A Class of Fractional Stochastic Differential Equations

Transcription

1 Vietnam Journal of Mathematics 36:38) Vietnam Journal of MATHEMATICS VAST 8 A Class of Fractional Stochastic Differential Equations Nguyen Tien Dung Department of Mathematics, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Ha Noi, Vietnam Received August 15, 7 Revised April 15, 8 Abstract In this paper we consider the fractional case of a class of stochastic differential equations that has many important applications Based on an approximation approach we solve the equation with polynomial drift and fractional noise An explicit solution is found and some applications are given Mathematics Subject Classification: 9E3, 6K99 Keywords: Fractional Brownian motion, Black-Scholes, Ginzburg-Landau equation, Verlhust equation, ruin probability 1 Introduction The fractional Brownian motion fbm) of Hurst parameter H, 1) is a centered Gaussian process B H = {Bt H,t } with the covariance function R H t, s) =E[Bt H Bs H ] R H t, s) = 1 t H + s H t s H) In the case where H = 1, the process BH is a standard Brownian motion If H 1,BH is neither a semimartingale nor a Markov process and the stochastic calculus developed by Itô cannot be applied There are various approaches to fractional stochastic calculus by using some difficult tools such as: Malliavin calculus see, for instance [1, 4]), theory of Wick product [5] However, it is not easy to find explicit solutions from these methods for many practical problems

2 7 Nguyen Tien Dung In this paper using an approximation approach in L Ω), we investigate a class of fractional stochastic equations of the form dx t =ax n t + bx t )dt + cx t db t, 11) where n N,n and B t is a fractional Brownian motion of Liouville form that is defined below This equation is a generalization of many important equations such as: the Black-Sholes equation in mathematical finance, the Ginzburg-Landau equation in theoretical physics, the Verlhust equation in population study In [8] Mandelbrot has given a representation of B H of the form: where U t = B H t = 1 [ ] Ut + B t, Γ1 + α) t s) α s) α) dw s,b t = t s) α dw s and α = H 1 B t is a process possessing main properties of Bt H such as of long memory and it is called a fractional Brownian motion of Liouville form [, 6] It is known that B t is approximated in L Ω) by stochastic processes B t = t s + ε) α dw s, where B t is a semimartingale and the convergence is uniform in t [,T] The paper is organized as follows: In Sec we recall some important results from the approximation approach and formulate our approximation problem Section 3 contains main results of this paper In Sec 4 some applications to finance and physics are introduced An Approximation Method Our method is based on a result on approximation of the fractional process B t = t t s) α dw s by semimartingales given in [9] that we recall below: For every ε>, as in [1] we define: B t = Then we have t s + ε) α dw s, α = H 1 1, 1 ) 1)

3 A Class of Fractional Stochastic Differential Equations 73 Theorem 1 I The process { B t,t } is a semimartingale Moreover where ϕ ε t) = B t = αit)+ε α W t, ) t s + ε) α 1 dw s and It) = ϕ ε s)ds II The process B t converges to B t in L Ω) when ε tends This convergence is uniform with respect to t [,T], ie: lim ε t T B t B t = 3) Proof Refer to [9] Corollary For any p 1, the process B t uniformly converges in t to B t in L p Ω) when ε tends Proof Noting that B t N, σ t ) and B t N,σ t ) σ t := E B t = t + ε)h ε H H and σ t := E Bt) = th H Since t [,T] and ε +, it follows that both σ t and σ t are bounded by some positive constant Thus, the Gaussian processes B ε t) and Bt) have null means and finite variance Hence they have finite moments of any order If 1 p then by applying the Lyapunov inequality, we obtain E Bt B t p) 1 p E B t B t ) 1 when ε If p> then it follows from Hölder inequality E X + Y p p 1 E Y p + E Y p ) for any p 1 that E B t B t p E B t B t ) 1 E B t B t p ) 1 B t B t [ 1 p 3 E B t p + E B t p )] Since B t and B t have moments of any order so there exists some constant M p depending only p such that E B t B t p M p B t B t

