(t) db t and thus Y t = exs f(s) db s which is a martingale from previous work. 15. Lecture 15 - Tuesday 19 March Theorem 15.1 (Multivariate Itô s formula). Let B j (t) be a sequence of independent Brownian motions. Consider the Itô processes X i t, with dx i t = b i (t) dt + m σ ki (t) db k (t) Suppose f(t, x 1,..., x n ) is a continuous function of its components and has continuous partial derivatives f t, f f x i, 2 x i x j. Then We have df(t, X 1 t,..., X m t k=1 ) = f m t dt + f dxt i + 1 x i 2 dx i t dx j t = = i=1 m k,s=1 i, σ ki σ sj db k (t) db s (t) σ ki σ kj dt+ k=1 as d[b i, B j ](t) = when B i, B j are independent Example Let dx t = µ 1 (t)dt + σ 1 (t) db 1 (t) dy t = µ 2 (t)dt + σ 2 (t) db 2 (t) 2 f x i x j dx i t dx j t

26 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 26 with B 1, B 2 independent. Then and by independence, d[x, Y ](t) = dx t dy t =. d(x t Y t ) = Y t dx t + X t dy t + d[x, Y ](t) Example Let Then In particular, if µ 1 = µ 2 =, then X t Y t = dx t = µ 1 (t)dt + σ 1 (t) db 1 (t) dy t = µ 2 (t)dt + σ 2 (t) db 1 (t). d(x t Y t ) = Y t dx t + X t dy t + σ 1 (t)σ 2 (t) dt. [σ 1 (s)y s + σ 2 (s)x s ] db s + Thus, Z t = X t Y t σ 1(s)σ 2 (s) dt is a martingale. Theorem 15.4 (Tanaka s formula). We have σ 1 (s)σ 2 (s) dt where B t a = a + 1 L(t, a) = lim ϵ> 2ϵ sgn(b s a) db s + L(t, a) 1 Bs a ϵ ds Proof. Let Then we have and Then by Itô s formula, we have f ϵ (B t ) = f ϵ () + x a ϵ 2 x a > ϵ f ϵ (x) = 1 (x a)2 x a ϵ 2ϵ 1 x > a + ϵ f ϵ(x) = 1 ϵ (x a) x a ϵ 1 x < a ϵ f ϵ (x) = x a > ϵ 1 x a ϵ ϵ f ϵ(b s) db s f ϵ (B s ) ds

27 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 27 and Obviously 1 lim ϵ 2 f ϵ (B s ) ds = 1 ϵ Note that f ϵ(b s ) sgn(b s a) 2 ds = lim f ϵ() = a ϵ B s a ϵ 1 Bs a ϵ ds = L(t, a) 1 ϵ (B s a) sgn(b s a) 2 ds a.s. Theorem L(t, a) = Theorem If f is integrable on R, then L(t, s)f(s) ds = δ a (B s ) ds f(b s ) ds 16. Lecture 16 - Thursday 21 March Definition 16.1 (Linear cointegration). Consider two non stationary time series X t and Y t. If there exist coefficients α and β such that αx t + βy t = u t with u t stationary, then we say that X t and Y t are cointegrated. Definition 16.2 (Nonlinear cointegration). If Y t f(x t ) = u t is stationary, with f( ) a nonlinear function. 17. Lecture 17 - Tuesday 3 May Stochastic integrals for martingales. We now seek to define stochastic integrals with respect to processes other than Brownian motion. Example Let X t = Y s db s, and thus dx s = Y s db s. Then Z t = Y s dx s = Y s Y s db s Revus and Yov - Continuous Martingale and Brownian Motion Definition 17.2 (Martingale). A martingale with respect (1) M t adapted to F t. (2) E( M t ) <.

