Stochastic calculus Introduction I. Stochastic Finance. C. Azizieh VUB 1/91. C. Azizieh VUB Stochastic Finance
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1 Stochastic Finance C. Azizieh VUB C. Azizieh VUB Stochastic Finance 1/91
2 Agenda of the course Stochastic calculus : introduction Black-Scholes model Interest rates models C. Azizieh VUB Stochastic Finance 2/91
3 Stochastic calculus : introduction C. Azizieh VUB 3/91
4 Agenda Stochastic calculus Introduction I Stochastic calculus : introduction Probability space, stochastic process in continuous time, filtration, martingales, convergence of random variables Itô stochastic integral : motivation Itô stochastic integral : construction Itô lemma Stochastic differential equations Girsanov theorem 4/91
5 Probability space Stochastic calculus Introduction I We consider the set of all possible states of the world Ω, with a sigma-algebra F. We speak about the notion of measurable space (Ω, F). In financial modelling, Ω represents the different possible evolution scenarios in the market : each scenario ω Ω typically represents a possible evolution in time of the prices of a set of financial instruments. An element of the sigma-algebra, A F, represents an event, hence typically a set of scenarios, to which a probability can be attributed : 5/91
6 Probability space Stochastic calculus Introduction I A probability measure is a mapping : P : F [0, 1] : A P(A) such that : P( ) = 0, P(Ω) = 1, If A i (i IN 0 ) are 2 by 2 disjoint, then : P ( i IN0 A i ) = P(A i ) i=1 (sigma-additivity) We will always work on a probability space (Ω, F, P). 6/91
7 Stochastic process and filtration The prices of financial assets, or more generally, of market variables (e.g. : interest rates, exchange rates, etc), will be modelled by stochastic processes in continuous time, i.e. the given of a random variable at each instant t : X : [0, + ) Ω R : (t, ω) X(t, ω) such that X(t,.) : Ω R is a random variable for each fixed t. Hence we have : For a fixed scenario ω, we have a trajectory For a fixed instant t, we have a random variable Now, if t represents time, in order to model concepts like the progressive arrival of information, causality and predictability, we introduce the concept of filtration. 7/91
8 Stochastic process and filtration A filtration is an increasing family of sub-σ-algebras on the probability space : for all 0 < s < t. F 0 = {Ω, } F s F t F Interpretation : in a dynamical framework, when time passes, information is progressively revealed to the observer quantities seen as stochastic at t=0 are not stochastic anymore at a future instant t > 0 if their value is revealed to the observer in the mean time, i.e. given the available information at t > 0. The notion of filtration allows to enrich the modelling framework in order to take into account the time dimension and information flow. 8/91
9 Filtration and process Notions linked by two definitions : Definition (Natural filtration or filtration generated by a stochastic process) Every stochastic process X generates a filtration called the natural filtration of the process, and defined by : F t = σ(x(s) s t) t (filtration corresponding to the history of the process) Definition (Adapted process) A process X is said to be adapted to a filtration (F t ) if X(t) is F t measurable t 9/91
10 Filtration and process Concretely, if (F t ) represents the available information (in a market) through time, a process (X t ) is adapted if its value at t is known as soon as we have reach instant t. In finance, we will model prices of financial assets by adapted processes. Definition { A filtration {F t } is said to satisfy the usual conditions if F0 contains all sets of zero probability t 0 F t = F s (right continuous). s>t In what follows, we will always assume that the filtration satisfies the usual conditions (...) 10/91