4 74 Nguyen Tien Dung The proof is thus complete Next, let us consider the following fractional differential equation in a complete probability space Ω, F, P) dx t = ) axt n + bx t dt + cxt db t 4) X t t= = X or X t X = X t t= = X, where the stochastic integral ) t ax n s + bx s ds + c X s db s X s db s will be defined as the L -limit of 5) X s d B s when ε, if it exists The initial value X is a measurable random variable independent of {B t : t T } As we said in Introduction, for the fractional stochastic calculus it is not good enough to find explicit solutions of fractional stochastic differential equations In order to avoid this difficulty and moreover, because B t B t it will be fully natural to approximate 4) by the following equation d X t = ) a X n t + b X t dt + c X t d B t 6) X t t= = X And then the solution of equation 4) will be the limit in L Ω) of the solution of 6) when ε 3 Main Results The equation 6) is a stochastic differential equation driven by a semimartingale with a polynomial drift and a constant volatility So the existence and uniqueness of its solution are assured Using formula ) we can rewrite equation 6) as follows d X t = a X n t + b X ) t + cαϕ ε t) X t dt + cε α Xt dw t 31) X t t= = X We have the following theorem Theorem 31 The solution of equation 6) can be explicitly given by X t = e b 1 c ε α )t+c B t X t +)a ) ) 1 e n 1) b 1 c ε α )s+c B s ds

5 A Class of Fractional Stochastic Differential Equations 75 Proof Put Y t = e cεα W t According to the Itô formula we have dy t = Y t 1 c ε α dt cε α dw t ) 3) We consider Z t = X t Y t and we get by applying the integration of parts formula dz t = X t dy t + Y t d X t c ε α Xt Y t dt = {ae n 1)cεα W t Z t ) n + b + cαϕ ε t) 1 c ε α) } Z t dt 33) This is an ordinary Bernoulli equation of the form and the solution Z t is given by Z t = e Qu)du Z t = P t)z n t + Qt)Z t Z + 1 n)p s)e s n 1) Qu)du ) 1 ds, where P t) =ae n 1) cεα W t,qt) =b 1 c ε α + cαϕ ε t) and b 1 c ε α )t + cαit) Qu)du = Hence the solution Z t of equation 33) can be expressed as Z t = e b 1 c ε α )t+cαit) Z +)a ) ) 1 e n 1) b 1 c ε α )s+cαis)+cε α W s) ds Combining the last expression and B t = αit) +ε α W t we obtain a solution of the approximation equation 6) X t = e b 1 c ε α )t+c B t X t +)a ) ) 1 e n 1) b 1 c ε α )s+c B s ds The proof is thus complete Theorem 3 Suppose that X is a random variable such that X > as and E[X n ] < If H> 1 and a then the stochastic process X t defined by Xt = X ebt+cbt +)a ) 1 e n 1) bs+cbs) ds 34)

6 76 Nguyen Tien Dung is the limit in L Ω) of X t This limit is uniform with respect to t [,T] Proof Put θ ε t) =e b 1 c ε α )t+c B t and θt) =e bt+cbt Then it is clear that for each m 1 there exists a finite constant M m > such that E[θε m t)] M m,e[θ m t)] M m for every t [,T] Indeed, E[θ m t)] = e mbt E[e mcbt ]=e mbt e 1 mcbt) = e mbt+ 1 m c b t <, and E[θ m ε t)] = e mb 1 c ε α )t+ 1 m c b t < Moreover, by applying Hölder inequality we have the following estimates for any m, k 1: E[θ m ε t) θ k t)] E θ ε t) m ) 1 E θt) k ) 1 So there exists a finite constant M m,k > such that E[θ m ε t) θ k t)] M m,k, t [,T] 35) We now can prove that θ ε t) L θt) uniformly with respect to t [,T], ie: Indeed, we see that lim ε t T θ ε t) θt) = 36) θ ε t) θt) θt) 4 exp 1 c ε α t + c B t B t ) ) 1 4 M 4 exp 1 c ε α t + c B t B t ) ) ) Using the relation e x 1=x + ox), we obtain exp 1 c ε α t + c B t B t ) ) c ε α t + c B t B t ) 4 + o) 4 38) 1 c ε α T + c B t B t ) 4 + o) 4 and thus 36) follows from Corollary We have also that θε n 1 s)ds L θ n 1 s)ds uniformly with respect to t [,T] Indeed, we have the following estimates: E t θε n 1 s)ds θ n 1 s)ds E θ n 1 ε t) θ n 1 t) ds t [,T] 39)