28 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 28 (3) E(M t F s ) = M s a.s. A process is a submartingale if (3) is replace with E(M t F s ) M s. A process is a supermartingale if (3) is replaced with E(M t F s ) M s. Example Let N t be a poisson process with intensity λ. Then (1) N t is a submartingale with respect to the natural filtration. (2) N t λt is a martingale with respect to the natural filtration. Theorem If E(H2 (s)) ds < and H(s) is adapted to F s = σ(b t, t s), then Y t = H(s) db s, t is a continuous, square integrable martingale - that is, E(Y 2 t < ). Theorem Let M t be a continuous, square integrable martingale with respect to F t. Then there exists an adapted process H(s) such that E(H2 (s)) ds < and M t = M + where B t is a Brownian motion with respect to F t. H(s) db s Theorem M t, t is a Brownian motion if and only if it is a local continuous martingale with [M, M](t) = t, t under some probability measure Q. Proof. A local continuous martingale is of the form M t = M + H(s) db s. Then we have [M, M](t) = H 2 (s) ds = t H(s) = 1a.s. M t = B t. Theorem Let M t, t e a continuous local martingale such that [M, M](t). Let τ t = inf{s : [M, M](s) t}. Then M(τ t ) is a Brownian motion. Moreover, M(t) = B([M, M](t)), t. This is an application of the change of time method. Example B t is a Brownian motion - and then Y t = Bt 2 t is a martingale. We have dy t = H(s) db s = 2B s db s. Thus, B 2 t t = 2 B s db s.

29 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 29 Definition 17.9 (Predictability of a stochastic process). A stochastic process X t, t is said to be predictable with respect to F t if X t F t for all t, where ( ) F t = F t+h, F t = σ F t h. h h> Example Let g t be a step process, with g t = ζ j 1 [tj,t j+1 )(t) Then g t is not predictable. Let g t be a step process, with g t = Then g t is predictable. i=1 ζ j 1 (tj,t j+1 ](t) i=1 Example Let N t be a Poisson process. Then N t is predictable, but N t is not predictable. From now on, assume M t, t is right continuous, square integrable martingale with left hand limits. Lemma M 2 t is a submartingale. Proof. E(M 2 t F s ) = E(M 2 s + 2(M t M s ) + (M t M s ) 2 F s ) = M 2 s + E((M t M s ) 2 F s ) M 2 s Theorem (Doob-Myer decomposition). By Doob-Myer we can write M 2 t = L t + A t where L t is a martingale, and A t is a predictable process, right continuous, and increasing, such that A =, E(A t ) <, t. A t is called the compensator of Mt 2, and is denoted by M, M (t). Example Consider Bt 2 = Bt 2 t + t. Then B, B (t) = t = [B, B](t) Theorem If M t is continuous then [M, M](t) = M, M (t).

30 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 3 Example Let N t be a Poisson process. Then we know Ñt = N t λt is a martingale. We may prove Ñt 2 λt is a martingale, that is, Ñ, Ñ (t) = λt. However, [Ñ, Ñ](t) = λt + Ñt Ñ, Ñ (t) Example X t = f(s) db s is a continuous martingale. Thus, [X, X](t) = f 2 (s) ds = X, X (t) Theorem If M t is a continuous, square integrable martingale, then Mt 2 [M, M](t) is a martingale, and so which implies [M, M](t) = M, M (t) + martingale E[M, M](t) = E M, M (t) = EM 2 t We now turn to defining integrals such as where M s is a martingale. X s dm s Definition Let L 2 pred be the space of all predictable stochastic process X s satisfying the condition X 2 s d M, M (s) <. Then the integral X s dms is defined as before in two steps. (1) If X s L 2 pred and X s = n ζ j1 [tj,t j+1). Define I(X) = ζ j (M tj+1 M tj ). (2) For all X s L 2 pred, there exists a sequence of step process Xn s such that X n s X s in L 2. Define I(x) to be the limit in such situations such that E(I(X) I(X n )) 2. Proposition Properties of the integral.