11 Continuous stochastic process A continuous stochastic process is such that t X t (ω) is continuous for all ω Ω 0 Ω with P [Ω 0 ] = 1. In other words : a process whose trajectories are almost all continuous X(t) Time t 11/91
12 Continuous time martingale The notion of martingale is central in market finance. Definition (continuous time martingale) A stochastic process (M t ) t 0 is a martingale w.r.t. a filtration {F t } t if for all t : 1 M t is F t -measurable (adapted process) 2 M t is integrable (hence IE [ M t ] < ) 3 IE [M t F s ] = M s for all s < t. Interpretation : accumulated gains of a player in an equilibrated game In particular, E[M t ] = M 0. 12/91
13 Martingales - Doob s results in continuous time Definition (Stopping time in a continuous time framework) A random variable τ with values in R + { } is a stopping time w.r.t. the filtration {F t } t if {τ t} F t for all t 0. Interpretation : we don t need information on the future to know at each instant if τ has already occurred. Example : the first instant at which a stock price reaches some fixed level Counter-example : the last instant before a future date at which a stock reaches a given level. Other examples? 13/91
14 Martingales - Doob s results in continuous time An example of link between stopping time and stochastic process : Definition (Process stopped by a stopping time) Let (Y t ) t be an adapted process to the filtration {F t } t and let τ be a stopping time. The process stopped at instant τ, denoted by (Y τ t ) t or (Y t τ ) t is defined by Y τ t (ω) = Y t τ(ω) (ω) t. 14/91
15 Martingales - Doob s results in continuous time Example : Let Y be a stochastic process and let us define the stopping time : T = inf{t 0 Y (t) M} for M R (M = barrier). The process Y stopped at M is then : { Y (t) if t T Z (t) = Y (t T ) = M if t > T 15/91
16 Martingales - Doob s results in continuous time When we stop a (continuous) martingale with a stopping time, we still have a (continuous) martingale : Theorem (Doob s stopping time theorem) Let (M t ) t be a continuous martingale w.r.t. filtration {F t }. Let τ be a stopping time w.r.t. {F t }. Then the process X t = M t τ is a continuous martingale w.r.t. {F t }. Proof : see [Steele, section 4.4]. 16/91
17 Martingales - Doob s results in continuous time Theorem (Doob s maximal inequalities in continuous time) Let (M t ) t be a continuous non-negative sub-martingale and λ > 0.. Then for all p 1, we have : ( ) λ p P sup M t > λ IE [ M p ] T (1) {t:0 t T } and, if M T L p (dp) for some p > 1, then : sup M t p p p 1 M T p. (2) {t:0 t T } Proof : see [Steele, section 4.4]. 17/91
18 Convergence of random variables Let {X n n IN} be a sequence of random variables. What is lim n X n?? One can actually define different types of convergence : almost sure convergence quadratic convergence (or L 2 ) convergence in probability convergence in distribution convergence in L p norm... 18/91
19 Almost sure convergence X n p.s. X iff P[ω Ω X n (ω) X(ω)] = 1 Example : Strong law of large numbers : If (X n ) is a sequence of random variables i.i.d. with finite expectation IE[X 1 ] = µ, then : 1 n p.s. X i E[X 1 ] = µ n i=1 19/91
20 Quadratic convergence (or in L 2 ) X n X in quadratic mean (or in L 2 ) iff X n X 2 L = IE[(X 2 n X) 2 ] = (X n (ω) X(ω)) 2 dp(ω) 0 Ω 20/91
21 Convergence in probability P X n X iff ɛ > 0 : P[ω Ω : Xn (ω) X(ω) ɛ] 0 Link between these different types of convergence : Almost sure convergence convergence in probability Convergence in L 2 convergence in probability 21/91
22 Convergence in distribution X n D X iff f bounded continuous : E[f (Xn )] E[f (X)] In particular this is equivalent to the convergence of characteristic functions : Definition : characteristic function of a random variable X : z R : Φ X (z) := E[exp(izX)] = e izx(ω) dp(ω) = e izx dµ X (x) (characteristic function = inverse Fourier transform of the density of X if X has such a density (up to some normalization constant)...) Result : (X n ) converges in distribution iff we have convergence of characteristic functions : X n D X iff ΦXn (z) Φ X (z) for all z R Ω R 22/91
