Ch 1. Wiener Process (Brownian Motion)

Transcription

1 Ch. Wiener Process (Brownian Motion) I. Introduction of Wiener Process II. Itô s Lemma III. Stochastic Integral IV. Solve Stochastic Differential Equations with Stochastic Integral his chapter introduces the stochastic process (especially the Wiener process), Itô s Lemma, and the stochastic intergral. he knowledge of the stochastic process is the foundation of derivative pricing and thus indispensable in the field of financial engineering. his course, however, is not a mathematic course. he goal of this chapter is to help students to build enough knowledge about the stochastic process and thus to be able to understand academic papers associated with derivative pricing. I. Introduction of Wiener Process he Wiener process, also called Brownian motion, is a kind of Markov stochastic process. Stochastic process: whose value changes over time in an uncertain way, and thus we only know the distribution of the possible values of the process at any time point. (In contrast to the stochastic process, a deterministic process is with an exact value at any time point.) Markov process: the likelihood of the state at any future time point depends only on its present state but not on any past states. In a word, the Markov stochastic process is a particular type of stochastic process where only the current value of a variable is relevant for predicting the future movement. he Wiener process Z(t) is in essence a series of normally distributed random variables, and for later time points, the variances of these normally distributed random variables increase to reflect that it is more uncertain (thus more difficult) to predict the value of the process after a longer period of time. See Figure - for illustration. -

2 Figure - Instead of assuming Z(t) N(, t), which cannot support algebraic calculations, the Wiener process dz is introduced. Z ε t (change in a time interval t) ε N(,) Z follows a normal distribution { E[ Z] = var( Z) = t std( Z) = t Z( ) Z() = n ε i t = Z i, where n = t Z( ) Z() also follows a normal distribution { E[Z( ) Z()] = var(z( ) Z()) = n t = std(z( ) Z()) = std(z( )) = Variances are additive because any pair of Z i and Z j (i j) are assumed to be independent. Z() = if there is no further assumption. As n, t converges to and is denoted as dt, which means an infinitesimal time interval. Correspondingly, Z is redenoted as dz. In conclusion, dz is noting more than a notation. It is invented to simplify the representation of a series of normal distributions, i.e., a Wiener process. -2

3 he properties of the Wiener process {Z(t)} for t : (i) (Normal increments) Z(t) Z(s) N(, t s). (ii) (Independence of increments) Z(t) Z(s) and Z(u) are independent, for u s < t. (iii) (Continuity of the path) Z(t) is a continuous function of t. Other properties: Jagged path: not monotone in any interval, no matter how small a interval is. None-differentiable everywhere: Z(t) is continuous but with infinitely many edges. Infinite variation on any interval: V Z ([a, b]) = variation of a real-valued function g on [a, b]: V g ([a, b]) = sup g(t i ) g(t i ), a = t < t 2 < < t n = b, P where P is the set of all possible partitions with mesh size going to zero as n goes to infinity Quadratic variation on [, t] is t [Z, Z](t) = [Z, Z]([, t]) = sup P Z(t i ) Z(t i ) 2 cov(z(t), Z(s)) = E[Z(t)Z(s)] E[Z(t)]E[Z(s)] = E[Z(t)Z(s)] (If s < t, Z(t) = Z(s) + Z(t) Z(s).) = E[Z 2 (s)] + E[Z(s)(Z(t) Z(s))] = E[Z 2 (s)] = var(z(s)) = s = min(t, s) (he covariance is the length of the overlapping time period (or the sharing path) between Z(t) and Z(s).) Generalized Wiener process dx = adt + bdz { E[dX] = adt var(dx) = b 2 dt std(dx) = b dt dx N(adt, b 2 dt) X( ) X() = n X i N(a, b 2 ) -3