31 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 31 (1) If M s is a (local) martingale, then is a (local) martingale. f(s) dm s Proof. E( s f(u) dm u F s ) = (2) If M t is a square integrable martingale and satisfies then E( f 2 (s) d M, M (t)) < I(f) = f(s) dm s is square integrable with E(I(f)) =, and E(I 2 (f)) = f 2 (s) d M, M (s). In particular, if M(s) = s σ(u) db u, then X s dm s = X s σ(s) db s provided X2 s σ 2 (s) ds < and σ2 (s) ds (3) If X t = f(x) dm s and M s is a continuous, square integrable martingale, then [X, X](t) = f 2 (s) d[m, M](s) = X, X (t) 18. Lecture 18 - Tudsay 1 May Let M t be a martingale. Then M 2 t [M, M](t) is a martingale, and M 2 t = martingale + M, M (t) Recall that if M t is a continuous square integrable martingale, then M, M (t) = [M, M](t) Generally speaking, Theorem If [M, M](t) = M, M (t) + martingale t f 2 (s) d M, M (s) <

32 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 32 a.s. then is well defined, and and EY 2 t = = Y t = f(s) dm s f 2 (s) d M, M (t) f 2 (s) d[m, M](t) Y, Y (t) = [Y, Y ](t) = if M t -continuous f 2 (s) d[m, M](s). Proof. By Itô s formula, we also have Hence dy t = f(t) dm t d[y, Y ](t) = dy t dy t [Y, Y ](t) = = f 2 (t) dm t dm t = f 2 (t) d[m, M](t) Y 2 t = Y dy 2 t = 2 f 2 (s) d[m, M](s) = Y, Y (t) Y s dy s + 1 [Y, Y ](t) Y s f(s) dm s + Y, Y (t) = 2Y t f(t) dm t + d Y, Y (t) Y, Y (t) = Y 2 t 2 Y s f(s) dm s = [Y, Y ](t) since Y sf(s) dm s is a martingale Itô s integration for continuous semimartingales. Definition Let (X t, F t ) be a continuous semimartingale. Then X t = M t + A t n where M t is a martingale and A t is a continuous adapted process of bounded variation (lim δ A t j+1 A tj < ).

33 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 33 Definition 18.3 (Integrals for semimartingales). c s dx s = c s dm s + c s da s and since A t has bounded variation, the second integral is defined in the Riemann-Stiljes (R.S.) sense. Theorem 18.4 (Itô s formula). Let X t be a continuous semimartingale. Let f(x) have twice continuous derivatives. Then f(x t ) = f(x ) + f (X s ) dx s f (X s ) d[x, X](s) Proof. Partition the interval [, t], and use a Taylor expansion to express f(x) = f(x ) + f (x )(x x ) f (x )(x x ) Stochastic differential equations. Consider the equation We seek to solve for a function f(t, x) such that dx t = σ(t, X t ) db t = µ(t, X t ) dt X t = f(t, B t ). Such an f(t, B t ) is a solution to the stochastic differential equation. Definition 18.5 (Strong solution). X t = X + σ(s, X s) db s + µ(s, B s) ds 19. Lecture 19 - Thursday 12 May Theorem Let dx t = a(t, X t ) dt + b(t, X t ) db t Assume EX <. X is independent of B s and there exists a constant c > such that (1) a(t, x) + b(t, x) C(1 + x ). (2) a(t, x), b(t, x) satisfy the Lipschitz condition in x, i.e. a(t, x) a(t, y) + b(t, x) b(t, y) C x y for all t (, T ). Then there exists a unique (strong) solution. Example Let with c 1, c 2 constants. dx t = c 1 X t dt + c 2 X t db t,