23 Convergence in distribution convergence in probability convergence in distribution Theorem (Central limit theorem) If (X n ) is a sequence of i.i.d. random variables with finite variance σ 2, then n i=1 X i ne[x 1 ] σ D N(0, 1) n 23/91
24 Brownian motion : Motivation Suppose that we would like to define a stochastic model for the evolution of a financial asset in continuous time. Discrete time Continuous time Deterministic model S n = S 0 (1 + i) n S(t) = S 0 e δt Stochastic model S n = S 0 n k=1 (1 + i k)?? 24/91
25 : Motivation Continuous deterministic model We consider a financial asset which evolves deterministically in time : constant rate of return : δ value of the asset at t : S(t) we assume a double linearity hypothesis : S(t) = δs(t) t the evolution of the price is supposed to be proportional to time and to the invested amount 25/91
26 : Motivation Continuous deterministic model If we take the limit for t 0, we obtain the differential equation : ds(t) = δs(t)dt whose solution (passing by (0, S(0))) is given by the exponential function : S(t) = S(0)e δt Now, if we assume that the rate of return is not constant anymore, i.e. if δ = δ(t), then the differential equation becomes : whose solution is given by : ds(t) = δ(t)s(t)dt t0 δ(s)ds S(t) = S(0)e 26/91
27 : Motivation Binomial model on several periods (discrete time model) : Deterministic model : Stochastic model : S(n) = S(0) n (1 + i k ) = S(0) k=1 S(n) = S(0) n k=1 e Y k n k=1 où Y 1,..., Y n are i.i.d. : { δ + σ p = 1/2 Y i = δ σ q = 1 p = 1/2 e δ k We can rewrite : Y i = δ + σx i where X i = { 1 1/2 1 1/2. 27/91
28 : Motivation We hence have : S(n) = S(0) n k=1 e δ+σx k = S(0)e nδ+σ n k=1 X k We hence obtain the cumulative log-return until n : log(s(n)/s(0)) = n Y i = i=1 }{{} δn + σ determ. trend n i=1 X i }{{} random walk In particular, the moments of cumulative log-returns are given by : E[log(S(n)/S(0))] = δn var[log(s(n)/s(0))] = σ 2 n (variance proportional to time) 28/91
29 : Motivation We will now consider passing to a continuous time model. This will be done in two steps : 1 Modification of the (space and time) scale of the random walk : The time periods of 1 are replaced by t, and jumps of +1 and -1 are replaced by jumps of + x and x on each period of time. The random walk then becomes : W (n) = x. and in terms of moments : E[W (n)] = 0 m X i, with m = n/ t i=1 Var[W (n)] = m( x) 2 = n ( x)2 t 29/91
30 : Motivation 2 Passing to the limit x, t 0 We will now consider a sequence of x k, t k with x k, t k 0 in such a way that the limiting process is non-trivial. For this purpose, we will require : (and not x k t k 1!) ( x k ) 2 t k 1 30/91
31 : Motivation Construction of BM as limit of a random walk : illustration X k (t) Time t 31/91
32 : Motivation Construction of BM as limit of a random walk : illustration X k (t) Time t 32/91
33 : Motivation Construction of BM as limit of a random walk : illustration X k (t) Time t 33/91
34 : Motivation Construction of BM as limit of a random walk : illustration X k (t) Time t 34/91
35 : Motivation Construction of BM as limit of a random walk : illustration X k (t) Time t 35/91
36 : Motivation Construction of BM as limit of a random walk : illustration X k (t) Time t 36/91
37 : Motivation Construction of BM as limit of a random walk : illustration X k (t) Time t 37/91
38 : Motivation Construction of BM as limit of a random walk : illustration X k (t) Time t 38/91
39 : Motivation One can show that the sequence of processes (W (k) (t)) defined above converges in distribution to a process with continuous trajectories called Brownian motion : where W (t) : t/ t k W (k) (t) = x k is a continuous time process is called Standard Brownian motion, or Wiener process, or infinitesimal random walk i=1 X i W (t) 39/91