4 Itô process (also called diffusion process) (Kiyoshi Itô, a Japanese mathematician, deceased in 28 at the age of 93.) dx=a(x, t)dt+b(x, t)dz drift and volatility are not constants, so it is no more simple to derive E[dX] and var(dx) (Both generalized Wiener processes and Itô process are called stochastic differential equation (SDE).) For the stock price, it is commonly assumed to follow an Itô process ds = µsdt + σsdz ds S ds S d ln S = µdt + σdz (also known as the geometric Brownian motion, GBM) N(µdt, σ2 dt) = d ln S = ds (WRONG!) (Note that this differential result is true only ds S S when S is a real-number variable. his kind of differentiation CANNO be applied to stochastic processes. he stochastic calculus is not exactly the same as the calculus for real-number variables.) In fact, the stock price follows the lognormal distribution based on the assumption of the geometric Brownian motion, but it does not mean d ln S N(µdt, σ 2 dt). (Advanced content) Stochastic volatility (SV) process for the stock price (Heston(993)): ds = µsdt + V SdZ S, dv = κ(θ V )dt + σ V V dzv, and corr(dz S, dz V ) = ρ SV. (Advanced content) Jump-diffusion process for the stock price (Merton(976)): ds = (µ λe[y S ])Sdt + σsdz + (Y S )Sdq, where dq is a Poisson (counting) process with the jump intensity λ, i.e., the probability of an event occuring during a time interval of length t is Prob {the event does not occur in (t, t + t], i.e., dq = } = λ t λ 2 ( t) 2... Prob {the event occurs once in (t, t + t], i.e., dq = } = λ t, Prob {the events occur twice in (t, t + t], i.e., dq = 2} = λ 2 ( t) 2. -4

5 and the random variable (Y S ) is the random percentage change in the stock price if the Poisson events occur. Merton (976) considers ln Y S N(µ J, σj 2 ). Note that dz, Y S, and dq are mutually independent. he introduction of the term (λe[y S ]) in the drift is to maintain the growth rate of S to be µ. his is because E[(Y S )dq] = E[Y S ] E[dq] = E[Y S ] λdt. If Y S follows the lognormal distribution, E[Y S ] = E[Y S ] = e E[ln Y S]+ = e µ J + 2 σ2 J. 2 var(ln Y S) II. Itô s Lemma Itô s Lemma is in essence the aylor series. aylor series: f(x, y) = f(x, y ) + f (x x x ) + f (y y y ) [ ] + 2 f (x x 2! x 2 ) f (x x x y )(y y ) + 2 f (y y y 2 ) 2 + Using Itô s Lemma to derive a stochastic differential equation: Given dx = a(x, t)dt + b(x, t)dz, and f(x, t) as a function of X and t, the stochastic differential equation for f can be derived as follows. df = ( f + f a + 2 f b 2 )dt + ( f b)dz, t X 2 X 2 X where a and b are the abbreviations of a(x, t) and b(x, t). he Itô s Lemma holds under the following approximations: (i) (dt) dt (dt).5 (dt) 2. (ii) dz dz =? By definition, dz dz = ε 2 dt. ε N(, ) var(ε) = E[ε 2 ] (E[ε]) 2 = E[ε 2 ] = E[(dZ) 2 ] = dt In addition, var((dz) 2 ) = var(ε 2 dt) = (dt) 2 var(ε 2 ) (because (dt) 2 ) dz dz a.s. = dt ( a.s. means almost surely ) -5

6 Itô s Lemma vs. differentiation of a deterministic function of time. For a deterministic function of time f(t), if df = g(t), we can interpret that with an dt infinitesimal change of dt, the change in f is g(t)dt, which is deterministic. he interpretation of the Itô s Lemma: with a infinitesimal change of dt, the change in f is ( f + f a + 2 f b 2 )dt + ( f b)dz. Note that the first term plays a similar role t X 2 X 2 X as g(t)dt, but the second term tells us that the change in f is random. o apply the Itô s Lemma is similar to taking the differentiation for stochastic processes. Based on the result of dz dz = (dz) 2 = dt, it is straightforward to infer that the quadratic variation of the Wiener process over [, t], i.e., [Z, Z](t) = [Z, Z]([, t]) = sup Z(t i ) P Z(t i ) 2, equals t. Similar to the derivation of the Itô s Lemma that E[(dZ) 2 ] = dt and var((dz) 2 ) when n (dt ), (Z(t i ) Z(t i )) 2 converges to t i t i almost surely if (t i t i ) is very small. his is because E[(Z(t i ) Z(t i )) 2 ] = E[ε 2 (t i t t )] = t i t t, and var((z(t i ) Z(t i )) 2 ) = var(ε 2 (t i t t )) = (t i t t ) 2 var(ε 2 ). So, we can conclude that when n (t i t t ), sup P (Z(t i ) Z(t i )) 2 = t. Example of applying the Itô s Lemma: f = ln S, ds = µsdt + σsdz d ln S = ( + µs σ 2 S 2 )dt + σsdz S 2 S 2 S = (µ σ2 )dt + σdz 2 ln S = (µ σ2 ) t + σ Z 2 ln S t+ t ln S t = (µ σ2 σ2 ) t + σ Z N((µ ) t, 2 2 σ2 t) ln S t+ t N(ln S t + (µ σ2 2 ) t, σ2 t) Consider t = t, n ln S t+ t ln S t N((µ σ2 ) t, 2 σ2 t) ln S t+2 t ln S t+ t N((µ σ2 ) t, 2 σ2 t). ln S ln S t N((µ σ2 ) t, 2 σ2 t) ln S ln S t N((µ σ2 2 )n t, σ2 n t) -6

