Lecture 12: Diffusion Processes and Stochastic Differential Equations

Lecture 12: Diffusion Processes and Stochastic Differential Equations 1. Diffusion Processes 1.1 Definition of a diffusion process 1.2 Examples 2. Stochastic Differential Equations SDE) 2.1 Stochastic differential equations 2.2 Approximation and simulation for SDE 3. Solving of Stochastic Differential Equations 3.1 Linear SDE 3.2 General linear SDE of diffusion and non-diffusion types 3.3 Transformations of SDE 4. LN Problems 1

1. Diffusion Processes 1.1 Definition of a diffusion process Definition 12.1. A Stochastic real-valued process Xt), t is said to be a diffusion process if it satisfies the following conditions: A: Xt) is a Markov process. B: There exist limits: 1 { µx, t) = lim E Xt + ) Xt)) Xt) = x }, σ 2 1 { x, t) = lim E Xt + ) Xt)) 2 Xt) = x }. C: Xt), t is a continuous process. µx, t) is called a drift coefficient, parameter); σ 2 x, t) is a diffusion coefficient, parameter). 1.2 Examples 1) A Brownian Motion A BM Xt) = X) + µt + σw t), t is a diffusion process with the initial state X) = const, drift µx, t) = µ and diffusion σ 2 x, t) = σ 2. 2

2) A Geometrical Brownian Motion A GBM St) = S)e µt+σw t), t is a diffusion process with the initial state X) = const, µx, t) = xµ + σ2 2 ) and σ 2 x, t) = x 2 σ 2. a) By the definition, µt+s)+σw t+s) St + s) = S)e µs+σw t+s) W t)) = St)e b) Su), u t and W t + s) W t), s are independent. c) Thus St) is a Markov process. Indeed, P{St + s) u St) = v, St k ) = v k, k = 1,..., n} = P{St)e µs+σw t+s) W t)) u St) = v, St k ) = v k, k = 1,..., n} = P{ve µs+σw t+s) W t)) u St) = v, St k ) = v k, k = 1,..., n} = P{ve µs+σw t+s) W t)) u} ) lnu/v) µs = Pt, v, t + s, u) = Φ σ, s where Φu) = 1 2π u e v2 2 dv. d) Also, { 1 } E St + s) St)) St) = x = 1 ) } {x E e µ +σ[w t+ ) W t)] St) 1 = x = x ) e µ + σ2 2 1 = x ) ) 1 + µ + σ2 2 ) + o ) 1 x µ + σ2 2 3 as.

d) Thus, e) Also, µx, t) = xµ + σ2 2 ). { 1 2 } St) E St + ) St)) = x = 1 { ) } 2 St) E x e 2 µ +σw t+ ) W t)) 1 = x { } = x2 E e 2µ +2σZt+ ) Zt)) 2e µ +σw t+ ) W t)) + 1 ) = x2 e 2µ + 4σ2 2 2e µ + σ2 2 + 1 = x2 1 + 2µ + 2σ 2 ) + o ) ) ) 2 1 + µ + σ2 2 ) + o ) + 1 ) = x2 σ 2 + o ) x 2 σ 2 as. f) Thus, σ 2 x, t) = x 2 σ 2. g) St) = S)e µt+σzt) is a continuous process. h) Therefore, the GMB St) ) is a diffusion process with the drift term µx, t) = x µ + σ2 2 and diffusion coefficient σ 2 x, t) = x 2 σ 2. 3). Let W t), t be a standard BM. Consider the process Y t) = W t) 2, t. It is a diffusion process with the initial 4