40 : Motivation This convergence is actually the dynamic equivalent of the central limit theorem in the static case : If (X n ) is a sequence of i.i.d. random variables of mean µ and finite variance σ 2, then we note S n = n i=1 X i, S n nµ σ n D N(0, 1) 40/91
41 : Motivation By construction, this process W (t) has the following properties : (i) E[W (k) (t)] = 0 E[W (t)] = 0 (ii) var[w (k) (t)] = t ( X k ) 2 t k var[w (t)] = t (iii) W (k) (t) = sum of i.i.d. rvs W (t) N(0, t) (Central Limit Theorem) (iv) W is a process with independent and stationary increments 41/91
42 : Definition All these lead to the following definition : Definition (Standard ) A stochastic process {B t : 0 t < T } in continuous time is standard Brownian motion on [0, T ) if 1 B 0 = 0. 2 The increments of B t are independent, i.e. for all finite set of instants : 0 t 1 < t 2 <... < t n < T, the random variables are independent. B t2 B t1, B t3 B t2,..., B tn B tn 1 3 for all 0 s < t T, the increments B t B s have a Gaussian distribution of mean 0 and variance t s. 4 B t (ω) is a continuous function of t for almost all ω. 42/91
43 : Illustration A typical sample path : X(t) Time t 43/91
44 : back to the stochastic continuous model The random walk W (k) (t) converges to the standard Brownian motion W (t). The price of the asset S(t) hence becomes at the limit : S (k) (t) = S(0)e tδ+σ x t/ tk k X k=1 k δt+σw (t) S(t) = S(0)e This is what will be called later geometric Brownian motion. 44/91
45 : Historical Perspective 1829 : Brown : movement of pollen particles in suspension 1900 : Bachelier models financial stock prices by a Brownian motion 1905 : Einstein models particles in suspension in a liquid or a gaz, subject to collisions 1923 : Wiener proposes a rigorous construction of Brownian motion (also called Wiener process ) 1944 : Ito contributes to define a stochastic integral w.r.t. a Brownian motion BM has become a central process in finance for modelling the uncertainty present in the markets 45/91
46 Brownian motion and the Markov property By using the independence of disjoint increments we directly obtain : i.e. Markov property. P[X t+s B X u, 0 u t] = P[X t+s B X t ]. We also get : P[X t+s B X t = x] = P[X t+s X t B x X t X 0 = x] = P[X t+s X t B x] = P[X t+s X t + x B] = P[Y B], where Y N (x, s). We also have : P[X t+s B X t = x] = B 1 2πs e (y x)2 /(2s) dy. 46/91
47 Non standard Brownian motion from level a Z (t) = a + B(t) where (B t ) is a standard BM Brownian motion from level a, with drift µ and volatility σ Z (t) = a + µt + σb t In particular, E[Z (t)] = a + µt var[z (t)] = σ 2 t 47/91
48 Brownian motion and Gaussian processes Definition ( Gaussian processes) {X t : 0 t < } is a Gaussian process iff every linear combination of values of X at different instants is normally distributed : n α k X tk k=1 has a normal distribution for all coefficients α k and instants t k. Remark : such a process is characterized by : a mean function t E[X t ] and an autocovariance function (s, t) Cov[X s, X t ]. Example of Gaussian process : Brownian motion. 48/91
49 Brownian motion and Gaussian processes Let us compute the covariance function of a Brownian motion. for all s t : cov (B s, B t ) = IE [(B t B s + B s ) B s ] = IE[B t B s ]IE[B s ]+IE [ B 2 s ] = IE [ B 2 s ] = s = cov (B s, B t ) = min (s, t) 0 s, t <. 49/91
50 Brownian motion and Gaussian processes Lemma Let {X t : 0 t T } be a Gaussian process with IE [X t ] = 0 for all 0 t T and let cov (X s, X t ) = min (s, t) for all 0 s, t T, then the process {X t } has independent increments. Moreover, if the process has continuous trajectories and X 0 = 0, then {X t } t is a standard Brownian motion on [0, T ]. 50/91
51 Martingales and Brownian motion Theorem Let (B t ) be a standard Brownian motion. Then : 1 (B t ) t 2 ( B 2 t t ) t ( ( 3 exp αbt α 2 t/2 )) t are martingales w.r.t. à {F t } t (= the family of sub-sigma algebra σ (B s, s t) completed by adding the sets of zero probability). 51/91
