ITÔ-SKOROHOD STOCHASTIC EQUATIONS AND APPLICATIONS TO FINANCE

AND APPLICATIONS TO FINANCE

CIPRIAN A. TUDOR

Received 6 November 2003 and in revised form 8 June 2004

We prove an existence and uniqueness theorem for a class of It ˆo-Skorohod stochastic equations. As an application, we introduce a Black-Scholes market model where the price of the risky asset follows a nonadapted equation.

1. Introduction

The introduction of the anticipating (or Skorohod) integral in [8] and of the anticipat- ing stochastic calculus in [7] has opened the question of solving anticipating stochastic differential equations. In general, the existence and uniqueness of the solution for these equations is not known. The difficulty of solving such equations is due to the fact that the classical method of Picard iterations cannot be applied because the mean square formula for the Skorohod integral involves the Malliavin derivation in a such way that we cannot find “closed” formulas. Only in few particular cases do some results exist; see, for exam- ple, [1,2,3]. We have recently proved in [9] that the set of Skorohod integrals coincides with a set of integrals of It ˆo type. In the present work, using this correspondence between Skorohod integrals and It ˆo-Skorohod integrals, we introduce a class of anticipating equa- tions (calledItˆo-Skorohod equations) that can be solved using standard techniques. As an application, we introduce a market model where the price of the risky asset follows such an equation with a random initial condition (the price at the transaction time). We prove that our model is complete and has no arbitrage opportunities and we derive a Black- Scholes formula when the initial price of the risky asset is given by a standard normal random variable.

We organized the paper as follows.Section 2contains some preliminaries on the an- ticipating stochastic calculus. InSection 3, we define the class of It ˆo-Skorohod equations and we prove the existence and uniqueness of the solution. InSection 4, we introduce a market model with price dynamics following an It ˆo-Skorohod equation and we obtain a Black-Scholes option valuation formula and the expression of the replicant portfolio.

Copyright©2004 Hindawi Publishing Corporation

Journal of Applied Mathematics and Stochastic Analysis 2004:4 (2004) 359–369 2000 Mathematics Subject Classification: 60H05, 60H07

URL:http://dx.doi.org/10.1155/S1048953304311044

2. Preliminaries

We start with some elements of the Malliavin calculus. We refer to [6] for a complete presentation of this topic. Let (W(t))_t∈[0,1]be a standard Wiener process on the canonical Wiener space (Ω,F,P) and let (Ft)t∈[0,1]be the filtration generated byW. A functional of the Brownian motion of the form

F=fWt1

,. . .,Wt_n, (2.1)

witht1,. . .,tn∈[0, 1] and f ∈C^∞_b(Rⁿ), is called a smooth random variable and this class is denoted by᏿. The Malliavin derivative is defined on᏿as

DtF= n i=1

∂ f

∂xi

Wt1

,. . .,Wtn

1[0,ti](t), t∈[0, 1], (2.2)

ifFhas the form (2.1). The operatorDis closable and can be extended to the closure of

᏿with respect to the seminorm

F_k,p^p =E|F|^p+ k j=1

ED^(j)F_L^p2([0,1]), (2.3)

whereD⁽ⁱ⁾ denotes theith iterated derivative. The adjoint ofDis denoted byδ and is called the Skorohod integral. That is,δis defined on its domain

Dom(δ)=

u∈L²[0, 1]×Ω/E¹

0u_sD_sF ds≤CFL²(Ω)

(2.4) and is given by the duality relationship

EFδ(u)=E¹

0u_sD_sF ds, u∈Dom(δ),F∈᏿. (2.5) Recall that the variance of the Skorohod integral is

Eδ²(u)=E¹

0u²_αdα+E¹

0

1

0D_βu_αD_αu_βdα dβ. (2.6) ByL^k,p, we denote the setL²([0, 1];D^k,p), fork≥1 and p≥2, and we note thatL^k,pis a subset of the domain ofδ. The following version of the Ocone-Clark formula was given in [7]:

F=EF/F[s,t]^c

+ _t

sEDαF/F[α,t]^c

dW(α), forF∈D^1,2. (2.7)

We will need the integration-by-parts formula Fδ(u)=δ(Fu) +

[0,1]D_sFu_sds (2.8)

if all above terms are defined. Recall also that ifFis a random variable, Malliavin differ- entiable, and measurable with respect to aσ-algebraFA,A∈Ꮾ(R), then

DF₌0, onA^c_×Ω. (2.9)

We define, fork≥1 andp≥2, the sets of processes ᏹ^k,p=

X=

X_tt∈[0,1],X_t= _t

0u_sdW_s,u∈L^k,p, ᏺ^k,p=

Y=

Yt

t∈[0,1],Yt= _t

0E vs/F[s,t]^c

dWs,v∈L^k,p.