state Y ) =, the drift coefficient µx, t) 1 and the diffusion coefficient σ 2 x, t) = 4x 2. a) Y t) = W t) 2 is a Markov process. The proof is similar to the proof given in example 2). b) Also, 1 { E W t + ) 2 W t) 2 } W t) 2 = x = 1 {W E t + ) W t) + W t)) 2 W t) 2 } W t) 2 = x = 1 { E W t + ) W t)) 2 + 2W t + ) W t))w t)) } W t) 2 = x = 1 { E W t + ) W t)) 2 + 2W t + ) W t)) x) } W t) 2 = x = 1 E W t + ) W t) ) 2 + 2 xe W t + ) W t) )) = 1 + ) 1. c) Thus, µx, t) 1. 5

d) Also, 1 { W E t + ) 2 W t) 2) 2 } W t) 2 = x W ) 2 ) ) 2 } Zt) = E{ t + ) W t) + 2 W t + ) W t) x 2 = x = 1 W t + ) W t) ) 4 2 E + E4x W t + ) W t) ) 3 + E4x2 W t + ) W t) ) ) 2 = 1 e) Thus, ) 2 3 + 4x + 4x 2 4x 2 as. σ 2 x, t) = 4x 2. f) Y t) is a continuous process. g) Therefore, the process Y t) is a diffusion process with the initial state Y ) =, the drift coefficient µx, t) 1 and the diffusion coefficient σ 2 x, t) = 4x 2. 2. Stochastic Differential Equations SDE) 2.1. Stochastic differential equations 1) A discrete time analogue is a stochastic difference equa- 6

tion: Xn ) Xn 1) ) = µ X n 1) ) ), n 1) + + σ X n 1) ) ) ), n 1) W n ) W n 1) ), n = 1, 2,..., N, where: a) >, N = T ; b) W n ) W n 1) ), n = 1,..., N are independent increments of a standard BM, i.e., independent normal random variables with mean and variance ; c) X) = const R 1. 2) The above stochastic difference equation can be rewritten in the equivalent integral form, Xn ) = X) + n k=1 µ Xk 1) ), k 1) ) + n k=1 X σ k 1) ) ) ), k 1) W k ) W k 1) ), n = 1, 2,..., N, A Continuous time analogues of the above equations are used to define a continuous time stochastic differential equation. 3) Let a real-valued stochastic process Xt), t [, T ] with the initial value X) = const and a standard BM W t), t [, T ] are defined on the same probability space < Ω, F, P > with a filtration F t, t [, T ]. Definition 12.2. We say that the process Xt), t [, T ] 7

satisfies a stochastic differential equation, { dxt) = µ Xt), t ) dt + σ Xt), t ) dw t), t [, T ] 1) iff it satisfies the following stochastic integral equation, { t Xt) = X) + µxs), s)ds + t σxs), s)dw s), t [, T ], 2) which should be understood in the following way: a): Processes Xt) and W t) are adapted to filtration {F t }. b): σ-algebra F t and the process W t + s) W t), s [, T t] are independent, for every t [, T ], c): µx, t) and σx, t) are non-random measurable functions such that stochastic process µxt), t) and σxt), t) belong to the class H 2 [, T ] i.e., T µxt), t) dt and T σ2 Xt), t)dt < 1 with probability 1. d): The stochastic process Xt) has a stochastic differential dxt) = at)dt+bt)dw t) with coefficients at) = µxt), t) and bt) = σxt), t). 4) The first question that should be answered is to give reasonable conditions, under which a solution of equations 1) and 2) exists and is unique. The answer to this and related questions give the following important theorem. Theorem 12.1. Let the coefficients of equation 1) satisfy the following conditions: 8