52 Martingales and Brownian motion Proof 1) B t F t-measurable, IE [ ] Bt 2 = t <, IE [B t B s F s ] = 0 since IE [B t B s] = 0. 2) ( Bt 2 t ) F t-measurable, IE [ B 2 t t ] IE[B t 2 ] + t = 2t, IE [ ] [ ] Bt 2 t F s = IE (B t B s) 2 Bs 2 + 2B tb s t F s ] = IE [(B t s) 2 F s t Bs 2 + 2Bs 2 = t s t + Bs 2 = Bs 2 s. 3) IE [ exp ( αb t α 2 t/2 ) ] F [ ( s ) = IE exp α (B t B s) α2 (t s) 2 = IE [exp (αb t s)] exp ( = exp αb s α2 s 2 ( α2 2 (t s) ) exp ] ( F s exp ( ) αb s α2 s 2 ) αb s α2 s 2 ). 52/91
53 Martingales and Brownian motion Remark : We can actually show that the converse is also true : if (X t ) is a continuous process such that (X t ) and (Xt 2 t) are martingales and X(0) = 0, then (X t ) is a standard Brownian motion Brownian motion appears as the fundamental example of continuous martingale 53/91
54 Brownian motion and bounded variation Let us consider a sequence of subdivisions of interval [0,t] : 0 = t 0 < t 1 <... < t n = t with thikness tending to 0 (lim n max 0 i n t i+1 t i = 0). We know that [ n ] n E ( B(t i )) 2 = t i = t. i=1 If f is a deterministic function with bounded variation, lim n n i=1 f (t i) exists, and n ( f (t i )) 2 = 0. lim n i=1 This implies that trajectories of a Brownian motion cannot be of bounded variation. In particular, they cannot be differentiable (in the classical sense). Hence, these trajectories are continuous everywhere but nowhere differentiable. i=1 54/91
55 Brownian motion and quadratic variation Lemma (Quadratic variation of a BM) Let t j = j 2 n t (j = 0,..., 2 n ) be a partition of [0,t]. Then Z n = (B(t j ) B(t j 1 )) 2 t in L 2 2 n j=1 Proof (i) E[Z n ] = t (ii) IE[(Z n t) 2 ] = Var[Z n ] = 2 n j=1 Var( B(t j) 2 ). Now, if X N(0, σ 2 ), then Var(X 2 ) = 2σ 4 (exercise), which implies : 2 n ( ) 2 t Var(Z n ) = 2 2 n = 2 n+1 t n j=1 55/91
56 Let us consider a market composed of d assets of price at t : S t = (S 1 t,..., S d t ), modeled like a vector process A trading strategy can be modelled by a vector φ t describing the quantities invested in each of the assets at instant t : φ t = (φ 1 t,..., φ d t ) The value at t of the portfolio obtained by following this strategy is then given by : d V t(φ) = φ k t St k = φ t.s t k=1 We will denote by 0 = T 0, T 1,..., T n+1 = T the rebalancing instants : between 2 such instants, the composition of the portfolio is supposed unchanged. 56/91
57 We can then denote by φ i = (φ 1 i,..., φ d i ) the composition of the portfolio between T i and T i+1, and rewrite φ t as : n φ t = φ 0 I t=0 + φ i I ]Ti,T i+1 ] i=0 ( ) Generally, the rebalancing instants T i are random (depending e.g. from the levels reached by some assets in the market...). The strategy φ t appears as a Stochastic process. Definition (Simple predictable process) A process φ t (t [0, T ]) that can be represented by ( ) is called a simple predictable process if moreover the stochastic instants T i are stopping times, and if φ i are bounded random variables with φ i F Ti, (which means that the value φ i is revealed at instant T i ). 57/91
58 Example of simple predictable process (one of its component) : 58/91
59 Gain process for a simple trading strategy If φ t is simple (we speak about simple trading strategy), then the gain realized by the trader between T i and T i+1 is equal to the scalar product φ i.(s Ti+1 S Ti ) Hence the accumulated gain on [0, T ] of the trader beginning with an initial composition given by φ 0 can be written as : G T (φ) = φ 0.S 0 + n φ i.(s Ti+1 S Ti ) i=0 59/91
60 Definition (Stochastic integral of a simple predictable process) G T is called the stochastic integral of the simple predictable process φ w.r.t. the price process (S t ), and will be denoted by : T 0 φ(u).ds(u) In finance, the cumulative gains of a trader following a strategy lead naturally to a new concept of integral : the stochastic integral 60/91
