Lecture 7: Stochastic integrals and stochastic differential equations

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1 Lecture 7: Stochastic integrals and stochastic differential equations Eric Vanden-Eijnden Combining equations ( and (2 from Lecture 6, one sees that W N t satisfies the recurrence relation W N t n = W N t n + ξ n+ t, W N =. ( where t n = n/n, t = /N and {ξ n } n N are i.i.d. random variables taking values ± with probability 2 as before. A natural generalization of this relation is X N t n+ = X N t n + b(x N t n,t n t + σ(x N t n,t n ξ n+ t, X = x (2 If the last term were absent, this would be the forward Euler scheme for the ordinary differential equation (ODE Ẋt = b(x t,t. If b(x,t and σ(x,t meet appropriate regularity requirements, it can be shown that Xt N converges to a stochastic process X t as N (i.e. as t with n t t. The limiting equation for X t is denoted as the stochastic differential equation (SDE dx t = b(x t,tdt + σ(x t,tdw t, X = x, (3 as a remainder that the last term in (2 divided by t does not have a standard function as limit. The notation dw t comes from ( since this equation can be written as Wt N n+ Wt N n = ξ n+ t. We note that the convergence of X N t to X t holds provided only that the ξ n s are i.i.d. random variables with mean zero, Eξ n =, and variance one, Eξn 2 =. The standard choice in numerical schemes is to take ξ n = N(,, in which case t ξn+ d = Wtn+ W t n. In the discussion below, however, we will stick to the choice where {ξ n } n N are i.i.d. random variables taking values ± with probability 2 since it facilitates the calculations. Next, we study the properties of X t solution of (3 and introduce some nonstandard calculus due to Itô to manipulate this solution. Itô isometry and Itô formula Consider the recurrence relation X N t n+ = X N t n + f(w N t n ξ n+ t, X N =. 57

2 Let us investigate the properties of the limit of Xn t N as N, assuming that this limit exists. The limiting form of the recurrence relation above is traditionally denoted as dx t = f(w t,tdw t, X =, which can also be expressed as the stochastic integral X t = f(w s,sdw s. Stochastic integral have special properties called the Itô isometry E f(w s,sdw s =, ( 2 t E f(w s,sdw s = Ef 2 (W s,sds. The first of these identity is often written and used in differential form Ef(W s,sdw s =. The Itô isometry is easy to demonstrate. The first identity is implied by EX N t n = E n m= n = m= f(w N t m,t m ξ m+ t Ef(W N t m,t m Eξ m+ t =, where we used the independence of the ξ m s and Eξ m =. The second identity is implied by n E(Xt N n 2 = E f(wt N m,t m f( W t N p,t p ξ m+ ξ p+ t = m,p= n m= Ef 2 (W N t m,t m t, where we use the fact that ξ m and ξ p are independent unless m = p, and ξ 2 m = by definition. Going back to (3, a very important formula to manipulate the solution of this equation is Itô formula which states the following. Assume that X t is the solution of (3 and let f be a smooth function. Then g(x t satisfies the SDE dg(x t = g (X t dx t + 2 g (X t σ 2 (X t,tdt = ( g (X t b(x t,t + 2 g (X t σ 2 (X t,t dt + g (X t σ(x t,tdw t. If g depends explicitly on t, then an additional term g/ tdt is present at the right handside. Itô formula is the analog of the chain rule in ordinary differential calculus. However ordinary chain rule would give dg(x t = g (X t dx t. 58

3 Here because of the non-differentiability of X t, we have the additional term that depends on g (x. The proof of Itô formula can be outlined as follows. We Taylor expand g(x N t n+ g(x N t n using the recurrence relation (2 for X N t n and keep terms up to O( t: g(x N t n+ g(x N t n = g (X N t n (X N t n+ X N t n + 2 g (X N t n (X N t n+ X N t n 2 + = g (Xt N n (Xt N n+ Xt N n ( g (Xt N n b(xt N n,t n t + σ(xt N n,t n ξ n+ t + O( t 3/2 = g (X N t n (X N t n+ X N t n + 2 g (X N t n σ 2 (X N t n,t n ξ 2 n+ t + O( t3/2 = g (X N t n (X N t n+ X N t n + 2 g (X N t n σ 2 (X N t n,t n t + O( t 3/2, where in the last equality we used ξn+ 2 =. The Itô formula follows in the limit as t. 2 Examples The Itô isometry and the Itô formula are the backbone of the Itô calculus which we now use to compute some stochastic integrals and solve some SDEs. As an example of stochastic integral, consider Taking f(x = x 2 in Itô formula gives Therefore W s dw s. 2 dw t 2 = W t dw t + 2 dt. W s dw s = 2 W 2 t 2 t. Notice that the second term at the right hand-side would be absent by the rules of standard calculus. Yet, this term must be present for consistency, since the expectation of the left hand-side is E W s dw s =, using the first Itô isometry, and the expectation of the right hand-side is zero only with the term 2 t included since 2 EW t 2 = 2 t. As a first example of SDE, consider dx t = γx t dt + σdw t, X = x This is the Ornstein-Uhlenbeck process. Using Itô formula with f(x,t = e γt x, we get (this is Duhammel principle d ( e γt X t = γe γt X t dt + e γt dx t = σe γt dw t. 59

4 Figure : Three realizations of the Ornstein-Uhlenbeck process with X = and γ = σ =. Integrating gives X t = e γt x + σ e γ(t s dw s. This process is Gaussian being a linear combination of the Gaussian process W t. Its mean and variance are (using the Itô isometry Thus when γ > EX t = e γt x E(X t EX t 2 = σ 2 (e γ(t s 2 ds = σ2 2γ ( e 2γt. ( d X t N, σ2, 2γ as t. As a second example of SDE, consider the so-called geometric Brownian motion dy t = Y t dt + αy t dw t, Y = y. This process has some application in mathematical finance. Itô s formula with f(x = log x gives dlog Y t = (Y t dt + αy t dw t Y t 2Y 2 α 2 Yt 2 dt. Integrating we get Y t = ye t 2 α2 t+αw t. Note that by the rules of standard calculus, we would have obtained the wrong answer t Y t = ye t+αwt (wrong! 6