7 A Class of Fractional Stochastic Differential Equations 77 An application of Hölder inequality again yields for every t [,T] E θε n 1 t) θ n 1 t) = E[ θ ε t) θt) A ε t)] θ ε t) θt) A ε t), 31) where A ε t) = θ ε t) θt) θε n t)+θε n 3 t) θt) + + θ n t) ) Using inequalities of the form 35) we see that there exists a finite constant M n > such that A ε t) M n, t [,T] 311) It follows from 39),31) and 311) that E M n t M n T θε n 1 s)ds t T t T θ n 1 s)ds θ ε t) θt) θ ε t) θt) t [,T] 31) The last inequality assures that θε n 1 s)ds θ n 1 s)ds as ε t T Put η ε t) =X +a) θε n 1 s)ds and ηt) =X +a) θ n 1 s)ds From the results above we can see that η ε t) L ηt) uniformly with respect to t [,T] Next we will show that η 1 ε t) L η 1 t) uniformly with respect to t [,T] 313) Indeed, since a, we have η ε t) X and ηt) X as for every t [,T] The theorem of finite increments applied to the function gx) =x 1 η 1 ε t) η 1 t) 1 n 1 Xn η ε t) ηt)) By an argument analogous to the previous one, we get η 1 ε t) η 1 t) M η ε t) ηt), t [,T], yields

8 78 Nguyen Tien Dung where M> is a finite constant And 313) now follows from this estimate As a consequence we have the following assertion X t = θ ε t)η 1 ε t) L θt)η 1 t) =X t) The proof of the theorem is thus complete 4 Applications We can check that some famous equations can be considered as particular cases of our fractional equation studied in this paper 1 The fractional Black-Scholes model given by dx t = µx t dt + νx t db t, a =,b= µ, c = ν, X t = X e νbt+µt The fractional Verhulst equation a = 1,b= λ, c = σ, n =, dx t = X t + λx t) dt + σx t db t, X t = e σbt+λt X 1 + 1) 1) e σbs+λs ds ) 1, X t = X e σbt+λt t 1+X e σbs+λs ds 3 The fractional Ginzburg-Landau equation dx t = X 3 t +α + σ )X t) dt + σx t db t, a = 1,b= α + σ,c= σ, n =3, X t = σ σbt+α+ X e )t 1+X t σ σbs+α+ e )s) 1

9 A Class of Fractional Stochastic Differential Equations 79 References 1 E Alòs, O Mazet, and D Nualart, Stochastic calculus with respect to fractional Brownian motion with Hurst paramenter less than 1, J Stochastic processes and their applications 86 ), F Comte and E Renault, Long Memory in Continuous-Time Stochastic Volatility Models, Mathematical Finance ), K Dȩbicki, Z Michna, and T Rolski, On the remum from Gaussian processes over infinite horizon, Probability and Math Stat ), L Decreusefond and A S Üstünel, Stochastic Analysis of the Fractional Brownian Motion, J Potential Analysis ), T E Duncan, Y Hu, and B Pasik Duncan, Stochastic Calculus for Fractional Brownian Motion, SIAM Control and Optimization 38 ), D Feyel de la A Pradelle, Fractional integrals and Brownian processes, Potential Analysis ) J Jacques and R Manca, Semi-Markov Risk Models For Finance, Insurance And Reliability, Springer, 7 8 B Mandelbrot and van J Ness, Fractional Brownian motions, Fractional Noises and Applications, J SIAM Review ), Tran Hung Thao, An approximate approach to fractional analysis for finance, Nonlinear Analysis 7 6), 14 13