61 In view of getting some results concerning option prices in given models we cannot limit ourselves to the case where φ is a simple strategy. Hence we need to give a sense to this stochastic integral / gain process, also in the case where the composition continuously changes with time. We will first give a sense to this integral in the case where S t is a Brownian motion of dimension 1, and in the case where φ t is a more general process (not necessarily simple) : t 0 φ(s)db s 61/91
62 A first naive idea would consist to define the stochastic integral as a limit obtained along each trajectory : ( ) t n [ ] φ(s)db s (ω) = lim φ ti 1 (ω) B ti (ω) B ti 1 (ω) max(t i t i 1 ) 0 0 The problem is that (almost all) trajectories of a Brownian motion have no bounded variation... and that this limit hence does not exist in general for a given trajectory! i=1 62/91
63 Itô integral - Construction The stochastic integral will be defined as a limit of Riemann sums, but not in the sense of a.s. convergence, but in quadratic convergence (L 2 ) Objective : give a mathematical sense to : I (X) (ω) = T 0 X (ω, t) db t. where X(t) is a general stochastic process. 63/91
64 Contrarily to the Riemann integral, the result can depend on the point y i [t i 1, t i ). In the Itô integral, we will fix y i = t i 1 ; The final objective is to develop an integral allowing to introduce and to study stochastic evolution equations in continuous time ( Śtochastic differential equations, describing the dynamics of market variables) 64/91
65 Notations B [0, T ] = the set of Borel sets on [0, T ] (Borel sigma-algebra, i.e. the smallest σ algebra containing all open sets in [0, T ].) {F t } t the standard Brownian filtration. for all fixed t : F t B = the smallest σ algebra containing the product sets A B où A F t and B B. We say that a stochastic process X (.,.) is measurable if X (.,.) is F T B-measurable. We say that a process X (.,.) is adapted if X (., t) F t, t (0, T ). 65/91
66 In summary, Itô stochastic integral will be defined by following the subsequent steps : Definition of Itô integral for a simple process Itô isometry : the stochastic integral preserves the L 2 norm (in other words, is continuous for that norm) Density of the set of simple processes within the set of adapted square-integrable stochastic processes (with the L 2 ) Extension by density of the Itô integral on the set of square-integrable adapted stochastic processes thanks to the Itô isometry 66/91
67 Construction - Step 1 - Integral of indicator functions If X = I (a,b] is an indicator function, with (a, b] [0, T ], then the integral of X w.r.t. a Brownian motion is defined as : I (X) (ω) = T 0 X(s)dB(s) = B b (ω) B a (ω). (3) 67/91
68 Construction - Step 2 - Integral of step functions If X is a step function (or staircase function), i.e. n 1 X (ω, t) = c i I ]ti,t i+1 ] où 0 = t 0 < t 1 <... < t n = deterministic subdivision, and c i are constant, then by definition (additivity of the integral) : i=0 T 0 n 1 T X(s)dB(s) = c i I ]ti,t i+1 ](s)db(s) = i=0 0 i c i (B(t i+1, ω) B(t i, ω)) 68/91
69 Construction - Step 3 - Integration of simple stochastic processes X is square-integrable simple process if it can be written as : n 1 X (ω, t) = a i (ω) 1 (ti <t t i+1 ) with a i F ti, IE [ ] ai 2 <, et 0 = t0 < t 1 <... < t n 1 < t n = T (we will denote by H0 2 the set of such processes). Then we define the stochastic integral of X H0 2 as : i=0 I (X) (ω) = T 0 n 1 X(s)dB(s) = a i (ω) (B(t i+1, ω) B(t i ; ω)). (4) i=0 69/91
70 Construction - Step 4 - Extension by density X is a square-integrable process on (0,T) if : X is an adapted process [ ] T IE 0 X 2 (ω, s)ds <. The set of square-integrable adapted processes will be denoted by H 2 = H 2 [0, T ] Mathematicians show that H 2 is a closed vector subspace of L 2 (dp dt). We will denote by H0 2 (0, T ) the set of simple square-integrable (cf. preceding slide). 70/91
71 Construction - Step 4 - Extension by density We will now extend the definition for all square-integrable processes X H 2. The key of this extension is Itô isometry Lemma (Ito Isometry on H 2 0 ) X H0 2, we have In other words : I (X) L 2 (dp) = X L 2 (dp dt) T IE((I(X)) 2 ) = IE( X 2 (s)ds). 0 71/91