(2.10)

We will refer to the elements ofᏺ^k,p asItˆo-Skorohod integral processes and to the ele- ments ofᏹ^k,pasSkorohod integral processes.It has been proved in [9] that for sufficiently regular integrands, the two classes coincide. As a consequence, to study Skorohod inte- gral processes, it suffices to study It ˆo-Skorohod integral processes, which have two inter- esting properties. Firstly, note that the integralYt=_t

0E[uα/F[α,t]^c]dWα exists even for u∈L²([0, 1]×Ω) and has similarities to a classical It ˆo integral. Observe, by (2.6), that this integral is an “isometry”:

E^t

0E u_α/F[α,t]^c

dW_α 2

=E^t

0

E u_α/F[α,t]^c

2

dα. (2.11)

Secondly, if we define, for everyλ≤t,Y_t^λ=_λ

0E[uα/F[α,t]^c]dWα, then the process (Y_t^λ)λ≤t

is anF(λ,t]^c-martingale and we have

λ→limt,λ≤tY_t^λ=Y_t a.s. and inL². (2.12) We will now define the stochastic integral with respect to Itˆo-Skorohod integral processes.

Definition 2.1. Letu,v∈L²([0, 1]×Ω) be adapted processes and consider moreYt=Y0+ _t

0E[u_α/F[α,t]^c]dW_α+₀^tE[v_α/F[α,t]^c]dα. By definition, for any adapted square integrable processX,

_t

0X_sdY_s:= _t

0X_sd_sY_t^s, (2.13)

where

Y_t^λ=Y0+ _λ

0E u_α/F[α,t]^c

dW_α+ _λ

0E v_α/F[α,t]^c

dα (2.14)

and the integral on the right-hand side of (2.13) is understood in the semimartingale sense.

3. Itˆo-Skorohod stochastic equations

In this section, we state and prove an existence and uniqueness theorem for a class of anticipating stochastic differential equations using the method of Picard iterations. It is known that in the anticipating stochastic calculus, this method cannot be applied because the formula of the mean square of the Skorohod integral involves the Malliavin deriva- tive and one cannot find “closed” formulas. We define here a new class of anticipating equations, located “between” It ˆo and Skorohod equations, that can be solved by classical techniques. Consider the following stochastic differential equation:

X_t=Z+ _t

0σs,EX_s/F[s,t]^c

dW_s+ _t

0bs,X_sds. (3.1) Note that the stochastic integral above is a Skorohod integral since the integrand is not adapted and the initial condition is anticipating. The solution will also be anticipating.

In what follows, the coefficientsσ(t,x),b(t,x) : [0, 1]×R→Rare given and satisfy the following standard conditions.

(H1) (Measurability):σandbare jointly measurable in (t,x)∈[0, 1]×R.

(H2) (Lipschitz condition): there exists aD >0 such that for allt∈[0, 1] andx∈R, σ(t,x)−σ(t,y)+b(t,x)−b(t,y)≤D|x−y|. (3.2) (H3) (Linear growth condition): there exists a C >0 such that for all t∈[0, 1] and

x∈R,

σ(t,x)²+b(t,x)²≤C²1 +|x|²

. (3.3)

We also make a hypothesis concerning the initial valueZ.

(H4)Zis a random variable withE|Z|²<∞.

A square integrable process that satisfies a.s. (3.1) is called a solution of (3.1). For given coefficientsσandb, any solutionXwill depend on the initial valueZ. We will say that the solution is unique if, for everyt∈[0, 1],P(X_t¹=X_t²)=1 for any two solutionsX¹andX² with the same initial condition.