34 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 34 Example Let dx t = [c 1 (t)x t + c 2 (t)] dt + [σ 1 (t)x t + σ 2 (t)] db t Let a(t, x) = c 1 (t)x + c 2 (t)x, b(t, x) = σ 1 (t)x + σ 2 (t). Just follow Kuo p Let H t = e Y t, Y t = Then by the Itô product formula, we have σ 1 (s) ds + c 1 (s) db s 1 2 c 2 1(s) ds d(h t X t ) = H t (dx t σ 1 (t)x t dt c 1 (t)x t db t c 2 (t)c 1 (t) dt) Then by definition of the X t, we obtain which can be integrated to yield d(h t X t ) = H t (c 2 (t) db t + σ 2 (t) dt c 1 (t)c 2 (t) dt) H t X t = C + H s c 2 (s) db s + Dividing both sides by H t we obtain our solution X t. H s (σ 2 (s) c 1 (t)c 2 (t)) ds Theorem The solution to the linear stochastic differential equation is given by X t = Ce Yt + where Y t = c 1(s) db s + dx t = [c 1 (t)x t + c 2 (t)] dt + [σ 1 (t)x t + σ 2 (t)] db t e Yt Ys c 2 (t) db s + ( σ1 (s) 1 2 c2 1(s) ) ds e Yt Ys (σ 2 (s) c 1 (t)c 2 (t)) dt 2. Lecture 2 - Tuesday 17 May 2.1. Numerical methods for stochastic differential equations. Theorem 2.1 (Euler s method). For the stochastic differential equation dx t = a(x t ) dt + b(x t ) db t, we simulate X t according to X tj = X tj 1 + a(x tj 1 ) t j + b(x tj 1 ) B tj Theorem 2.2 (Milstein scheme). For the stochastic differential equation dx t = a(x t ) dt + b(x t ) db t,

35 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 35 we simulate X t according to X tj = X tj 1 + a(x tj 1 ) t j + b(x tj 1 ) B tj b (X tj 1 )( B 2 t j t j ) 2.2. Applications to mathematical finance Martingale method. Consider a market with risky security S t and riskless security β t. Definition 2.3 (Contingent claim). A random variable C T : Ω R, F T -measurable is called a contingent claim. If C T is σ(s T )-measurable it is path-independent. Definition 2.4 (Strategy). Let a t represent number of units of S t, and b t represent number of units of β t. If a t, b t are F t -adapted, then they are strategies in our market model. Our strategy value V t at time t is V t = a t X t + b t β t Definition 2.5 (Self-financing strategy). A strategy (a t, b t ) is self financing if dv t = a t ds t + b t dβ t The intuition is that we make one investment at t =, and after that only rebalance between S t and β t. Definition 2.6 (Admissible strategy). (a t, b t ) is an admissible strategy if it is self financing and V t for all t T. Definition 2.7 (Arbitrage). An arbitrage is an admissible strategy such that V =, V T and P(V T > ) >. Alternatively, an arbitrage is a trading strategy with V =, and E(V T ) >. Definition 2.8 (Attainable claim). A contingent claim C T is said to be attainable if there exists an admissible strategy (a t, b t ) such that V T = C T. In this case, the portfolio is said to replicate the claim. By the law of one price, C t = V t at all t. Definition 2.9 (Complete). The market is said to be complete if every contingent claim is attainable Theorem 2.1 (Harrison and Pliska). Let P denote the real world measure of the underlying asset price X t. If the market is arbitrage free, there exists an equivalent measure P, such that the discounted asset price ˆX t and every discounted attainable claim Ĉt are P -martingales. Further, if the market is complete, then P is unique. In mathematical terms, C t = β t E (β 1 T C T F t ). P is called the equivalent martingale measure (EMM) or the risk-neutral measure.

36 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS Lecture 21 - Thursday 19 May For a trading strategy (a t, b t ), then the value V t satisfies V t = V + where B s is the riskless asset. α s ds s + b s dβ s To price an attainable option X, let (a t, b t ) be a trading strategy with value V t that replicates X. Then to avoid arbitrage, the value of X at time t = is given by V Change of Measure. Let (Ω, F, P) be a probability space. Definition 21.1 (Equivalent measure). Let P and Q be measures on (Ω, F). Then for any A F, if P(A) = Q(A) = then we say the measures P and Q are equivalent. If P(A) = Q(A) =, we write Q << P. Theorem 21.2 (Radon-Nikodyn). Let Q << P. Then there exists a random variable λ such that λ, E P (λ) = 1 and Q(A) = for any A F. λ is P=almost surely unique. A dp = E p (λ1 A ) Conversely, if there exists λ such that λ 1, E P (λ) = 1, then defining Q(A) = λ dp and then Q is a probability measure and Q << P. Consequently, if Q << P, then whenever E Q ( Z ) <. A E Q (Z) = E P (λz) The random variable λ is called the density of Q with respect to P, and denoted by λ = dq dp Example Let X N(, 1) and Y N(µ, 1) under probability P. Then there exists a Q such that Q is equivalent to P and Y N(, 1) under Q.