7 ln S ln S t N((µ σ2 2 )( t), σ2 ( t)) ln S N(ln S t + (µ σ2 2 )( t), σ2 ( t)) he stock price is lognormal distributed. Another derivation: apply the stochastic integral on the both side of the equation d ln S t τ = σ2 (µ )dτ + σdz(τ) t 2 t Since the integrand is a constant and the variable τ is a real-number variable, it is simply the integral for a real-number variable. ln S τ t = (µ σ2 )( t) + 2 σ(z(τ) t ) Z( ) Z(t) Z( t) N(, t) ln S ln S t N((µ σ2 )( t), 2 σ2 ( t)) Example 2: f = S Ke r( t) (f is the value of a forward agreement) df = (µs rke r( t) )dt + σsdz Example 3: F = Se r( t) (F is the forward price of a stock) df = (µ r)f dt + σf dz Itô s Lemma for multiple variates ds S dx X = µ Sdt + σ S dz S (foreign stock price) = µ Xdt + σ X dz X (exchange rate: foreign dollar = X domestic dollars) Define f = S X (the value of a foreign stock share in units of domestic dollars) df=[ f t + f S µ SS + f X µ XX f S 2 σ 2 S S f X 2 σ 2 X X2 + 2 f ρ S X XS σ S σ X S X]dt + f σ S SSdZ S + f σ X XXdZ X df = [µ S XS + µ X XS + ρ XS σ S σ X SX]dt + σ S XSdZ S + σ X XSdZ X df f = (µ S + µ X + ρ XS σ S σ X )dt + σ S dz S + σ X dz X (because f = SX) dz S dz X = ε S dt εx dt = εs ε X dt E[dZ S dz X ] = E[ε S ε X ]dt = ρ XS dt var(dz S dz X ) = (dt) 2 var(ε S ε X ) a.s. dz S dz X = ρ XS dt -7

8 (Advanced content) Given ds = (µ λk Y )Sdt + σsdz + (Y S )Sdq, where K Y = E[Y S ] and f(s, t) as a function of S and t, the Itô s Lemma implies df = { f + f (µ λk t S Y )S f S 2 σ 2 S 2 + λe[f(sy S, t) f(s, t)]}dt + f S σsdz + (Y f )fdq, where λdte[f(sy S, t) f(s, t)] is the expected jump effect on f, and (Y f )dq is introduced to capture the unexpected (zero-mean) jump effect on f, where (Y f ) is the random percentage change in f if the Poisson event occurs. Note that λdte[f(sy S, t) f(s, t)] + (Y f )fdq represents the total effect on f if the Poisson event occurs. Suppose f = ln S, the Itô s Lemma implies d ln S = (µ λk Y 2 σ2 )dt + σdz + J ln S, where J ln S represents the total effect on ln S due to the random jump in S. If the jump occurs in S at t, we can obtain S(t + ) S(t) S(t) = (Y S ), since (Y S ) is the precentage change if the jump occurs. Rewriting the above eguation leads to S(t + ) S(t) = (Y S )S(t) = Y S S(t) S(t) S(t + ) = Y S S(t). he random jump in ln S at t, if the Poisson event occurs, is ln S(t + ) ln S(t) = ln Y S + ln S(t) ln S(t) = ln Y S. According to the above inference, we can express the total jump effect by J ln S = ln Y S dq, and thus d ln S = (µ 2 σ2 λk Y )dt + σdz + ln Y S dq. -8