72 Construction - Step 4 - Extension by density Proof of Itô isometry We first calculate X 2 L 2 (dp dt). X is a simple process, hence can be written in the form : X (ω, t) = n 1 a i (ω)1 (ti <t t i+1 ) i=0 with a i F ti, IE [ a 2 i Let us take the square of X : such that : IE ] < and 0 = t0 < t 1 <... < t n = T X 2 (ω, t) = n 1 ai 2 (ω) 1 (ti <t t i+1 ) i=0 [ ] T 0 X 2 (ω, t) dt = n 1 IE [ ai 2 i=0 ] (ti+1 t i ). 72/91
73 Construction - Etape 4 - Extension by density Let us compute now I (X) 2 L 2 (dp) : IE [ I (X) 2] = IE n 1 = IE i=0 a i (ω) ( ) ) 2 B ti+1 B ti ( n 1 i=0 [ a 2 i ( Bti+1 B t i ) 2 ] as the double products have a zero expectation. Then, as B ti+1 B ti is independent of a i F ti, we have : IE [I (X) 2] = n 1 IE [ ] ai 2 (ti+1 t i ). i=0 73/91
74 Construction - Step 4 - Extension by density We will now extend by density the stochastic integral to the set of square integrable processes. We will use the following result : Lemma (density of H 2 0 in H2 ) For each process X H 2, there exists a sequence {X n } with X n H0 2 such that [ ] T X X n 2 L 2 (dp dt) = IE (X(s) X n (s)) 2 ds 0 for n. 0 74/91
75 Itô integral - Construction - Step 4- Extension by density Consequence of the Itô isometry and the density result For each simple process of the sequence, we can define its stochastic integral : I(X n ) = T 0 X n(s)db(s) The idea is to define I (X) for a general process X (not necessarily simple) as the limit of the sequence I (X n ) n in L 2 : I (X) def = lim n (I (X n )) where I (X) L 2 (dp) and the convergence is such that I (X) I (X n ) L 2 (dp) 0. 75/91
76 Construction - Step 4 - Extension by density Let us check that the stochastic integral is well defined 1, i.e. that (1) the limit of the sequence I(X n ) exists and (2) does not depend on the considered sequence X n tending to X : (1) X X n L 2 (dp dt) 0 implies that (I (X n)) converges in L 2 (dp) : Indeed, the convergence of the sequence (X n) in L 2 (dp dt) implies that this is a Cauchy sequence in L 2 (dp dt), which thanks to the Itô isometry, implies that (I (X n)) is also a Cauchy sequence L 2 (dp). As L 2 (dp) is a complete metric space (convergence of Cauchy sequences towards limits belonging to the space), the Cauchy sequence (I (X n)) converges to an element of L 2 (dp), that we denote by I(X). 1. The argument developed here is classical in functional analysis 76/91
77 Construction - Step 4 - Extension by density (2) Is I(X) well defined? i.e. for another choice of the sequence (X n) n converging to X : X X n L2 (dp dt) 0, does the new sequence I (X n) converges to the same limit in L 2 (dp) as the initial sequence I (X n )? The answer is yes, since X n X n L2 (dp dt) 0 thanks to the triangle inequality, and Itô isometry implies that : I (X n ) I (X n) L2 (dp) 0. 77/91
78 Itô integral - Itô isometry Now, we will show that when extended by density to the whole space H 2, Itô integral is still an isometry : Theorem (Itô isometry in H 2 (0, T )) For X H 2 [0, T ], we have that I (X) L2 (dp) = X L 2 (dp dt). In other words, ( T ) 2 [ T ] IE X(s)dB(s) = IE X 2 (s)ds /91
79 Itô integral - Itô isometry Proof First, we chose (X n ) n H0 2 such that X n X L 2 (dp dt) 0 for n. The triangle inequality for the L 2 (dp dt) norm implies : X n L 2 (dp dt) X L 2 (dp dt). Similarly, since (I (X n )) n L 2 (dp) I (X), I (X n ) L2 (dp) I (X) L 2 (dp). But we know that on H0 2, Itô isometry holds : I (X n ) L2 (dp) = X n L2 (dp dt) n, and the uniqueness of the limit completes the proof. 79/91
80 Itô integral as a stochastic process I : H 2 L 2 (dp) has been defined, and maps a stochastic process on a random variable (and not a process). Now, we would like to be able to consider the evolution in time of the gains of the trader, i.e. be able to consider t X(u)dB(u) as a stochastic process (in mathematical 0 terms, we need a mapping which maps a process to a process). For that purpose, we introduce a truncation function m t H 2 [0, T ] defined by : { 1 if s [0, t] m t (ω, s) = 0 else. 80/91