37 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 37 Proof. P(X A) = 1 2π Define Q(A) = A = 1 2π = 1 2π e t2 2 A dt µ2 µx e 2 dp A A µ2 µx e 2 e x2 2 dx e (µ+x)2 2 dx Then Q << P, P << Q and let λ = dq dp = µ 2 e µx 2 Then λ satisfies the conditions of Radon-Nikodyn theorem. Then we have E Q (Y ) = E P ((X + µ)λ) µ2 µx = (X + µ)e 2 dp = 1 2π (x + µ)e (µ+x)2 2 dx = 22. Lecture 22 - Tuesday 24 May Theorem Let λ(t), t T be a positive martingale with respect to F t such that E P (λ(t )) = 1. Define a new probability measure Q by Q(A) = λ(t ) dp Then Q << P and for any random variable X, we have A E Q (X) = E P (λ(t )X) E Q (X F t ) = E P ( λ(t )X λ(t) F t ) = E P(λ(T )X F t ) E P (λ(t ) F t ) a.s. and if X F t, then for any s t, we have ( ) λ(t)x E Q (X F s ) = E P λ(s) F s ( ) ( )

38 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 38 Consequently a process S(t) is a Q-martingale if and only if S(t)λ(t) ( ) is a P-martingale Proof. ( ). We have Q(E P (λ(t )) F t = ) = E p (λ(t )1 EP (λ(t ) F t=)) = We have ( ). We have ( ) ( EP (λ(t )X F t ) E Q E P (λ(t ) F t ) 1 A = E Q λ(t) E ) P(λ(T )X F t ) E P (λ(t ) F t ) 1 A = E P (E P (λ(t )X F t )1 A ) = E P (λ(t )X1 A ) = E Q (X1 A ) ( ) λ(t)x 1 E P λ(s) F s λ(s) E P(λ(t)X F s ) = 1 λ(s) E P (E P (λ(t )X F t ) F s ) = 1 λ(s) E P(λ(T )X F s ) = E Q (X F s ) because of ( ). ( ). We have E Q (S(t) F u ) = S(u) E P ( λ(t)s(t) λ(u) F u ) = S(u) E P (λ(t)s(t) F u ) = λ(u)s(u) as required. Theorem Let B s, s T be a Brownian motion under P. Let S(t) = B t + µ t, u. Then there exists a Q equivalent to P such thats(t) is a Q-Brownian motion and λ(t ) = dq dp = e µb T 1 2 µ 2T. Note that S(t) is not a martingale under P, but it is a martingale under Q.

39 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 39 Proof. Under Q, Q(B = ) = λ(t ) dp = 1 B = S(t) is a Q-martingale if and only if S(t)λ(t) is a P-martingale. But we have X t = S(t)λ(t) = (B t + µt) e µb t 1 2 µ2 t is a martingale. Finally, note that [S, S](t) = [B, B](t) = t Black-Scholes model. Definition 22.3 (Black-Scholes model). The Black-Scholes model assumes the risky asset S t follows the diffusion process given by ds t = µ dt + σ db t and the riskless asset follows the diffusion S t Define the discounted process as follows: dβ t β t = r dt Ŝ t = S t β t, ˆVt = V t β t, Ĉ t = C t β t. 23. Lecture 23 - Thursday 26 May Lemma (a) By a simple application of Itô s lemma, (b) Ŝt is a Q-martingale with with q = µ r σ. (c) Note that dŝt Ŝ t dŝt Ŝ t = (µ r) dt + σ db t. λ = dq dp = e qb T 1 2 q 2 T = σd(b t + µ r σ t) = σd ˆB t where ˆB t = B t + qt is a Brownian motion under Q. (d) dŝt = σŝt d ˆB t.

