Lecture 19. Stochastic integrals with respect to square integrable martingales

In the same way that a stochastic integral with respect to Brownian motion was constructed, a stochastic integral with respect to square integrable martingales may be defined. We shall not repeat this construction, since it was done in the Brownian motion case, but we point out the main results without proofs.

Let (M_t)_{t \geq 0} be a continuous square integrable martingale on a filtered probability space (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathcal{F},\mathbb{P}) that satisfies the usual conditions. We assume that \sup_{t \ge 0} \mathbb{E} \left( M_t^2 \right) <+\infty, and M_0=0. Let us denote by \mathcal{L}_M^2 (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathbb{P}) the set of processes (u_t)_{t \ge 0} that are progressively measurable with respect to the filtration (\mathcal{F}_t)_{t \ge 0} and such that
\mathbb{E} \left( \int_0^{+\infty} u_s^2 d\langle M \rangle_s \right)<+\infty.
We still denote by \mathcal{E} the set of simple and predictable processes, that is the set of processes (u_t)_{t \ge 0} that may be written as:
u_t=\sum_{i=0}^{n-1} F_i 1_{(t_i,t_{i+1}]} (t),
where 0\le t_0 \le ... \le t_n and where F_i is a random variable that is measurable with respect to \mathcal{F}_{t_i} and such that \mathbb{E}( F_i^2)<+\infty. We define an equivalence relation \mathcal{R} on the set \mathcal{L}_M^2 (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathbb{P}) as follows:
u\mathcal{R} v \Leftrightarrow \mathbb{E} \left( \int_0^{+\infty} (u_s-v_s)^2 d\langle M \rangle_s \right)=0.
and denote by
L_M^2 (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathbb{P})=\mathcal{L}_M^2 (\Omega, (\mathcal{F}_t)_{t \ge  0},\mathbb{P})/\mathcal{R},
the set of equivalence classes. It is easy to check that L_M^2 (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathbb{P}) endowed with the norm
\| u \|^2=\mathbb{E} \left( \int_0^{+\infty} u_s^2 d\langle M \rangle_s \right),
is a Hilbert space.

Theorem. There exists is a unique linear map
\mathcal{I}_M:L_M^2 (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathbb{P}) \rightarrow L^2 (\Omega, \mathcal{F},\mathbb{P})
such that:

  • For u=\sum_{i=0}^{n-1} F_i 1_{(t_i,t_{i+1}]} \in \mathcal{E}, \mathcal{I} (u)=\sum_{i=0}^{n-1} F_i (M_{t_{i+1}} -M_{t_i});
  • For u \in L_M^2 (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathbb{P}),
    \mathbb{E} \left( \mathcal{I}_M (u)^2\right)=\mathbb{E} \left( \int_0^{+\infty} u_s^2 d\langle M \rangle_s\right).

The map \mathcal{I}_M is called the Itō integral with respect to the continuous and square integrable martingale (M_t)_{t \geq 0} . We denote for u \in L_M^2 (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathbb{P}),
\mathcal{I}_M (u)=\int_0^{+\infty} u_s d M_s.

Proposition. Let (u_t)_{t \ge 0} be a stochastic process which is progressively measurable with respect to the filtration (\mathcal{F}_t)_{t \ge 0} and such that for every t \ge 0, \mathbb{E}\left( \int_0^t u_s^2 d\langle M \rangle_s \right) < +\infty. The process
\left( \int_0^t u_s dM_s \right)_{t \ge 0}=\left( \int_0^{+\infty} u_s 1_{[0,t]}(s)dM_s \right)_{t \ge 0}
is a square integrable martingale with respect to the filtration (\mathcal{F}_t)_{t \ge 0} that admits a continuous modification.

Proposition. Let (u_t)_{t \ge 0} be a stochastic process which is progressively measurable with respect to the filtration (\mathcal{F}_t)_{t \ge 0} and such that for every t \ge 0, \mathbb{E} \left( \int_0^t u_s^2 d\langle M \rangle_s \right)< +\infty. We have
\left\langle \int_0^{\cdot} u_s dM_s \right\rangle_t=\int_0^t u_s^2d\langle M \rangle_s.

Proposition. Let u \in L_M^2 (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathbb{P}) be a stochastic process whose paths are left continuous. Let t \ge 0. For every sequence of subdivisions \Delta_n [0,t] such that
\lim_{n \rightarrow +\infty}\mid\Delta_n [0,t]\mid=0,
the following convergence holds in probability:
\lim_{n \rightarrow +\infty} \sum_{k=0}^{n-1}u_{t^n_{k}} \left( M_{t^n_{k+1}}  M_{t^n_{k}}\right)=\int_0^t u_s dM_s.

Proposition. Let us assume that M_t=\int_0^t \Theta_s dB_s where (B_t)_{t \ge 0} is a Brownian motion on (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathcal{F},\mathbb{P}) and where \Theta \in L^2 (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathbb{P}). For u \in L_M^2 (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathbb{P}),
\int_0^t u_s dM_s=\int_0^t u_s \Theta_s dB_s.

Exercise. Let (M_t)_{t \ge 0} and (N_t)_{t \ge 0} be two square integrable martingales such that for every t \ge 0,
\mathbb{E} \left( \int_0^t M_s^2 d\langle N \rangle_s \right)<+\infty, \quad \mathbb{E} \left( \int_0^t N_s^2 d\langle M \rangle_s \right)<+\infty.
For t \ge 0,
M_tN_t =M_0 N_0+ \int_0^t M_s dN_s +\int_0^t N_s dM_s +\langle M,N \rangle_t.

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