9 III. Stochastic Integral Stochastic integral (or called Itô intergral or Itô calculus): allows one to integrate one stochastic process (the integrand) over another stochastic process (the integrator). Usually, the integrator is a Wiener process. b Integral over a stochastic process: X(τ)dZ(τ), where X(τ) can be a deterministic a function or a stochastic process, and dz(τ) is a Wiener process. (vs. integral over a realnumber variable: b f(y)dy, where f(y) is a deterministic function of the real-number a variable y) hree cases of X(τ) are discussed: simple deterministic processes, simple predictable processes, and general predictable processes (or Itô s processes). Stochastic integral for simple deterministic processes If X(τ) is a deterministic process, given any value of t, the value of X(τ) can be known exactly. herefore, in an infinitesimal time interval, (t i, t i ], the value of X(τ) can be approximated by a constant C i. he term simple means to approximate the process by a step function. (In contrast, if X(τ) is a stochastic process, given any value of τ, we only konw the distribution of possible values for X(τ).) Figure -2 X ( ) C 5 C C 3 C 2 C 4 t t t t t t 5 For simple deterministic processes, we can define the stochastic integral as follows. (his definition is similar to the rectangle method to define the integral over a real-number variable.) X(τ)dZ(τ) = n C i (Z(t i ) Z(t i )) N(, n Ci 2 (t i t i )) here should be a term lim in front of each n. It is omitted for simplicility. n -9

10 In the above equation, the reason for the final normal distribution:. he sum of normally distributed random variables is still a normally distributed random variable. 2. he mean for the resulting random variable is the sum of the mean of all normally distributed random variables. 3. he variance for the resulting random variable is the sum of the variances of all normally distributed random variables because all normally distributed random variables are independent. Note that the result of a stochastic intergral is a distribution, and we are interested in the mean and variance of this distribution. (i) According to the above definition, if X(t) =, the result of the stochastic integral is consistent with the definition of the Wiener process. X(τ)dZ(τ) = dz(τ) = Z(τ) = Z( ) Z() N(, ) = n (Z(t i ) Z(t i )) N(, n (t i t i )) = N(, ) (ii) Alternative way to calculate the variance of the result of the stochastic integral. var( XdZ) = E[( XdZ) 2 ] (E[ XdZ]) 2 = E[( XdZ) 2 ] = E[( n C i (Z(t i ) Z(t i ))) 2 ] = n C i C j E[(Z(t i ) Z(t i ))(Z(t j ) Z(t j ))] j= calculate the squared term in the expectation, and then apply the distributive property of the expectation over the addition and scaler multiplication = n Ci 2 (t i t i ) because cov(z(t i ) Z(t i ), Z(t j ) Z(t j )) =, and var(z(t i ) Z(t i )) = t i t i Simple predictable process: in the time interval (t i, t i ], the constant C i is replaced by a random variable ξ i, which depends on the values of Z(t) for t t i, but not on values of Z(t) for t > t i. herefore, X(t) is defined as follows. X(t) = ξi {t t=} + ξ i I {t ti <t t i }, where I is a indicator function and ξ is a constant. he corresponding stochastic intergral is defined as follows. X(τ)dZ(τ) ξ i (Z(t i ) Z(t i )). -

11 he reason for the name predictable :. he value of X(t) for (t i, t i ], ξ i, is determined based on the information set formed by {Z(t)} until t i, denoted by F ti. It is also called that ξ i is F ti -measurable. (See Figure -3) 2. In contrast, the value of Z(t i ) Z(t i ) will not realize until the time point t i, i.e., this value will be known based on the information set F ti. In other words, Z(t i ) is F ti -measurable. (See Figure -3) 3. herefore, we say that X(t) is predictable since we know its realized value just before the time point at which Z(t) is realized. 4. In the continuous-time model, Z(t) is F t -measurable (the realized value is known at t). For any process that we can know its realized value just before t, we call this process to be F t -measurable and thus predictable. Figure -3 i X () t ti t i Z( ti) Z( ti ) Z() t ti t i Stochastic integral of general predictable processes Let X n (t) be a sequence of simple predictable processes (which can be approximated by a step function with a series of predictable random variables) convergent in probability to the process X(t), which is general predictable (i.e., X(t) is predictable and X2 (τ)dτ< ). he sequence of their integrals Xn (τ)dz(τ) also converges to X(τ)dZ(τ) in probability, i.e., lim n X n (τ)dz(τ) = X(τ)dZ(τ). (In practice, the general predictable process is also known as the predictable process for short.) -