D: µx, t) and σx, t) satisfy the following inequalities; µx, t) µy, t) + σx, t) σy, t) K x y, µx, t) 2 + σx, t) 2 K 1 + x 2). E: µx, t) and σx, t) are continuous functions in x and t. Then: I: There exists the unique solution of the SDE 1) Xt), t [, T ] for any other solution Xt): P{ Xt) = Xt), t T } = 1). II: Xt) is a diffusion process with coefficients µx, t) and σx, t). III: sup t T EXt) 2 <. a) The proof can be found in SK: Chapter 31. It is based on the use of successive approximation method applied to the stochastic integral equation 2): b) Put X ) t) = X), t [, T ] and then, for n = 1, 2,..., X n) t) = X) + + T T µx n 1) s), s)ds σx n 1) s), s)dw s), t [, T ]. 3) c) Then, it is possible to prove that there exists a process Xt) such that T E X n) t) Xt) 2 dt as n. 4) 9

d) The propositions I III hold for the process Xt). 5) In an obvious way the definition of SDE and Theorem 12.1 can be translated to the interval [t, T ], { dxs) = µxs), s)ds + σxs), s)dw s), 5) s [t, T ]. To indicate the initial value Xt) = x = const one can use the notation X x,t s), s [t, T ] for the solution of the equation 5). Theorem 12.2. Let conditions D and E of Theorem 12.1 hold. Then the diffusion process Xt), which is the solution of equation 1), has transition probabilities, P t, x, s, u) = P{Xs) y/xt) = x} = P{X x,t s) y}, t s. 2.2 Approximation and simulation for SDE 1) One can approximate the SDE 1) by the following stochastic difference equation, X ) n ) X ) n 1) ) = µx ) n 1) ), n 1) ) + σx ) n 1) ), n 1) )W n ) W n 1) )), n = 1, 2,..., N, 6) where: >, N = T. 2) The stochastic difference equation 6) has a recurrence structure that let one solve it by computing sequentially the val- 1

ues X) = X ) ), X ) ),..., X ) N ). 3) There are theorems which let, under conditions analogous to those in Theorem 12.1, interpolate in a reasonable way for example using a linear interpolation) the processes X ) t) between points n, n =, 1,..., N and guarantee appropriate convergence of these interpolated processes to the solution of the SDE 1) as. 4) Note also that the recurrence relation 6) let one effectively simulate the trajectories of the processes X ) t). 3. Solving of Stochastic Differential Equations 3.1 Linear SDE 1) Let us consider a stochastic difference equation, St + ) St) = µst), t) St) + σst), t)w t + ) W t)), 7) where: a) S) = const; b) t =,, 2,..., N. A financial interpretation of this equation is that the rate of return of some pricing process in the interval [t, t + ] is proportional to the length of interval and is given by some normal random variable with mean µst), t) and variance σ 2 St), t) depending on the value of the process at moment t. The assumption is a natural one. 11

2) The continuous tine analogue of equation 7) is a stochastic differential equation dst) St) = µst), t)dt + σst), t)dw t). 8) This classical model was introduced by Samuelson 1965) and Merton 1973). 3) The important is a linear case when takes the form, dst) St) = µt)dt + σt)dw t), 9) where a) S) = const; b) µt) and σt) are non-random continuous functions. 4) Equation 9) can be rewritten as: dst) = St)µt)dt + St)σt)dW t). 1) So, it is a diffusion type SDE with coefficients µx, t) = xµt) and σx, t) = xσt) which are linear functions in x. St) = S)e t µs) 1 2 σ2 s))ds+ t σs)dw s), t [, T ]. 11) Theorem 12.3. The equation 1) has the unique solution which is a diffusion process given explicitly by the following formula, a) Conditions D and E obviously hold and functions µx, t) = xµt) and σx, t) = xσt) are continuous in x and t. So by Theorem 12.1 solution is unique and it is a diffusion process. 12

b) Let us calculate d ln St) using Itô formula. ft, x) = ln x we get, For function f t, f x = 1 x, f xx = 1 x 2 c) Thus, d ln St) = + 1 [ 1 ] St) 2 σt) 2 2 St) 2 + 1 ) St) St)µt) dt + 1 St)σt)dW t), 12) St) or d ln St) = d) This relation is equivalent to ln St) = ln S) + + µt) 1 ) 2 σ2 t) dt + σt)dw t). 13) t e) Relation 14) implies that { t St) = S)e t T. t µs) 1 ) 2 σ2 s) ds σs)dw s), t T. 14) µs) 1 2 σ2 s) ) ds+ t σs)dw s), 15) 5) Note that, by Theorem 12.1, we know that the stochastic process St) given by 13) is: a) diffusion process; b) sup s T ESt) 2 <. 13