81 Itô integral as a stochastic process For X H 2 [0, T ], the product m t X H 2 [0, T ] t [0, T ], hence I (m t X) = T 0 m t(u)x(u)db(u) is a well defined element of L 2 (dp) One can show that we can construct a continuous martingale M t such that t [0, T ], we have P (M t = I (m t X)) = 1. The process {M t, t [0, T ]} is then the Ito integral considered as a process. 81/91
82 Itô integral as a stochastic process Theorem (Itô integral as martingales) for all X H 2 [0, T ], there exists a process {M t, t [0, T ]} which is a continuous martingale w.r.t. the standard Brownian filtration (F t ) t such that the event : {ω : M t (ω) = I (m t X) (ω)} has a probability 1 for all t [0, T ]. Proof : [Steele, pg 83-84, thm 6.2] 82/91
83 Itô integral as a stochastic process The integral sign : notation for all X H 2 [0, T ] and if {M t : 0 t T } is a continuous martingale such that we write : P [M t = I (m t X)] = 1 for all 0 t T, M t (ω) = t 0 X (ω, s) db s 0 t T t 0 X (ω, s) db s is a notation for what is well defined in the left-hand side of the equation. 83/91
84 Itô integral as a stochastic process But the notation is well chosen since ( ) 2 [ t ] t X H 2 = IE X (ω, s) db s = IE X 2 (ω, s) ds 0 0 t [0, T ]. Moreover, we have the following result : Proposition for all 0 s t and for all b H 2, we have ( ) 2 [ t ] t IE b (ω, u) db u F s = IE b (ω, u) 2 du F s. (5) s s 84/91
85 Itô integral as a stochastic process Proof Inequality (5) is equivalent to A F s : [ ( ) ] t 2 [ IE 1 A s b (ω, u) db ] t u = IE 1 A s b2 (ω, u) du. This is true by Theorem 1.4 on slide 78 for the modified integrand : b (ω, u) = 0 u [0, s) (t, T ] 1 A b (ω, u) u [s, t] 85/91
86 Riemann s representation Theorem For all f : R R continuous, if we consider the partition of [0, T ] given by t i = i T n with 0 i n, we have : lim n i=1 n f ( ) ( ) T B ti 1 Bti B ti 1 = f (B s ) db s (6) where the limit is taken in the sense of the convergence in probability. 0 86/91
87 Stochastic integral : explicit calculation In some cases, this integral can be explicitly computed from the definition. Example : t 0 B s db s =? If we denote i B := B ti B ti 1 and i t := t i t i 1, we have : ( B ti 1 i B ) = ( Bti 1 B t i Bt 2 i 1) i i = 1 [ (B 2 2 ti B 2 ) ( ] 2 ti 1 Bti B ti 1) = 1 B 2 2( t B0 2 ) 1 2 i i ( i B ) 2 = 1 2 B2 t 1 2 ( i B ) 2, i 87/91
88 Stochastic integral : explicit calculation We have seen that ( i B ) 2 t i in L 2. We then get : t 0 B s db s = 1 2 B2 t 1 2 t. Remark : We just saw that ( i B) 2 behaves like i t i.e., formally : (db t ) 2 = dt 88/91
89 Riemann s representation Proposition (Gaussian integrals) Let f C[0, T ] (a deterministic function), then the process defined by X t = t 0 f (s)db s for all t [0, T ] is a Gaussian process of zero mean with indepent increments and with covariance function cov(x s, X t) = s t 0 f 2 (u)du. Moreover, if we consider the partition of [0, T ] given by t i = it for 0 i n n and ti [t i 1, t i ], then lim n n i=1 f (t i ) ( B ti B ti 1 ) = T 0 f (s)db s, where the limit is taken in the sense of the convergence in probability. 89/91
90 Interpretation trajectory by trajectory of the Ito integral Theorem (Interpretation trajectory by trajectory of the Ito integral on H 2 ) If f H 2 is bounded and if ν is a stopping time such that f (ω, s) = 0 for almost all ω {ω : s ν}, then for almost all ω {ω : t ν}. X t (ω) = t 0 f (ω, s) db s = 0 90/91
91 Interpretation trajectory by trajectory of the Ito integral Theorem (Persistence of the identity on H 2 ) If f and g H 2 and if ν is a stopping time such that f (ω, s) = g (ω, s) for almost all ω {ω : s ν}, then the integrals X t (ω) = t 0 f (ω, s) db s and Y t (ω) = t 0 g (ω, s) db s are equal for almost all ω {ω : t ν}. 91/91
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