40 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 4 (e) In a finite market, where S t takes only finitely many values, Ŝt is a Q-martingale is a necessary condition for no-arbitrage. (f) Note that ds t S t = µ dt + σ db t = r dt + σ d ˆB t Theorem A value process V t is self financing if and only if the discounted value process ˆV t is a Q-martingale. d ˆV t = a t dŝt dv t = αds t + b t dβ t V t = V + a s ds s + b s dβ s V t is self financing. Proof. By Itô s formula, we have d ˆV t = e rt dv t re rt V t dt = e rt (a t ds t + b t dβ t ) re rt (a t S t + b t β t ) dt a t (e rt ds t re rt dt) = a t dŝt Theorem In the Black-Scholes model, there are no arbitrage opportunities. Proof. For any admissible trading strategy (a t, b t ), we have that the discounted value process ˆV t is a Q-martingale. So if V =, then E( ˆV ) =, and we have E Q ( ˆV T ) = E Q ( ˆV T F ) = ˆV = which implies Q( ˆV T > ) =, which implies P(V T > ) =, which implies that E P (V T ) =, which then implies no arbitrage. Theorem For any self financing strategy, V t = a t S t + b t β t = V + And so a strategy is self financing if a u ds u + S t da t + β t db t + d[a, S](t) = We now consider several cases. (1) If a t is of bounded variation, then [a, S](t) =. Hence S t da t + β t db t =, b u dβ u

41 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 41 which implies Hence da t db t <. db t = S t β t da t (2) If a t is a semi-martingale a t = a 2 t +A t, then b t must be a semi-martingale, where b t = b 2 t +B t where a 2 t, b 2 t are the martingale parts and A t, B t are of bounded variation. 24. Lecture 24 - Tuesday 31 May Theorem Given a claim C T under the self-financing assumption, there exists a Q-martingale such that In particular, we have V t = E Q ( e r(t t)c T F t ), F t = σ(b s, s t). V = E Q (e rt C T ) Proof. For any Q-martingale ˆV t, we have ˆV t = E Q ( ˆV T F t ). Then V t = E Q (e r(t t) C T F t ), as V T = C T. Theorem A claim is attainable (there exists a trading strategy replicating the claim), that is, G(t) = V t = V + G(t) a u ds u + b u dβ u V T, V T = C T Theorem Suppose that C T is a non-negative random variable, C t F T and E Q (C 2 T f < ), where Q is defined as before. Then (a) The claim is replicable. (b) where ĈT = e rt C T. V t = E Q ( e r(t t) C T F t ) In particular, V = E Q (e rt C T ) = E Q (ĈT ). Theorem Assume ˆV t = E Q (ĈT F t ). exists an adapted process H(s) such that ˆV t = E Q (ĈT F t ) Using the martingale representation theorem, there ˆV t = ˆV H(s) d ˆB s d ˆV t = H(t) d ˆB t.