6) Equation dst) = St)µdt + St)σdW t) has the unique solution St) = S) exp{µ 1 2 σ2 )t + σw t)} which is a Geometrical Brownian Motion. 3.2 General linear SDE of diffusion and non-diffusion types 1) Method used in the proof of Theorem 12.3 can be applied to linear SDE of non-diffusion type, dst) St) = µt)dt + σt)dw t), t T 16) where S) = const, and we assume that: F: µt) and σt) are continuous stochastic processes such that µt) and σt) belong to H2 [, T ]. G: There is a filtration {F t } such that processes St), µt), σt) and W t) are adapted to the filtration{f t }. H: σ-algebra F t and process W t + s) W t), s T t are independent, for every t [, T ]. Theorem 12.4. Under conditions F H, SDE 16) has the unique solution which is a continuous process given by the following formula, St) = S)e t µs) 1 2 σ2 s))ds+ t σs)dw s), t T. 17) a) Let St) be a process given by 17), i.e, St) = S)e Xt) where Xt) = t µs) 1 2 σs)2 )ds + t σs)dw s). 14

b) The process Xt) has stochastic differential, dxt) = µt) 1 2 σ2 t) ) dt + σt)dw t). c) Let us calculate dfxt)) where fx) = S)e x using Itô formula, f t =, f x = S)e x, f xx = S)e x. d) Thus, dst) = + 1 2 S)eXt) σ 2 t) + S)e Xt) µt) 1 ) 2 σ2 t) dt or +S)e Xt) σt)dw t), dst) = St)µt)dt + St)σt)dW t). e) Therefore, St) = S)e Xt) is a solution of equation 17). f) Let St) be some other continuous solution of equation 17), i.e., d St) = St)µt)dt + St)σt)dW t). g) Then by repeating the calculations given in the proof of Theorem 12.3, we get, then, d ln St) = µt) 1 2 σ2 t) ) dt + σt)dw t), ln St) = ln S) + and finally, t 1 µs) 2 σ2 s) ) t ds + σs)dw s), St) = S)e t µs) 1 2 σ2 s))ds+ t σs)dw s). 15

h) Therefore, St) = St). 3.3 Transformations of SDE 1) Let consider SDE with continuous non-random coefficients µt) and σt), dxt) = Xt)µt)dt + Xt)σt)dW t). Let Y t) = X 2 t) and let us find SDE for Y t). Transformation function is fx) = x 2 that gives us or Thus by Itô formula, f t =, f x = 2x, f xx = 2. dy t) = + 1 2 2 Xt)2 σt) 2 + 2Xt) 2 µt) 2) dt+ +2Xt) Xt)σt)dW t), dy t) =Y t) 2µt) + σ 2 t) ) dt + Y t)2σt)dw t). Both processes Xt) and Y t) are, by Theorem 12.1, diffusion processes that are the unique solutions for the the corresponding SDE. 2) The following theorem presents the transformation method in a general form. Theorem 12.5. Let Xt) is a process, which is the solution of stochastic differential equation, dxt) = µxt), t)dt + σxt), t)dw t), t T. 18) 16