12 Any adapted and left continuous process is a predictable process. A process is an adapted process iff it is F t measurable. process Z(t) is an adapted process. For example, the Wiener A left-continuous function is a function which is continuous at all points when approached from the left. In addition, a function is continuous if and only if it is both right-continuous and left-continuous. Since Z(t) is a continuous function of t, it must be left-continuous. hus, we can conclude that Wiener process Z(t) is a predictable process, so Z(t) itself (or even all Itô processes) can be the integrand in a stochastic intrgral. his is also the reason for the name of the Itô integral. Figure -4 Left continuous Right continuous Solve Z(τ)dZ(τ), given Z() =. Define X n (t) = n Z(t i )I {t ti <t t i } ( lim X n (t) converges to Z(t) in probability) n Xn (τ)dz(τ) = n = 2 Z(t i )(Z(t i ) Z(t i )) [(Z(t i )) 2 (Z(t i )) 2 (Z(t i ) Z(t i )) 2 ] = 2 (Z( ))2 2 (Z())2 2 (Z(t i ) Z(t i )) 2 Z(τ)dZ(τ) = lim n Xn (τ)dz(τ) = (Z( 2 ))2 2-2

13 Properties of Itô Integral: (i) (αx(τ)+βy (τ))dz(τ) = α X(τ)dZ(τ)+β Y (τ)dz(τ) (distributive property) (ii) I [a,b](τ)dz(τ) = Z(b) Z(a), < a < b < (iii) E[ X(τ)dZ(τ)] = (iv) var( X(τ)dZ(τ)) = E[( X(τ)dZ(τ))2 ] = E[X(τ)2 ]dτ (Itô Isometry) Find E[ Z(τ)dZ(τ)] and var( Z(τ)dZ(τ)). (i) E[(Z( )) 2 ] = var(z( )) + E[Z( )] 2 = E[ Z(τ)dZ(τ)] = E[ 2 (Z( ))2 2 ] = (Property (iii) can be applied to obtaining the identical result directly.) (ii) var( Z(τ)dZ(τ)) = 4 var((z( ))2 ) Apply Property (iv) to finding var( = 4 {E[(Z( ))4 ] E[(Z( )) 2 ] 2 } = 4 {3 2 2 } = 2 If x N(µ, σ2 ), then E[x 4 ] = µ 4 + 6µ 2 σ 2 + 3σ 4. Since Z( ) N(, ), we can derive E[(Z( )) 4 ] = 3 2. Z(τ)dZ(τ)) as follows: var( Z(τ)dZ(τ)) = E[(Z(τ))2 ]dτ = τdτ = τ 2 2 =

14 IV. Solve Stochastic Differential Equations with Stochastic Integral How to solve X(t) systematically through the stochastic integral is the major application of the stochastic integral. Given dx(t) = αx(t)dt + σdz(t), solve X(t). }{{} Ornstein-Uhlenbeck process { αx(t) µ(x, t) σ σ(x, t) According to the stochastic integral, X(t) should satisfy X(t) = X()+ t µ(x, τ)dτ + t σ(x, τ)dz(τ) However, µ(x, t) is a function of X(t), so µ(x, t) is a stochastic process as well. Moreover, since the value of µ(x, t) is unknown due to the unsolved X(t). hus, we cannot derive X(t) by applying the stochastic integral directly. Define Y (t) = X(t)e αt dy (t) = e αt dx(t) + αe αt X(t)dt (through the Itô s Lemma) = e αt [ αx(t)dt + σdz(t)] + αe αt X(t)dt = σe αt dz(t) Y (t) = Y () + t σeατ dz(τ) a simple deterministic process X(t) = e αt (Y () + t σeατ dz(τ)), where Y () = X() Without the stochastic integral, as shown in the above example, different techniques should be employed to solve X(t). Later a systematical way to apply the stochastic integral to solving linear stochastic differential equations is introduced. It is worth noting that in the field of financial engineering, there is at least 95% of probability to consider linear stochastic differential equations. -4