We start by proving the existence and the uniqueness of the solution of (3.1).

Theorem3.1. Under assumptions (H1), (H2), (H3), and (H4), stochastic equation (3.1) has a unique solutionX_ton[0, 1]with

sup

0≤t≤1

EX_t²<∞. (3.4)

Proof. Throughout this proof,Kwill denote a generic constant depending only onDand E|Z|². We consider the usual Picard iterationsX_t⁽⁰⁾=Zand

X_t⁽ⁿ⁺¹⁾=Z+ _t

0σs,EX_s⁽ⁿ⁾/F[s,t]^c

dWs+ _t

0bs,X_s⁽ⁿ⁾ds. (3.5)

We first prove the existence of the solution. We have, from (2.6), (H3), and H¨older’s inequalities, that

EX_t⁽¹⁾−X_t⁽⁰⁾²≤2E^t

0σs,EZ/F[s,t]^c

dWs

²+ 2E^t

0b(s,Z)ds

2

≤2E^t

0

σs,EZ/F[s,t]^c2ds+ 2tE^t

0

b(s,Z)²ds

≤2C²E^t

0

1 +EZ/F[s,t]^c2

ds+ 2tC²E^t

0

1 +|Z|² ds

≤Kt.

(3.6)

Using the same arguments and condition (H4), we obtain

EX_t⁽ⁿ⁺¹⁾−X_t⁽ⁿ⁾²≤2E _t

0

σs,EX_s⁽ⁿ⁾/F[s,t]^c

−σs,EX_s⁽ⁿ⁻¹⁾/F[s,t]^c

dWs

² + 2E^t

0

bs,X_s⁽ⁿ⁾−bs,X_s⁽ⁿ⁻¹⁾ds

2

≤2E^t

0

σs,EX_s⁽ⁿ⁾/F[s,t]^c

−σs,EX_s⁽ⁿ⁻¹⁾/F[s,t]^c2

ds + 2tE^t

0

bs,X_s⁽ⁿ⁾−bs,X_s⁽ⁿ⁻¹⁾²ds

≤2D²(1 +t) _t

0EX_s⁽ⁿ⁾−X_s⁽ⁿ⁻¹⁾²ds.

(3.7)

By induction, one can show that there existsK >0 such that for allt∈[0, 1] andn≥1, EX_t⁽ⁿ⁺¹⁾−X_t⁽ⁿ⁾²≤(Kt)ⁿ⁺¹

(n+ 1)!. (3.8)

Relation (3.8) and standard arguments imply the convergence, inL²(Ω), of the successive approximationsX_t⁽ⁿ⁾to a limitXtdefined byXt=Z+^∞_n=0(X_t⁽ⁿ⁺¹⁾−X_t⁽ⁿ⁾).

To prove thatXis a solution, we take theL²(Ω)-limit in (3.5) asn→ ∞. Obviously,

2E^t

0

σs,EX_s⁽ⁿ⁾/F[s,t]^c

−σs,EX_s/F[s,t]^c

dW_s

2

≤K _t

0EX_s⁽ⁿ⁾−X_s²ds−→n→∞0, E^t

0

bs,X_s⁽ⁿ⁾−bs,X_sds

2

≤K _t

0EX_s⁽ⁿ⁾−X_s²ds−→n→∞0.

(3.9)

The uniqueness of the solution is given by Gronwall’s lemma since for any two solutions X,Y with the same initial condition and for everyt∈[0, 1], we have

EXt−Yt²≤K _t

0EXs−Ys²ds. (3.10) Concerning bound (3.4), we will only note that standard techniques apply (see, e.g., [4]).

Remark 3.2. We define the following stochastic differential equation:

Xt=EZ/Ft^c

+ _t

0σs,EXs/F[s,t]^c

dWs+ _t

0bs,EXs/F[s,t]^c

ds. (3.11)

Following the lines of the proof ofTheorem 3.1, one can show that (3.11) admits a unique solutionXwith sup_0≤t≤1E|X_t|²<∞.

In the particular case of linear coefficients, one can explicitly obtain the solution of (3.11).