42 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 42 On the other hand, d ˆV t = a t dŝt = a t σŝt d ˆB t. Hence we obtain our required result, and then solve for b t. a t = H(t), σŝt ( ) Example Let C T = f(s T ). Then V t = E Q e r(t t)f(s T ) F t. Since Ft = σ(b s, s t) = σ( ˆB s, s t), and Ŝt is a Q-martingale, we have Ŝ t = Ŝe σ2 2 t+σ ˆB t and so Then and so where S T = e rt Ŝ T = Ŝte rt e σ2 2 (T t)+σ( ˆB T ˆB t). V t = E Q [e r(t t) f F (t, x) = E Q [e r(t t) f = e r(t t) f (e rt Ŝ t e σ2 2 (T t)+σ( ˆB T ˆB t ) )] V t = F (t, S t ) R σ2 ( (e )] 2 )(T t)+σ T tz (xe σ2 2 (T t)+σz T t ) ϕ(z) dz where ϕ(z) = 1 2π e z2 2. Example In particular, if f(y) = (y K) +, then we obtain F (t, x) = e rθ d 1 xe σ2 2 θ+xz θ z2 2 dz K = xφ(d 1 + σ θ) Ke rθ Φ(d 1) = xφ(d 1 ) Ke rθ Φ(d 2 ), d 1 e rθ z2 2 dz where d 1 = log ( ) x K + (r + σ 2 2 )θ σ, d 2 = d 1 σ θ θ Theorem 24.7 (Black-Scholes model summary). V t = a t S t + b t β t, where ds t S t = µ dt + σ db t dβ t β t = r dt

43 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 43 (a) Ŝt = e rt S t is a Q-martingale, where dq dp = e qb T 1 2 q 2 T and q = µ r σ. Then ˆB t = B t + µ r σ t is a Q-Brownian motion, and dŝt = σŝt d ˆB t. (b) V t is self-financing if (a t, b t ) satisfies S t da t + β t db t + d[a, S](t) = which then implies ˆV t is a Q-martingale, ˆV t = e rt V t. (c) There are no arbitrage opportunities in the Black-Scholes model. Theorem 25.1 (Feyman-Kac formula). 25. Lecture 25 - Thursday 2 June (1) Suppose the function F (x, t) solves the boundary value problem F (t, x) t such that F (T, x) = Ψ(x). (2) Let S t be a solution of the SDE F (t, x) + µ(t, x) + 1 x 2 σ2 (t, x) 2 F (t, x) x 2 = ds t = µ(t, S t ) dt + σ(t, S t ) db t ( ) where B t is a Q-Brownian motion (3) Assume Then where F t = σ(b s, s t). T E(σ(t, S t ) 2 F (t, S t ) x 2 ) dt < F (t, S t ) = E Q (Ψ(S T ) F t ) = E Q (F (T, S T ) F t ). Proof. It is enough to show that F (t, S t ) is a martingale with respect to F t under Q. By Itô s lemma, we have df (t, S t ) = f(t, S t) t = which is a Q-martingale. dt + F (t, S t) x [ F t + µ(t, S t) F = F (t, S t) σ(t, S t ) db t x x + σ2 (t, S t ) 2 ds t F (t, S t ) 2 x 2 (ds t ) 2 ] 2 F (t, S t ) x 2 dt + F (t, S t) σ(t, S t ) db t x

44 MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS 44 Theorem 25.2 (General Feyman-Kac formula). Let S t be a solution of the SDE ( ). Assume that there is a solution to the PDE Then F (t, x) t Proof. Again by Itô s lemma, df (t, S t ) = F (t, x) + µ(t, x) + 1 x 2 σ2 (t, x) 2 F (t, x) x 2 = r(t, x)f (t, x). F (t, S t ) = E Q ( e T t r(u,s u) du F (T, S T ) F t ) ( F t + µ F x + σ2 2 ) F 2 x 2 dt + F x σ(t, S t) db t = r(t, x)f (t, S t ) dt + dm t where M t = F x σ(u, S u) db u. Hence we have df (t, S t ) = r(t, S t )X t dt + dm t d [e ] ν t r(u,su) du X ν = e ν t r(u,su) du dm ν ( ) e T t r(u,s u) du F (T, S T ) = F (t, S t ) + e ν t r(u,s u) du dm ν ( ) t ( E Q e ) T t r(u,su) du F (T, S T ) F t = F (t, S t ) ( ) ) and so we obtain our result, T + E Q ( T e ν t r(u,su) du F t x σ(ν, S ν) db ν F t }{{} = F (t, S t ) = E Q ( e T t r(u,su) du F (T, S T ) F t )