15 Solution of a linear stochastic differential equation: Given dx(t)=(α(t) + β(t)x(t))dt + (γ(t) + δ(t)x(t))dz(t), solve X(t). (i) Like solving a differential equation (it needs to solve the corresponding homogeneous differential equation first), we solve this SDE in the case of α(t) = γ(t) = first. du(t) = β(t)u(t)dt + δ(t)u(t)dz(t) du(t) U(t) = β(t)dt + δ(t)dz(t) (he U(t) is similar to S(t), so we can apply the result on p.-6 to solve U(t).) t U(t) = U() exp( (β(τ) 2 δ2 (τ))dτ + δ(τ)dz(τ)) }{{} () t (ii) Consider X(t) = U(t) V (t), and U() = and V () = X(), where du(t) = β(t)u(t)dt + δ(t)u(t)dz(t), dv (t) = a(t)dt + b(t)dz(t). he integration by parts for stochastic processes: U(t)V (t) U()V () = t V (τ)du(τ) + t U(τ)dV (τ) + [U, V ](t), where [U, V ](t)= lim (U(t i ) U(t i ))(V (t i ) V (t i )) (quadratic covariation). n In addition, d[u, V ](t) = du(t) dv (t) = σ U σ V dt. (there is no product of drift terms because they are all with (dt) 2 or (dt).5, which is too small relative to dt.) Stochastic product rule: dx(t) = du(t) V (t) + U(t) dv (t) + d[u, V ](t), where d[u, V ](t) = du(t) dv (t) = δ(t)u(t)b(t)dt Substitute du(t) and dv (t) into the above equation, and compare with dx(t). b(t) U(t) = γ(t), a(t) U(t) = α(t) δ(t) γ(t) b(t) = γ(t) U(t), a(t) = α(t) δ(t)γ(t) t U(t) α(τ) δ(τ)γ(τ) γ(τ) V (t) = V () + dτ + U(τ) U(τ) dz(τ), }{{} X(t) = U(t) V (t) = () (2) t (2) where V () = X() -5

16 Brownian bridge (pinned Brownian motion): dx(t) = b X(t) dt + dz(t), t, X() = a t α(t) = b, β(t) =, γ(t) =, δ(t) = t t U(t) = U()exp( t (β(τ) 2 δ2 (τ))dτ + t δ(τ)dz(τ)) = exp( t dτ) = exp(ln( τ τ) t ) = exp(ln t) = t b(t) = t a(t) = b t t = b ( t) 2 t V (t) = V () }{{} + b ( τ) dτ + t dz(τ) 2 τ }{{} b X() = a b t X(t) = U(t) V (t) = t b [a + b + t dz(τ)] t τ X(t) = a( t ) + b t + ( t) t Figure -5 Xt () dz(τ), t <, lim X(t) = b τ t b lim X ( t ) b t a E[X(t)] = a( t ) + b t var(x(t)) = t t2 = t t2 t( t) = cov(x(t), X(s)) = min(s, t) st/ t he Brownian bridge is suited to formulate the process of the zero-coupon bond price because the bond price today is known and the bond value is equal to its face value on the maturity date. he disadvantage of fomulating the bond price to follow the Brownian bridge is that the zero-coupon bond price could be negative due to the normal distribution of dz(t) in dx(t). -6

17 Given X(t) = a( t ) + b t + ( t) t dz(τ), τ t( t) prove (i) var(x(t)) =. (ii) cov(x(t), X(s)) = s st (if t > s). (i) According to the fourth property of Itô integral, that is, var( X(τ)dZ(τ)) = E[X(τ)2 ]dτ, we can derive var(x(t)) = ( t) 2 t ( τ )2 dτ = ( t) 2 (( τ) t ) = ( t) 2 ( t( t) ) = t (Note that a( t ) + b t in X(t) contributes nothing to var(x(t)).) (ii) cov(x(t), X(s)) = cov(x(s) + X(t) X(s), X(s)) (assume s < t) = var(x(s)) + cov(x(t) X(s), X(s)) = = = = = s( s) ( t) t + cov(( t) t dz(τ) ( s) s τ s= t+t s = ( t) t s s( s) s( s) (t s)( s) var( s dz(τ) ( s) s τ dz(τ) τ dz(τ) (t s) s τ dz(τ)) τ (t s)( s)( s ( τ )2 dτ) s( s) (t s)( s)( ( s) ) s s2 (t s)( s)( s ( s) ) = s s2 st+s 2 = s st dz(τ), ( s) s τ dz(τ) τ dz(τ)) τ Introduction to Stochastic Calculus with Applications, Klebaner, 25-7