Corollary3.3. Letσ,b∈RandX0∈L²(Ω). Consider the equation X_t=EX0/Ft^c

+ _t

0σEX_s/F[s,t]^c

dW_s+ _t

0bEX_s/F[s,t]^c

ds. (3.12)

Then the unique solution of (3.12) is given by Xt=EX0/Ft^c

e^{σWt+(b−σ2/2)t}. (3.13) Proof. DenoteM_t=e^{σWt+(b−σ2/2)t}. ThenM_tsatisfies the equation

Mt=1 + _t

0σMsdWs+ _t

0bMsds, (3.14)

and using (2.8) and (2.9), we obtain X_t=EX0/Ft^c

M_t

=EX0/Ft^c +

_t

0σEX0/Ft^c

MsdWs+ _t

0bEX0/Ft^c Msds

=EX0/F[s,t]^c

+ _t

0σEEX0/Fs^c

M_s/F[s,t]^c

dW_s+ _t

0bEEX0/Fs^c

M_s/F[s,t]^c

ds

=EX0/Ft^c +

_t

0σEXs/F[s,t]^c dWs+

_t

0bEXs/F[s,t]^c ds.

(3.15)

4. Black-Scholes model driven by Itˆo-Skorohod stochastic differential equations We introduce, in this section, a market model with price dynamics following an It ˆo- Skorohod stochastic equation. As usual, we will consider two assets on the probability

space (Ω,F,P, (Ft)t∈[0,1]): the safe investmentA=(A_t)t∈[0,1]satisfyingA_t=1 +r₀^tA_sds and the risky assetS=(S_t)_t∈[0,1]with price dynamics following the stochastic differential equation

St=ES0/Ft^c +

_t

0σESs/F[s,t]^c dWs+

_t

0bESs/F[s,t]^c

ds. (4.1)

Clearly,At=e^rtandCorollary 3.3implies that S_t=ES0/Ft^c

e^{σWt+(b−σ2/2)t}. (4.2) The value of the portfolio at the instanttis defined by

V_t=h_tA_t+H_tS_t, (4.3)

where the componentsh,H∈L²([0, 1]×Ω) are adapted to the Brownian filtration and represent the quantities of the safe asset and the risky asset at the instantt.

We say that the portfolio (ht,Ht)t∈[0,1]isself-financingif V_t=EV0/Ft^c

+ _t

0h_sdA_s+ _t

0H_sdS_s, (4.4)

where the differentialdSis understood in the sense ofDefinition 2.1.

Remark 4.1. Note thatDefinition 2.1can be used although the initial value depends ont because, by the Ocone-Clark formula (2.7), we can write

ES0/Ft^c

=S0− _t

0EDsS0/F[s,t]^c

dWs. (4.5)

In other words, the self-financing condition (4.4) can be written as Vt=EV0/Ft^c

+ _t

0hsre^rsds− _t

0HsEDsS0/F[s,t]^c dWs

+ _t

0H_sσES_s/F[s,t]^c

dW_s+ _t

0bH_sES_s/F[s,t]^c

ds.

(4.6)

In the following, we will denote by ˜S_t=e^−rtS_tthe discounted risky asset price. A nec- essary and sufficient condition for the portfolio, to be self-financing, is given in the next result.

Proposition4.2. Assume thath,H∈L²([0, 1]×Ω)and let the processVbe given by (4.3).

DenoteV˜_t=e^−rtV_t.Then the portfolio is self-financing if and only if V˜t=EV0/Ft^c

+ _t

0HsdS˜s for everyt∈[0, 1]. (4.7)

Proof. Suppose thatV satisfies (4.4). Define, for everyλ∈[0,t], Vλ,t=EV0/Ft^c

+ _λ

0hsre^rsds− _λ

0HsEDsS0/F[s,t]^c

dWs

+ _λ

0HsσESs/F[s,t]^c

dWs+ _λ

0bHsESs/F[s,t]^c

ds.

(4.8)

It is not difficult to check thatV_λ,t=E(V_λ/F[λ,t]^c).