5 Indeed the term 2 α2 t in the exponential is important for consistency since taking the expectation of the SDE for Y t using the first Itô isometry gives and hence dey t = EY t dt, EY t = ye t. The solution above is consistent with this since Ee αwt = e 2 α2t. 3 Generalization in multi-dimension The definition of Itô integrals and SDE s can be extended to multi-dimension in a straightforward fashion. The SDE dx j t = b j(x t,tdt + K k= σ jk (X t,tdw k t, j =,...,J where {Wt k } K k= are independent Wiener processes, defines a vector-valued stochastic process X t = (Xt,...,Xt J. The only point worth noting is the Itô formula, which in multidimension reads: df(x t = j= f(x t x j dx j t + 2 j,j = 2 f(x t x j x j ( K k= σ jk (X t,tσ kj (X t,t dt 4 Forward and backward Kolmogorov equations Consider the stochastic ODE dx t = b(x t dt + σ(x t dw t, X = y. Define the transition probability density ρ(x,t y via x2 x ρ(x,t ydx = P{X t+s [x,x 2 X s = y}. (ρ(x, t y does not depends on s because b(x and σ(x are time-independent. The transition probability density is an essential object because the process X t is Markov, in other words: for any t,s P(X t+s B[x,x 2 {X s } s s} = P(X t+s B[x,x 2 {X s }, i.e.the future behavior of X t given what has happened up to time s depends only on what X s was. We will derive equation for ρ. Let f be an arbitrary smooth function. Using Itô formula, we have f(x t f(y = f (X s dx s f (X s a(x s ds,

6 where a(x = σ 2 (x. Taking expectation on both sides, we get Ef(X t f(y = E or equivalently using ρ f(xρ(x,t ydx f(y = f (xb(xρ(x,s ydxds + 2 Since this holds for all smooth f, we obtain f (X s b(x s ds + 2 E f (X s (X s ds. t = x (b(xρ f (xa(xρ(x,s ydxds. x2(a(xρ (4 with the initial condition lim t ρ(x,t y = δ(x y. This is the forward Kolmogorov equation for ρ in terms of the variables (x,t. It is also called the Fokker-Planck equation. Equivalently, an equation for ρ in terms of the variables (y,t can be derived. The Markov property implies that ρ(x,t + s y = ρ(x, t zρ(z, s ydz. Hence ρ(x,t + t y ρ(x,t y = = ρ(x, t zρ(z, t ydz ρ(x, t y ρ(x,t z ( ρ(z, t y δ(z y dz. Dividing both side by t and taking the limit as t using the forward Kolmogorov equation one obtains ( t = ρ(x, t z z (b(zδ(z y z 2(a(zδ(z y dz, which by integration by parts gives t = b(y y + ρ 2 a(y 2 y2. (5 This is the backward Kolmogorov equation for ρ in terms of the variables (y, t. The operator L = b(y y + 2 2a(y y 2, is called the infinitesimal generator of the process. The coefficient b and a can be expressed as b(y = lim t t (E yx t y, 62 a(y = lim t t E y(x t y 2,

7 Figure 2: Snapshots of the density of the Ornstein-Uhlenbeck process at time t =. (blue, t =. (red, t = (green, and t = (magenta. Here X = y = and γ = σ =. The last snapshot at t = is very close to the equilibrium density. where E y denotes expectation conditional on X = y, Both the forward and the backward equations can be considered with different initial conditions. In particular, given a smooth function f, if we define u(y,t = E y f(x t, then u(y,t = f(xρ(x,t y and hence it satisfies u t = b(y u y + 2 a(y 2 u y 2, with the initial condition u(y, = f(y. In this sense, the SDE for X t is the characteristic equation that is associated with this parabolic PDE, much in the same way as the ODE Ẋ t = b(x t is the characteristic equation associated with the first order PDE u/ t = b(y u/ y. This can be generalized in many ways. For instance, the solution of v t = c(yv(y + b(y v y + 2 a(y 2 v y 2. with the initial condition v(y, = f(y, can be expressed as v(y,t = E y f(x t e t c(xsds. This is the celebrated Feynman-Kac formula in the context of SDEs. Let us consider an example. The forward differential equation associated with the Ornstein-Uhlenbeck process introduced in the last section is t = γ σ2 2 ρ (xρ + x 2 x 2 63

8 The solution of this equation is ρ(x,t y = ( πσ 2 ( e 2γt /γ exp γ(x ye γt 2 σ 2 ( e 2γt This shows that the Ornstein-Uhlenbeck process is a Gaussian process with mean ye γt and variance σ 2 ( e 2γt /2γ. It also confirms that this process tends to N(,σ 2 /2γ as t since ρ(x = lim ρ(x,t y = e γx2/σ2 t πσ 2 /γ. Generally, the limit of ρ(x,t y as t, when it exists, gives the equilibrium density ρ of the process. It satisfies = x (b(xρ x 2(a(xρ. Forward and backward Kolmogorov equations can also be derived for multi-dimensional processes. They read respectively and t = (b j (xρ + x j 2 j= t = b j (x + x j 2 j= where a jj (x = K k= σ jk(xσ j k(x. j,j = j,j = Notes by Walter Pauls and Arghir Dani Zarnescu. 2 x i x j (a jj (xρ 2 ρ a jj (x, x i x j. 64