Let also ft, x) is a non-random, monotonic in x for every t, continuous function that has continuous derivatives f t, f x and f x x and an inverse in x for every t function gt, x) ft, gt, x)) = x, gt, ft, x)) = x). Then the process Y t) = ft, Xt)) is a solution of the following SDE, dy t) = [f tt, gt, Y t))) + f xt, gt, Y t)))µt, gt, Y t))) + 1 2 f xxt, gt, Y t)))]dt + f xt, gt, Y t)))σt, gt, Y t)))dw t), t T. 19) a) Using Itô formula we get, dy t) = [f tt, Xt)) + f xt, Xt))µt, Xt)) + 1 2 f xxt, Xt))σt, Xt))]dt + f xt, Xt))σt, Xt))dW t). 2) b) By substituting Xt) = gt, Y t)) in relation 2) we get the SDE 19). 3) If coefficients of SDE 18) and 19) satisfy conditions D and E of Theorem 12.1, then the processes Xt) and Y t) are diffusion processes, which are unique solutions of the SDE, respectively, 18) and 19). In some cases transformation method let one solve SDE explicitly. 4) Let Xt) be a process which has a stochastic differential, dxt) = at)dt + bt)dw t). 21) 17

Relation 21) is actually the simplest SDE which has the solution, Xt) = X) + t as)ds + t 5) Let us now consider SDE of the form, bs)dw s). 22) dxt) = µxt))dt + σxt))dw t), 23) where a) µx) and σx) > are non-random functions that satisfy conditions D and E. Conditions D and E guarantee that there exists a unique solution of SDE 23), which is a diffusion process. 5) Let apply to process Xt) a transformation Y t) = fxt)), where fx) = x dy σy) assuming that integral defining fx) converges for every x. 6) We have, and, by Itô formula, or f t =, f x = 1 σx), f xx = σ x) σx) 2, dy t) = 1 σ Xt)) 2 σxt)) 2 σxt))2 1 + σxt)) µxt))) 1 dt + σxt))dw t) σxt)) dy t) = µxt)) σxt)) 1 ) 2 σ Xt)) dt + dw t). 24) 18

Note that the SDE 23) has the diffusion coefficient σt), while the transformed SDE 24) has the diffusion coefficient 1! 7) Let consider the case, where Then equation 24) takes the form, x µx) σx) 1 2 σ x) = a = const. 25) dy t) = adt + dw t). This equation has explicit solution Y t) = Y ) + at + W t) Let denote f 1 x) the inverse function for function fx) = determined by the relation, dy σy) x = f 1 x) dy σy). Then the solution of the SDE 23) is given by the formula, Example Xt) = f 1 Y t) ) = f 1 y) + at + W t) ). 26) 1) Let σx) = ɛ + x) 2. Then, x dy fx) = ɛ + y) = 1 2 ɛ + y 2) In this case, relation 25) takes the form µx) ɛ + x) 1 2ɛ + x) = a, 2 2 19 x = 1 ɛ 1 ɛ + x. 27)

or µx) = ɛ + x) 3 + aɛ + x) 2. 3) So, initial SDE has the non-linear form, dxt) = ɛ+xt)) 3 +aɛ+xt)) 2 )) ) dt+ɛ+xt)) 2 dw t) 28) 4) In this case, 1 ɛ 1 ɛ + x = y 1 ɛ + x = y + 1 ɛ = 1 ɛy ɛ ɛ + x = ɛ 1 ɛy x = ɛ 1 ɛy ɛ = ɛ2 y 1 ɛy f 1 x) = ɛ2 x 1 ɛx. 5) Thus, the solution of the SDE 28) has the following explicit form, Xt) = ɛ2 X) + at) + W t) ) 1 ɛ X) + at) + W t) ). 29) 4. LN Problems 4.1 Solve SDE dst) St) 4.2 Solve SDE dst) St) = t 2 dt + tdw t). = W t)dt + W t)dw t). 4.3 Let dxt) = e t dt + t 2 dw t). Find SDE for Y t) = Xt) 3. 2

4.4 Let dxt) = µxt))dt + X 3 dw t). Using the transformation Y t) = fxt)), where fx) = x dy y+ɛ), find µx) for 3 which this SDE has an explicit solution and. Find this solution. 4.5 Let dxt) = e t dt + t 3 dzt). Find SDE for Y t) = ln 1 + Xt) ). 4.6 Solve SDE dst) St) = e t dt + W t)dw t). 21