We can write It ˆo’s formula fore^−rλVλ,t since, for fixedt, the process (Vλ,t)λ∈[0,t]is a F[λ,t]^c-semimartingale. It holds, taking the limit (a.s. or inL²) asλ→t, that

V˜t=EV0/Ft^c

+ _t

0e^−rshsre^rsds− _t

0e^−rsHsEDsS0/F[s,t]^c

dWs

+ _t

0e^−rsσH_sES_s/F[s,t]^c

dW_s+ _t

0e^−rsbH_sES_s/F[s,t]^c

ds+ _t

0V_s,t−re^−rsds

=EV0/Ft^c

− _t

0e^−rsHsEDsS0/F[s,t]^c

dWs

+ _t

0e^−rsσH_sES_s/F[s,t]^c

dW_s+ _t

0e^−rs(b−r)H_sES_s/F[s,t]^c

ds.

(4.9)

On the other hand, writing It ˆo’s formula fore^−rλSλ,twith Sλ,t=ES0/Ft^c

+ _λ

0σESs/F[s,t]^c dWs+

_λ

0bESs/F[s,t]^c

ds=ESλ/F[λ,t]^c

, (4.10) we get

S˜_t=ES0/Ft^c

+ _t

0e^−rsσES_s/F[s,t]^c

dW_s+ _t

0e^−rs(b−r)ES_s/F[s,t]^c

ds. (4.11)

Identity (4.7) follows from (4.9) and the above equation usingDefinition 2.1. The proof

of the necessary part is not more difficult.

LetTbe the exercise time. In the classical Black-Scholes settings, to prove the nonexis- tence of arbitrage, it suffices to exhibit a probability measure equivalent toPunder which the discounted price ˜Sis a martingale. In our case, we have the following.

Proposition4.3. The unique probability measureP˜equivalent toPunder which the pro- cessS˜t/E(S0/Ft^c)is a martingale is given by the Radon-Nikodym derivative

dP˜

dP⁼expr−µ σ WT−1

2

(r−µ)²

σ² T P-a.s. (4.12)

Under probabilityP, the process˜ W˜t=Wt+ ((b−r)/σ)tis a standard Brownian motion and the discounted priceS˜satisfies the equation

S˜t=ES˜0/Ft^c

+ _t

0σES˜s/F[s,t]^c

dW˜s. (4.13)

Proof. DenoteZ_t=S˜_t/E(S0/Ft^c)=e^−rte^{σWt+(b−σ2/2)t}. It is well known that there exist the unique probability ˜P and the Brownian motion ˜W as above and it holds thatZ_t=1 + _t

0σZsdW˜s. Taking into account that the natural filtrations ofWand ˜Wcoincide, we get S˜_t=ES˜0/Ft^c

Z_t=ES˜0/Ft^c

+ _t

0σES˜0/Ft^c

Z_sdW˜_s=ES˜0/Ft^c

+ _t

0σES˜_s/F[s,t]^c

dW˜_s. (4.14)

Remark 4.4. Note that, byCorollary 3.3, we have ˜S_t=E( ˜S0/Ft^c)e^{σW˜t−(σ2/2)t}. Also, an im- mediate consequence of Propositions4.2and4.3is the fact that the market is complete and has no arbitrage opportunities.

ConsiderVT=(ST−K)⁺the payofffunction of the European call option with exercise timeT and strike priceK. Denote by ˜Ethe expectation with respect to ˜Pand by ˜Dthe Malliavin derivative with respect to ˜W. By formulas (4.7) and (4.13), we have

V˜_t=E˜V0/Ft^c

− _t

0H_sE˜D˜_sS0/F[s,t]^c

dW˜_s+ _t

0σH_sE˜S˜_s/F[s,t]^c

dW˜_s. (4.15)

Taking the conditional expectation with respect to theσ-algebraFt, we obtain E˜V˜t/Ft

=_E˜V0

− _t

0Hs_E˜D˜sS0/Fs

dW˜s+ _t

0σHs_E˜S˜s/Fs

dW˜s. (4.16)

Therefore, the process ( ˜E( ˜Vt/Ft))t∈[0,1]is a martingale and, for everyt≤T, it holds that E˜( ˜V_t/Ft)=_E˜( ˜V_T/Ft) or

E˜Vt/Ft

=_E˜e^−r(T−t)VT/Ft

. (4.17)

We have the following option valuation Black-Scholes formula.

Proposition 4.5. Assume that the terminal value is given by VT = f(ST)with f(x)= (x−K)⁺and the initial price of the risky asset isS0=W1+c, wherecis a positive constant.

Then

E˜Vt/Ft

=G

t, St

W1−Wt+c

, (4.18)

where

G(t,x)= 1 2π(1−T)

×

x

Re^{−u2/(1−T)}(u+c)Nd1

du

−Ke^−r(T−t)

Re^{−u2/(1−T)}Nd2

du

(4.19)

with

d1=d1(x,u)=lnK/x(u+c)+r+σ²/2(T−t)

σ^√T−t ,

d2=d2(x,u)=d1−σT−t,

(4.20)

andN(d)=(1/^√2π)_−∞^d e^−x2/2dx.

Proof. Using the fact that the increments of the Wiener process are independent on dis- joint intervals, the Markov property and (4.17) imply that

E˜Vt/Ft

=_E˜ e^−r(T−t)fST

/Ft

=_E˜ e^−r(T−t)fe^{σWt+(r−σ2/2)t}_E˜S0/FT^c

e^{σ(WT−Wt)+(r−σ2/2)(T−t)}

=G

t, St

E˜S0/Ft^c

,

(4.21)

where

G(t,x)=e^−r(T−t)_E˜ fxES0/FT^c

e^{σ(WT−Wt)+(r−σ2/2)(T−t)}. (4.22) SinceE(S0/FT^c)=W1−W_T+c, by the joint normal distribution of (W1−W_T,W_T−W_t),

G(t,x)= 1 2π(1−T)

×

Re^{−u2/2(1−T)}

e^−r(T−t) 2π(T−t)

Rfx(u+c)e^{σv+(r−σ2/2)(T−t)}e^{−v2/2(T−t)}dv

du.

(4.23) We refer to classical arguments (see [5]) to get

e^−r(T−t) 2π(T−t)

Rfx(u+c)e^{σv+(r−σ2/2)(T−t)}e^{−v2/2(T−t)}dv

=x(u+c)Nd1(x,u)−Ke^−r(T−t)Nd2(x,u),

(4.24)

whered1,d2are given by (4.20) and the conclusion follows.

Since the market is complete, every bounded contingent claim is attainable. There- fore, it is of importance to find the expression of the replicant portfolio. This is given in Proposition 4.6.

Proposition 4.6. Under the hypothesis of Proposition 4.5 and denoting g(t,x)= e^−rtG(t,e^rtx), the replicant portfolio is given by

Ht=

cσ S˜t

E˜S0/Ft^c−1 −1

σ∂g

∂x

t, S˜t

E˜S0/Ft^c

, ht=G

t, S_t W1−Wt+c

−cHt

S˜_t E˜S0/Ft^c

.

(4.25)

Proof. DenoteM_t=S˜_t/_E˜(S0/Ft^c). We utilize the classical procedure to determine the un- known quantitieshandH. We have that

E˜V˜t/Ft

=e^−rt_E˜Vt/Ft

=e^−rtG

t, St

E˜S0/Ft^c

=e^−rtG

t,e^rt S˜t

E˜S0/Ft^c

=e^−rtGt,e^rtM_t

(4.26)

withGaC^∞function on [0,T)×R. Writing It ˆo’s formula forg(t,M_t), we obtain gt,Mt

=g0,M0

+ _t

0σ∂g

∂x u,Mu

MudW˜u

+ _t

0

∂g

∂t

u,M_udu+1 2

_t

0

∂²g

∂x²

u,M_uσ²M_u²du.

(4.27)

Note first that the bounded variation part is zero. On the other hand, by (4.7), E˜V˜t/Ft

=_E˜V0

− _t

0HsED˜sS0/Fs +

_t

0σHs_E˜S˜s/Fs

dW˜s. (4.28) By (4.27) and (4.28), the natural candidate for H satisfies σ(∂g/∂x)(s,Ms)Ms= σHs_E˜( ˜Ss/Fs) and since ˜E( ˜Ss/Fs)=_E˜(S0)Ms, we obtain relation (4.25).

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Ciprian A. Tudor: Laboratoire de Probabilit´es et Mod`eles Al´eatoires, Universit´e de Paris 6, 4 Place Jussieu, 75252 Paris Cedex 5, France

E-mail address:[email protected]